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Comparison of operating systems
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These tables provide a comparison of operating systems, of computer devices, as listing general and technical information for a number of widely used and currently available PC or handheld (including smartphone and tablet computer) operating systems. The article "Usage share of operating systems" provides a broader, and more general, comparison of operating systems that includes servers, mainframes and supercomputers.
Because of the large number and variety of available Linux distributions, they are all grouped under a single entry; see comparison of Linux distributions for a detailed comparison. There is also a variety of BSD and DOS operating systems, covered in comparison of BSD operating systems and comparison of DOS operating systems. For information on views of each operating system, see operating system advocacy.
Contents
[hide]
1 General information
2 Technical information
3 Security
4 Commands
5 See also
5.1 Operating system comparisons
6 References
7 External links
General information[edit]
Name
Creator
Initial public release
Predecessor
Current stable version
Release date
Cost, availability
Preferred license[g 1]
Target system type
AIX
IBM 1986 UNIX System V Release 3 7.2 2015, October 5 Bundled with hardware Proprietary Server, NAS, workstation
Android
Android, Inc., Google 2008 Linux 8.1 Oreo 2018, April 9 Free Apache 2.0, GNU GPLv2 Smartphone, tablet computer, education
AmigaOS classic
Commodore International, Haage & Partner, Hyperion Entertainment 1985 TRIPOS (as the disk operating component of AmigaOS) 3.9 BB2 2002, March 20 Discontinued; Bundled with hardware up to version 3.0 (Amiga International hardware came with 3.1); versions 2.1, 3.0, 3.1, 3.5, 3.9 also available as separate packages Proprietary, open source clone available under AROS Public License Workstation, personal computer
AmigaOS 4
Hyperion Entertainment 2004 AmigaOS classic 4.1 Final Edition 2014 4.0 bundled with hardware; 4.0 for classic and 4.1 available as standalone package at €29 Proprietary Workstation, personal computer
Chrome OS
Google 2009 Gentoo[1] 62.0.3202.97 2017, October 10 Bundled with hardware Proprietary: Google OS Terms of Service Chromebook
DragonFly BSD
Matthew Dillon 2003 FreeBSD 4.8[2] 2017, March 27 Free BSD Server, workstation, NAS, embedded
FreeBSD
The FreeBSD Project 1993 386BSD 11.1 2017, July 26 Free BSD Server, workstation, NAS, embedded
GhostBSD
Eric Turgeon 2009 FreeBSD 10.1 2015, September 13 Free BSD Desktop, workstation
eComStation
Serenity Systems, Mensys BV 2001 OS/2 2.1 2011 Home-student edition (max. three per site) US$145.00 (equivalent to $157.74 in 2017)
business edition $290.00 Proprietary Server, workstation, personal computer
EPOC32
Psion PLC 1996 ER5 1999 Discontinued; Commercial Proprietary PDA
GNU/Linux
Notable contributors include: Richard Stallman for GNU Project and Linus Torvalds for Linux and the Unixes they emulated; Red Hat, Debian Project See: Comparison of Linux distributions and Linux Kernel#Development 1991 (kernel), See: Comparison of Linux distributions and History of Linux None 4.16.4 (kernel) 2018, April 24 (kernel) Free GNU GPLv2 (kernel) See: Comparison of Linux distributions
Haiku
Haiku Inc. 2009 BeOS R5 R1/Alpha4 2012 Free / Use latest night build, not Alpha4 MIT Personal computer
HP-UX
Hewlett-Packard 1983 UNIX System V 11.31, 11i v3 Update 14 2015, March US$400 Proprietary Server
IBM i
IBM 1988 OS/400 7.3 2016, April 15 Bundled with hardware Proprietary Server
Inferno
Bell Labs 1997 Plan 9 Fourth Edition 2009, June 30 Free MIT, GNU GPL, GNU LGPL, LPL NAS, server, embedded
iOS
Apple Inc. 2007 macOS 11.3 2018, March 29 Bundled with hardware and free updates given to most existing users, subject to hardware requirements Proprietary higher level API layers; open source core system (ARM versions): APSL, GNU GPL, others Smartphone, music player, tablet computer
IRIX
SGI 1988 UNIX System V 6.5.30 2006 Discontinued; Bundled with hardware Proprietary Server, workstation
Classic Mac OS
Apple Inc. 1984 None[g 2][g 3] 9.2.2 2000 Discontinued; Was bundled with 68k and PowerPC Macs;
versions 7-9 sold as retail upgrades[g 4]
Proprietary Workstation, personal computer
MVS
IBM 1972 OS/360 MVS/ESA SP - JES3 Version 5 R2.2 1995, September 29 Bundled with hardware Proprietary IBM mainframe
macOS
Apple Inc. 2001 NeXTSTEP, BSD 10.13.4 High Sierra 2018, March 29 Bundled with hardware; No-cost update via Mac App Store for users of Mac OS X 10.6 or later, assuming hardware requirements are met Proprietary higher level API layers; open source core system (Intel-PowerPC versions): APSL, GNU GPL, others Workstation, personal computer, embedded
macOS Server
Apple Inc. 2001 NeXTSTEP, BSD 10.12 / September 20, 2016 2016, September 20 Previously bundled with hardware; No longer a separate operating system, but a group of services installed atop any current version of Mac OS X; US$19.99 on the Mac App Store Proprietary higher level API layers; open source core system (Intel-PowerPC versions): APSL, GNU GPL, others Server
MPE
HP 1974 None MPE-V 1988 Discontinued; Was bundled with HP-3000 CISC hardware "Classic" Proprietary Server
MCP
Unisys 1961 None CP OS 17.0 2015, April Bundled with hardware Proprietary Server
MPE/XL
HP 1987 MPE 7.5 2002 Discontinued; Was bundled with HP-3000 PA-RISC hardware Proprietary Server
MINIX 3
Andrew S. Tanenbaum 2005 Minix2 3.3.0 2014 Free BSD Workstation
NetBSD
The NetBSD Project 1993 386BSD 7.1.1[3] 2017, December 22[3] Free BSD NAS, server, workstation, embedded
NetWare
Novell 1985 S-Net 6.5 SP8 2009, May 6 Superseded by Novell Open Enterprise Server; Was US$184 (equivalent to $209.89 in 2017) (one-user) Proprietary Server
NeXTSTEP
NeXT 1989 Unix 3.3 1995 Discontinued; Was bundled with hardware, then sold separately Proprietary Workstation
OpenBSD
The OpenBSD Project 1995 NetBSD 1.0 6.3 2018, April 15 Free ISC Server, NAS, workstation, embedded
OpenIndiana
Many, based on software developed by Sun Microsystems and many others 2010 OpenSolaris 2017, May 3[4] Free CDDL Server, workstation
OpenVMS
DEC (now VSI) 1977 RSX-11M 8.4-2L2 2017, July 10 Commercial, free non-commercial use Proprietary Server, workstation
OS/360
IBM 1966 None Operating System/360 R21.8 1972, August Bundled with hardware Proprietary IBM mainframe
OS/390
IBM 1995 MVS OS/390 Version 2 R10 2000, September 29 Bundled with hardware Proprietary IBM mainframe
OS 2200
Unisys 1967 as Exec 8e Exec 8, OS 1100 CP OS 16 (Exec 49.2) 2015, February 27 Bundled with hardware Proprietary Server
OS/2
IBM and Microsoft 1987 Windows 3.x 4.52 2001 Discontinued (see eComStation successor); Was US$300 (equivalent to $414.62 in 2017) Proprietary Personal computer, server
Plan 9
Bell Labs 1993 Unix Fourth Edition 2003 (except for minor later updates) Free LPL Workstation, server, embedded, HPC
QNX
QNX Software Systems 1982 Unix, POSIX 7.0.0 2017, March Bundled with BlackBerry 10 and PlayBook devices. Commercial; an academic version exists that needs authorization code before installing Proprietary Automotive, medical, smartphones, consumer, industrial, embedded, safety
Solaris
Sun, Oracle Corporation 1992 SunOS 11.3 2015, October 26 Commercial; (but free/no-cost perpetual license when used "for the purpose of developing, testing, prototyping and demonstrating your applications"[5]) CDDL Server, workstation
Symbian
Symbian Ltd. 1998 EPOC32 9.5 2009 Discontinued; Commercial Proprietary Phones
Symbian platform
Symbian Foundation 2010 (initially 1998 as Symbian) Symbian 3.0.4 2010 Free EPL embedded
Windows Server (NT family)
Microsoft 1993 OS/2, Windows 3.x and MS-DOS Windows Server 2016 (Version 1709 (10.0.16299)) 2017, October US$1050 5 CALs server; other editions dependent on number of CALs purchased Proprietary; Shared source Server, NAS, embedded
Windows (NT family)
Microsoft 1993 OS/2, Windows 9x and MS-DOS Windows 10 (Version 1709 (10.0.16299.402) (April 23, 2018; 3 days ago (2018-04-23)[6]) [±]) 2017, August 8 Windows 10 Home US$119, Windows 10 Pro US$199[7] Proprietary; Shared source Workstation, personal computer, media center, Tablet PC, embedded
Windows (classic 9x family)
Microsoft 1995 MS-DOS, Windows NT 3.5 Windows Me (Win 4.90.3000) 2000 Discontinued Proprietary Personal computer, media center
RISC iX
Acorn Computers 1988 BSD 4.3 1.21c 1993 Discontinued; Was bundled with hardware Proprietary Workstation
RISC OS
Acorn Computers 1987 Arthur, also the BBC Master OS 3.71 1997 Discontinued; Was bundled with hardware Proprietary Education, personal computer
RISC OS 4
RISCOS Ltd, Pace plc 1999 RISC OS 4.39 2004 Bundled with hardware, then sold separately at £70 (US$127) Proprietary Education, personal computer
RISC OS 5
Castle Technology, RISC OS Open 2002 RISC OS 4 5.22[8] 2015 Free for non-commercial use (recent releases); formerly bundled with hardware Shared source Education, personal computer
RISC OS 6
RISCOS Ltd 2006 RISC OS 4 6.20 2009 Bundled with hardware, then sold separately at £70 (US$127) Proprietary Education, personal computer
ZETA
yellowTAB 2005 BeOS R5 1.5 2007 Discontinued Proprietary Personal computer, media center, workstation
STOP 6, XTS-400
BAE Systems 2003 STOP 5, XTS-300 6.5 2008, August US$60,000 (equivalent to $68,198 in 2017)+; bundled with XTS hardware and OEM licensed Proprietary Server, workstation
ReactOS
ReactOS development team 1996 Windows NT 0.4.8 2018, April 14 Free GNU GPL, GNU LGPL Workstation, personal computer
TrueOS
PC-BSD Software 2006 FreeBSD[g 5] 10.1[9] 2014, November 16 Free BSD Personal computer, workstation, server
VxWorks
Wind River Systems 1987 VRTX 7 2014 March Paid Proprietary Embedded Real-time systems
z/OS
IBM 2000 OS/390 Version 2.3 (V2R3) 2017, September 29 Monthly license fee, about US$130 and up Proprietary IBM mainframe
z/VSE
IBM 2007 VSE/ESA 6.1 2015, October 5 Monthly license fee Proprietary IBM mainframe
z/VM
IBM 2000 VM 6.4 2016, November 11 Monthly license fee Proprietary IBM mainframe
HP NonStop
HP 1974 Guardian H06.24/J06.13 2012 Non-free Proprietary HP Nonstop Servers
Name
Creator
Initial public release
Predecessor
Current stable version
Release date
Cost, availability
Preferred license[g 1]
Target system type
^ Jump up to: a b Most OS distributions include bundled software with various other licenses.
Jump up ^ The Original Macintosh Anecdotes. Although Lisa OS ran on the same, but a slower variant, microprocessor and was developed by Apple Inc. at the same time as Classic Mac OS, they were developed as different projects, only sharing a similar GUI between them.
Jump up ^ Mac OS 7.6 was the first Macintosh system software to be labeled Mac OS. Operating systems before this were named Macintosh System Software through System Software 7.5, and known as System #.# for short.
Jump up ^ "Official Apple Support". apple.com.
Jump up ^ PC-BSD uses FreeBSD as a base system with custom configuration and several desktop-oriented tools to make an easy to use FreeBSD system for desktops and workstations.
Technical information[edit]
Name
Computer architectures supported
File systems supported
Kernel type
Source lines of code
GUI default is on[t 1]
Package management
Update management
Native APIs[t 2]
Non-native APIs supported through subsystems
AIX
POWER, PowerPC-AS, PowerPC, Power Architecture JFS, JFS2, ISO 9660, UDF, NFS, SMBFS, GPFS Monolithic with modules No installp, RPM Service Update Management Assistant (SUMA) SysV/POSIX
AmigaOS Classic
68k, PowerPC Proprietary (OFS, FFS, SFS, PFS), FAT, ISO 9660, UDF, many others via 3rd party drivers, such as SMBFS, etc. Microkernel Yes Installer[t 3] (almost not needed)[t 4] Proprietary BSD subset (available through 3rd party ixemul.library)
AmigaOS 4
PowerPC Proprietary (OFS, FFS, SFS, PFS), JXFS, FAT, ISO 9660, UDF, many others via 3rd party drivers, such as SMBFS, etc. Microkernel Yes Installer[t 3] (almost not needed)[t 4] AmiUpdate (almost not needed)[t 5] Proprietary BSD subset (available through 3rd party ixemul.library)
Chrome OS
ARM, x86 eCryptfs, NTFS, FAT, FAT16, FAT32, exFAT, ext2, ext3, ext4, HFS+, MTP (read and write), ISO9660 (read-only), UDF (read-only) Monolithic with modules ~ 17 million[15] Yes Portage Linux/POSIX
DragonFly BSD
x86-64 UFS1, MFS, ext2, FAT (16/32), HAMMER, ISO 9660 Hybrid No dports, pkg git, cvsup, rsync, pkg BSD/POSIX Mono, Java, Win16,[t 6] Win32,[t 6] Linux
eComStation
x86 HPFS (default), FAT, JFS, UDF, FAT32, NTFS (read only) Hybrid Yes WarpIN, Feature Install, others Maintenance Tool Proprietary, DOS API, Win16 POSIX, Java, others
FreeBSD
x86, x86-64, ARM, MIPS, PowerPC, others UFS2, ZFS, ext2, ext3, FAT, ISO 9660, UDF, NFS, others Monolithic with modules 6.25 million[16] No Ports collection, packages by source, network binary update (freebsdupdate) BSD/POSIX Mono, Java, Win16,[t 6] Win32,[t 6] Linux
GhostBSD
x86, x86-64 UFS2, ext2, ext3, FAT, ISO 9660, UDF, NFS, ReiserFS (read only), XFS (experimental), ZFS, others Monolithic with modules Yes Ports collection, packages by source, network binary update (freebsdupdate) BSD/POSIX Mono, Java, Win16,[t 6] Win32,[t 6] Linux
Linux
x86, x86-64, ARM, PowerPC, SPARC, others ext2, ext3, ext4, btrfs, ReiserFS, FAT, ISO 9660, UDF, NFS, and others Monolithic with modules ~15 million (kernel)[17]
lines of code for userland libraries and applications vary depending on the distribution
Yes Portage Linux/POSIX Mono, Java, Win16,[t 6] Win32[t 6]
Haiku
x86, PowerPC BFS (default), FAT, ISO 9660, ext3, NTFS Hybrid ~5.2 million[citation needed] Yes Ports collection (haikuport) pkgman, HaikuDepot POSIX, BeOS API Java, Qt
HP-UX
PA-RISC, IA-64 VxFS, HFS, CDFS, EVFS, NFS, CIFS Monolithic with modules No SD, swinstall swa (HP-UX Software Assistant) SysV/POSIX
Inferno
x86, PowerPC, SPARC, Alpha, MIPS, others Styx/9P2000, kfs, FAT, ISO 9660 Monolithic with modules, user space file systems Yes ? ? Proprietary
iOS
ARM HFS+, FTP Hybrid ~80 million[citation needed] Yes ? Software Update Cocoa, BSD-POSIX ?
Classic Mac OS
68k, PowerPC HFS+, HFS, MFS (Mac OS 8.0 and before), AFP, ISO 9660, FAT(System 7 and later), UDF Monolithic with modules Yes None Software Update (only in Mac OS 9) Toolbox, Carbon (from version 8.1)
macOS
PowerPC, x86, x86-64, (see also iOS for ARM) HFS+ (default), HFS, UFS, AFP, ISO 9660, FAT, UDF, NFS, SMBFS, NTFS (read only), FTP, WebDAV, APFS (experimental), ZFS (experimental) Hybrid with modules ~86 million[18] Yes macOS Installer Software Update Carbon, Cocoa, Java, BSD-POSIX Toolbox (only in versions up to Mac OS X 10.4, not supported on x86 architecture), Win16,[t 6] Win32[t 6]
MINIX 3
x86 Microkernel ~12,000 (C) + ~1,400 (Assembly)[19] No POSIX
NetBSD
x86, x86-64, ARM, MIPS, PowerPC, sparc64, others UFS, UFS2, ext2, FAT, ISO 9660, NFS, LFS, and others Monolithic with modules No[t 7] pkgsrc by source or binary (using sysinst) BSD-POSIX Linux, others
NetWare
x86 NSS, NWFS, FAT, NFS, AFP, UDF, CIFS, ISO 9660 Hybrid Yes NWCONFIG.NLM, RPM, X11-based GUI installer binary updates, ZENWorks for Servers, Red Carpet Proprietary
OpenBSD
x86, x86-64, SPARC, 68k, Alpha, VAX, others ffs, ext2, FAT, ISO 9660, NFS, some others Monolithic No[t 7] Ports collection, packages by source or binary (packages via pkg_add) BSD-POSIX
OpenVMS
VAX, Alpha, IA-64 Files-11 (ODS), ISO 9660, NFS, CIFS Monolithic with modules No PCSI, VMSINSTAL ? Proprietary POSIX
OS/2
x86 HPFS, JFS, FAT, ISO 9660, UDF, NFS Monolithic with modules Yes Feature Install and others ? Proprietary, DOS API, Win16 Win32
Plan 9
x86, Alpha, MIPS, PowerPC, SPARC, others fossil/venti, 9P2000, kfs, ext2, FAT, ISO 9660 Hybrid, user space file systems ~ 2.5 Million /sys/src (complete source of all supported architectures, kernels, commands and libraries) Yes None replica Proprietary (Unix-like) POSIX compatibility layer
QNX
x86, SH-4, PowerPC, ARM, MIPS QNX4FS, QNX6, ext2, FAT, ISO 9660, Joliet, NFS, CIFS, ETFS, UDF, HFS, HFS+, NTFS, others Microkernel POSIX, Java
ReactOS
x86, PowerPC, ARM FAT, NTFS (read only) Hybrid nearly 8 million[20] Yes None None Win32, NT API
RISC OS
ARM (both 26 and 32-bit addressing modes) Acorn ADFS, Econet ANFS, FAT, ISO 9660, many others as loadable filesystems Monolithic with modules. Cooperative multitasking with limited memory protection.[21] Yes Applications self-contained; hardware drivers often in ROM !IyoUpWtch Huge number of SWI calls; extensive C libraries
Solaris
x86, x86-64, SPARC UFS, ZFS, ext2, FAT, ISO 9660, UDF, NFS, QFS, some others Monolithic with modules Yes SysV packages (pkgadd)
Image Packaging System (pkg) (Solaris 11 and later) Image Packaging System (Solaris 11 and later) SysV/POSIX, GTK, Java Win16,[t 6] Win32,[t 6] Mono, Linux[t 8]
OpenSolaris
x86, x86-64, SPARC(AI) UFS, ZFS, ext2, FAT, ISO 9660, UDF, NFS, QFS, some others Monolithic with modules ~18.8 million[22] Yes Image Packaging System (pkg), SysV packages (pkgadd) Image Packaging System SysV/POSIX, GTK, Java Win16,[t 6] Win32,[t 6] Mono, Linux[t 8]
STOP 6, XTS-400
x86 Proprietary Monolithic No RPM for some untrusted applications Binary updates via postal mail and proprietary tools Some: SysV, POSIX, Linux, proprietary
Symbian
ARM FAT Microkernel Yes SIS files FOTA Proprietary POSIX compatibility layer
TrueOS
x86[t 9] UFS2, ext2, ext3, FAT, ISO 9660, UDF, NFS, ReiserFS (read only), XFS (experimental) and others Monolithic with modules Yes Ports collection, packages, PBI Graphical Installers by PBI updates, source, network binary update (freebsdupdate) BSD-POSIX Win16,[t 6] Win32[t 6]
Windows Server (NT family)
x86, x86-64, IA-64 NTFS, FAT, ISO 9660, UDF; 3rd-party drivers support ext2, ext3, ReiserFS,[t 10] and HFS Hybrid with modules ~45 million[23] Yes MSI, custom installers Windows Update Win32, NT API DOS API, Win16 (only in 32-bit versions), POSIX, .NET
Windows (NT family)
x86, x86-64, ARM NTFS, FAT exFAT ISO 9660, UDF; 3rd-party drivers support ext2, ext3, ReiserFS,[t 10] HFS+, FATX, and HFS (with third party driver) Hybrid with modules ~40 (XP)/64 (Vista and later) million[citation needed] Yes MSI, custom installers Windows Update Win32, NT API DOS API, Win16 (only in 32-bit versions), POSIX, .NET
ZETA
x86 BFS (default), FAT, ISO 9660, UDF, HFS, AFP, ext2, CIFS, NTFS (read only), ReiserFS (read only, up to v3.6) Hybrid Yes SoftwareValet, script-based installers None POSIX, BeOS API
z/OS
z/Architecture VSAM, BDAM, QSAM, BPAM, HFS, zFS, etc. Protected, multithreading, multitasking nucleus with programmable/user replaceable extensions. Not kernel-based.[clarification needed] No None, SMP/E SMP/E Filesystem access methods, Systems Services, etc. POSIX, many others.
Name
Computer architectures supported
File systems supported
Kernel type
Source lines of code
GUI default is on[t 1]
Package management
Update management
Native APIs[t 2]
Non-native APIs supported through subsystems
^ Jump up to: a b Operating systems where the GUI is not installed and turned on by default are often bundled with an implementation of the X Window System, installation of which is usually optional.
^ Jump up to: a b Most operating systems use proprietary APIs in addition to any supported standards.
^ Jump up to: a b Amiga OS features since OS 2.0 version a standard centralized Install utility called Installer, which could be used by any software house to install programs. It works as a Lisp language interpreter, and install procedures could be listed as simple text. AmigaOS can also benefit of a 3rd party copyrighted library called XAD that is available for all POSIX (Unix, Linux, BSD, and for AmigaOS, MorphOS, etc.). This library is freely distributable and publicly available on Aminet Amiga centralized repository of all Open Source or Free programs and utilities. XAD.Library, complete with GUI Voodoo-X, is based on modules and capable to manage over 300 compression methods and package systems (Voodoo-X GUI supports 80 package systems), including those widely accepted as standards such as .ZIP, .CAB, .LHA, .LZX, .RPM, etc.
^ Jump up to: a b A standard AmigaOS installation requires usually only few files (typically 3 to 10 files) to be copied in their appropriate directory, and libraries and language files for national localization to be put in their standard OS directories. Any Amiga user with some minimal experience knows where these files should be copied and could perform programs installations by hand.
Jump up ^ AmiUpdate is capable to update AmigaOS files and also all Amiga programs which are registered to use the same update program that is standard for Amiga. Updating AmigaOS requires only few libraries to be put in standard OS location (for example all libraries are stored in "Libs:" standard virtual device and absolute path finder for "Libs" directory, Fonts are all in "Fonts:" absolute locator, the files for language localization are all stored in "Locale:" and so on). This leaves Amiga users with a minimal knowledge of the system almost free to perform by hand the update of the system files.
^ Jump up to: a b c d e f g h i j k l m n o p using Wine
^ Jump up to: a b NetBSD and OpenBSD include the X Window System as base install sets, managed in their respective main source repository, including local modifications. Packages are also provided for more up-to-date versions which may be less tested.
^ Jump up to: a b "BrandZ (Community Group brandz.WebHome) - XWiki". Opensolaris.org. 2009-10-26. Retrieved 2011-12-18.
Jump up ^ only i686 CPU
^ Jump up to: a b Windows can read and write with Ext2 and Ext3 file systems only when a driver from FS-driver or Ext2Fsd is installed. However, using Explore2fs, Windows can read from, but not write to, Ext2 and Ext3 file systems. Windows can also access ReiserFS through rfstool and related programs.
Security[edit]
Name
Resource
access
control
Subsystem
isolation
mechanisms
Integrated
firewall
Encrypted
file
systems
No execute (NX)
page flag
Manufacturer acknowledged unpatched vulnerabilities (by severity)[s 1]
Secunia
Security-
Focus
Hard-
ware
Emula-
tion
Extremely critical
(number / oldest)
Highly
critical
(number / oldest)
Moderately critical
(number / oldest)
Less
critical
(number / oldest)
Not
critical
(number / oldest)
Total
(number / oldest)
AIX 7.1
POSIX, ACLs, MAC, Trusted AIX - MLS, RBAC chroot IPFilter, IPsec VPNs, basic IDS Yes Yes[s 2] N/A Unknown 0
FreeBSD 10.1
POSIX, ACLs, MAC chroot, Jails, MAC partitions, multilevel security, Biba Model, BSD file flags set using chflags, Capsicum Capability-based security IPFW2, IPFilter, PF, IPsec Yes Yes Yes[s 3] 0 0 0 0 0 >0
GhostBSD 3.1
POSIX, ACLs, MAC chroot, jail, MAC partitions, BSD file flags set using chflags IPFW2, IPFilter, PF Yes Yes Yes 0 0 0 0 0 >0
HP-UX 11.31
POSIX, ACLs chroot IPFilter Yes ? ? 0 0 3
June 30, 2004;
13 years ago (2004-06-30) 2
December 12, 2002;
15 years ago (2002-12-12) 0 >0
Inferno
POSIX Namespaces,[34] capability-based security, no superuser or setuid bit ? ? No No Unknown >0
Linux-based 2.6.39
POSIX, ACLs,[s 4] MAC chroot,[s 5] seccomp, Namespaces, SELinux, AppArmor Netfilter, varied by distribution Yes Yes Yes 0 0 0 6
June 24, 2004;
13 years ago (2004-06-24) 11
April 4, 2005;
13 years ago (2005-04-04) >0
Mac OS 9.2.2
No No No No No No 0 0 0 0 0 >0
OS X 10.10.5
POSIX, ACLs[s 6] chroot, BSD file flags set using chflags ipfw Yes Yes (as of 10.5, X64 only) Yes (Intel only) 0 0 1
April 14, 2009;
9 years ago (2009-04-14) 2
January 8, 2007;
11 years ago (2007-01-08) 5
November 22, 2006;
11 years ago (2006-11-22) >0
NetBSD 6.1.2
POSIX, Veriexec, PaX, kauth chroot, kauth, BSD file flags set using chflags IPFilter, NPF, PF Yes Yes No Unknown >0
NetWare 6.5 SP8
Directory-enabled ACLs Protected address spaces IPFLT.NLM Yes Yes No 0 0 1
August 31, 2010;
7 years ago (2010-08-31) 2
October 30, 2003;
14 years ago (2003-10-30) 0 0
OES-Linux
Directory-enabled ACLs chroot IPFilter Yes Yes No Unknown >0
OpenBSD 4.8
POSIX chroot, systrace, BSD file flags set using chflags PF Yes Yes Yes Unknown >0
OpenVMS 8.4
ACLs, privileges logical name tables ? ? Yes ? 0 0 0 0 0 Unknown
OS/2, eComStation
ACLs[s 7] No IPFilter No ? ? 0 0 0 0 0 0
TrueOS 8.1
POSIX, ACLs, MAC chroot, jail, MAC partitions IPFW2, IPFilter, PF Yes[s 8] ? ? 0 0 0 0 0 >0
Plan 9
POSIX ? Namespaces,[34] capability-based security, no superuser or setuid bit ipmux Yes No No Unknown >0
QNX 6.5.0
POSIX ? PF, from NetBSD ? ? ? 0 0 0 5
November 20, 2002;
15 years ago (2002-11-20) 1
November 7, 2002;
15 years ago (2002-11-07) Unknown
RISC OS
No No No No No No Unknown
Solaris 10
POSIX, RBAC, ACLs, least privilege, Trusted Extensions chroot, Containers,[s 9] Logical Domains IPFilter Yes[s 10] Yes No 0 2
October 31, 2007;
10 years ago (2007-10-31) 5
October 23, 2007;
10 years ago (2007-10-23) 3
September 10, 2009;
8 years ago (2009-09-10) 2
November 6, 2006;
11 years ago (2006-11-06) >0
OpenSolaris 2009.06
POSIX, RBAC, ACLs, least privilege, Trusted Extensions chroot, Containers,[s 9] Logical Domains IPFilter Yes[s 10] Yes No 0 0 0 0 0 >0
Windows Server 2012
ACLs, privileges, RBAC Win32 WindowStation, desktop, job objects Windows Firewall Yes Yes Yes 0 0 0 0 0 [Unknown]
Windows 8.1
ACLs, privileges, RBAC Win32 WindowStation, desktop, job objects Windows Firewall Yes Yes Yes 0 0 0 0 1
May 30, 2014;
3 years ago (2014-05-30) [Unknown]
ZETA
POSIX[s 11] No No No No No Unknown
STOP 6, XTS-400[s 12]
POSIX, multilevel security, Biba Model mandatory integrity, ACLs, privileges, subtype mechanism Multilevel security, Biba Model, subtype mechanism No No No No Unknown
z/OS 1.11
RACF RACF, low storage protection, page protection, storage protect key, execution key, subspace group facility, APF, ACR (alternate CPU recovery), more z/OS IPSecurity Optional Yes (storage protect key, execution key, APF, more) Yes 0 0 0 0 0 Unknown
Name
Resource
access
control
Subsystem
isolation
mechanisms
Integrated
firewall
Encrypted
file
systems
Hard-
ware
Emula-
tion
Extremely critical
(number / oldest)
Highly
critical
(number / oldest)
Moderately critical
(number / oldest)
Less
critical
(number / oldest)
Not
critical
(number / oldest)
Total
(number / oldest)
No execute (NX)
page flag
Secunia
Security-
Focus
Known unpatched vulnerabilities (severity is accounted for)[s 1]
^ Jump up to: a b Comparison of known unpatched vulnerabilities based on Secunia & SecurityFocus reports with severity of Not critical & above. Update lists manually with oldest published date(s).
Jump up ^ AIX use the PowerPC architecture which offer page-level protection mechanism. Since AIX version 5300-03 (5.3), this feature can be activated using the sedmgr command.
Jump up ^ The GCC stack protection (a.k.a. ProPolice stack-smashing protector) has been enabled in base system since FreeBSD 8.0-release.
Jump up ^ Support for the in 1997 withdrawn POSIX ACL draft is included in Linux 2.6, but requires a file system able to store them (such as ext3, XFS or ReiserFS).
Jump up ^ A jail mechanism is available separately in the Linux-VServer project, but is not integrated into any mainline Linux kernel.
Jump up ^ ACLs were added to Mac OS X starting with version 10.4.
Jump up ^ ACLs are available only in OS/2 Server versions with HPFS386 filesystem.
Jump up ^ Additionally swap space may be encrypted during installation, uses memory based tmp file storage by default.
^ Jump up to: a b "Solaris Containers" (including "Zones") are a jail-type mechanism introduced with Solaris 10.
^ Jump up to: a b Through ZFS
Jump up ^ Zeta has full Unix file permissions, but the OS is single user, and users always run as superuser.
Jump up ^ STOP 6 is certified under Common Criteria at EAL5+.
Commands[edit]
For POSIX compliant (or partly compliant) systems like FreeBSD, Linux, macOS or Solaris, the basic commands are the same because they are standardized.
Feature
AROS
FreeBSD
Linux-based
HP-UX
OpenVMS
macOS
Solaris
Windows (cmd)
Windows (PowerShell)
Windows (cygwin, SFU or MKS)
List directory
list, dir ls ls ls dir ls ls dir Get-ChildItem ls
Clear console
clear clear clear clear clear clear cls Clear-Host clear
Copy file(s)
copy cp cp cp copy cp cp copy Copy-Item cp
Move file(s)
move mv mv mv mv mv move Move-Item mv
Rename file(s)
rename mv mv, rename mv ren mv mv ren (rename) Rename-Item mv
Delete file(s)
delete rm rm rm del rm rm del (erase) Remove-Item rm
Delete directory
delete rmdir rmdir rmdir del rmdir rmdir rd (rmdir) Remove-Item rmdir
Create directory
makedir mkdir mkdir mkdir create/dir mkdir mkdir md (mkdir) New-Item mkdir
Change current directory
cd [c 1] cd cd cd set def cd cd cd (chdir) Set-Location cd
Run shell script with new shell
shell file.shell sh file.sh sh file.sh sh file.sh @ file.com sh file.sh sh file.sh cmd /c file.cmd powershell.exe file.ps1 sh file.sh
Kill processes
kill, killall killall, pkill, kill, skill kill stop kill, killall kill, pkill taskkill Stop-Process kill
Change process priority
changetaskpri nice nice, chrt nice set proc/prio nice nice start /low, start /normal, start /high, start /realtime Start-Process, wmic nice
Change I/O priority
[c 2] ionice set proc/prio nice[c 3] ? ? ? ?
Create file system
format newfs mkfs newfs init mkfs newfs,zpool / zfs create format Format-Volume ?
File system check and recovery
fsck fsck fsck analyze/disk fsck fsck,n/a chkdsk Repair-Volume ?
Create software raid
atacontrol, gmirror, zfs create (mdadm—create) diskutil appleRAID metainit, zpool create diskpart (mirror only) diskpart (mirror only) ?
Mount device
mount mount mount mount mount mount, diskutil mount mount mountvol New-PSDrive ?
Unmount device
assign drivename: dismount umount umount umount dismount umount, diskutil unmount(disk) umount mountvol /d Remove-PSDrive ?
Mount file as block device
mdconfig + mount mount -o loop hdid lofiadm + mount ? ? ?
Show network configuration
ifconfig ip addr, ifconfig ifconfig, lanadmin tcpip sh net (sh net) ifconfig ifconfig ipconfig Get-NetIPInterface, ipconfig ?
Show network route
netstat -r, route get, route monitor ip route, route netstat -r tcpip sh route netstat -r, route get, route monitor netstat -r route Get-NetRoute ?
Trace network route
traceroute traceroute traceroute tcptrace traceroute traceroute tracert Test-NetConnection ?
Trace network route with pings
traceroute -I traceroute -I & mtr tcptrace traceroute -I traceroute -I pathping pathping ?
Feature
AROS
FreeBSD
Linux-based
HP-UX
OpenVMS
macOS
Solaris
Windows (cmd)
Windows (PowerShell)
Windows (cygwin, SFU or MKS)
NOTE: Linux systems may vary by distribution which specific program, or even 'command' is called, via the POSIX alias function. For example, if you wanted to use the DOS dir to give you a directory listing with one detailed file listing per line you could use alias dir='ls -lahF' (e.g. in a session configuration file).
Jump up ^ May be omitted. Simply entering the directory name will change to it.
Jump up ^ This feature is still in development, see [1].
Jump up ^ The nice command utilizes the setpriority() system call, which affects I/O priority, see OS X man page .
See also[edit]
Comparison of command shells
Comparison of file systems
List of operating systems
Lightweight Linux distribution
MacvsWindows
Operating system advocacy
Security-focused operating system
Timeline of operating systems
Usage share of operating systems
Operating system comparisons[edit]
Comparison of BSD operating systems
Comparison of DOS operating systems
Comparison of IPv6 support in operating systems
Comparison of operating system kernels
Comparison of Linux distributions
Comparison of netbook-oriented Linux distributions
Comparison of Microsoft Windows versions
Comparison of mobile operating systems
Comparison of open-source operating systems
Comparison of real-time operating systems
Comparison of OpenSolaris distributions
Comparison of Windows Vista and Windows XP
References[edit]
Jump up ^ Steven J. Vaughan-Nichols (March 6, 2013). "The secret origins of Google's Chrome OS". ZDNet.
Jump up ^ "DragonFly BSD: release48".
^ Jump up to: a b "NetBSD 7.1.1 released (December 22, 2017)".
Jump up ^ Alexander Pyhalov (May 3, 2017). "OpenIndiana Hipster 2017.04 is here".
Jump up ^ "Oracle Solaris OTN License". Oracle.com. Retrieved 2013-10-04.
Jump up ^ "April 23, 2018—KB4093105 (OS Build 16299.402)". Microsoft Support. Microsoft.
Jump up ^ "Windows 10 release date, price, news and features". Techradar.com. Retrieved 2015-07-29.
Jump up ^ "RISC OS Open: News: RISC OS 5.22 stable is now available". www.riscosopen.org. Retrieved 2018-04-24.
Jump up ^ "Official PC-BSD Blog » PC-BSD 10.1-RELEASE Now Available". Official PC-BSD Blog.
Jump up ^ "The Chromium (Google Chrome) Open Source Project on Open Hub". openhub.net.
Jump up ^ Arno Wagner (December 11, 2009). "(S)LOC Count Evolution for Selected OSS Projects" (PDF). data for 2009, fig 1.
Jump up ^ Ryan Paul (2012-04-04). "Linux kernel in 2011: 15 million total lines of code and Microsoft is a top contributor". arstechnica.com. Retrieved 2012-08-21.
Jump up ^ Jobs, Steve (7 August 2006). "Live from WWDC 2006: Steve Jobs Keynote". Retrieved 2007-02-16. 86 million lines of source code that was ported to run on an entirely new architecture with zero hiccups.
Jump up ^ Tanenbaum, Andrew S. (2015). Modern Operating Systems: Global Edition. Pearson Education Limited. ISBN 9781292061955.
Jump up ^ "ReactOS Change Log".
Jump up ^ "RISC OS Memory Protection - Drobe.co.uk archives". drobe.co.uk.
Jump up ^ find usr/src -type f -exec wc -l {} + | grep total | awk '{ sum += $1 } END {print sum }' on [2] results in 18793105
Jump up ^ Ben Liblit; Andrew Begel; Eve Sweetser. "Cognitive Perspectives on the Role of Naming in Computer Programs" (PDF). Retrieved 2007-12-26.
^ Jump up to: a b Rob Pike; Dave Presotto; Ken Thompson; Howard Trickey; Phil Winterbottom. "The Use of Name Spaces in Plan 9".
External links[edit]
Wikibooks has a book on the topic of: A Neutral Look at Operating Systems
"Operating System Technological Comparison". Retrieved May 9, 2005.
[hide]
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Square root
From Wikipedia, the free encyclopedia
Jump to: navigation, search
"Square roots" redirects here. For other uses, see Square Roots (disambiguation).
The mathematical expression "The (principal) square root of x"
In mathematics, a square root of a number a is a number y such that y2 = a; in other words, a number y whose square (the result of multiplying the number by itself, or y⋅y) is a.[1] For example, 4 and −4 are square roots of 16 because 42 = (−4)2 = 16. Every nonnegative real number a has a unique nonnegative square root, called the principal square root, which is denoted by √a, where √ is called the radical sign or radix. For example, the principal square root of 9 is 3, denoted √9 = 3, because 32 = 3 • 3 = 9 and 3 is nonnegative. The term whose root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this example 9.
Every positive number a has two square roots: √a, which is positive, and −√a, which is negative. Together, these two roots are denoted ± √a (see ± shorthand). Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root. For positive a, the principal square root can also be written in exponent notation, as a1/2.[2]
Square roots of negative numbers can be discussed within the framework of complex numbers. More generally, square roots can be considered in any context in which a notion of "squaring" of some mathematical objects is defined (including algebras of matrices, endomorphism rings, etc.)
Contents
[hide]
1 History
2 Properties and uses
3 Computation
4 Square roots of negative and complex numbers
4.1 Square root of an imaginary number
4.2 Principal square root of a complex number
4.3 Algebraic formula
4.4 Notes
5 Square roots of matrices and operators
6 In integral domains, including fields
7 In rings in general
8 Principal square roots of the positive integers
8.1 As decimal expansions
8.2 As expansions in other numeral systems
8.3 As periodic continued fractions
9 Geometric construction of the square root
10 See also
11 Notes
12 References
13 External links
History[edit]
The Yale Babylonian Collection YBC 7289 clay tablet was created between 1800 BC and 1600 C, showing √2 and √2/2 = 1/√2 as 1;24,51,10 and 0;42,25,35 base 60 numbers on a square crossed by two diagonals.[3]
The Rhind Mathematical Papyrus is a copy from 1650 BC of an earlier Berlin Papyrus and other texts – possibly the Kahun Papyrus – that shows how the Egyptians extracted square roots by an inverse proportion method.[4]
In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, dated around 800–500 BC (possibly much earlier).[citation needed] A method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra.[5] Aryabhata in the Aryabhatiya (section 2.4), has given a method for finding the square root of numbers having many digits.
It was known to the ancient Greeks that square roots of positive whole numbers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers (that is to say they cannot be written exactly as m/n, where m and n are integers). This is the theorem Euclid X, 9 almost certainly due to Theaetetus dating back to circa 380 BC.[6] The particular case √2 is assumed to date back earlier to the Pythagoreans and is traditionally attributed to Hippasus.[citation needed] It is exactly the length of the diagonal of a square with side length 1.
In the Chinese mathematical work Writings on Reckoning, written between 202 BC and 186 BC during the early Han Dynasty, the square root is approximated by using an "excess and deficiency" method, which says to "...combine the excess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend."[7]
Mahāvīra, a 9th-century Indian mathematician, was the first to state that square roots of negative numbers do not exist.[8]
A symbol for square roots, written as an elaborate R, was invented by Regiomontanus (1436–1476). An R was also used for Radix to indicate square roots in Gerolamo Cardano's Ars Magna.[9]
According to historian of mathematics D.E. Smith, Aryabhata's method for finding the square root was first introduced in Europe by Cataneo in 1546.
According to Jeffrey A. Oaks, Arabs used the letter jīm/ĝīm (ج), the first letter of the word “جذر” (variously transliterated as jaḏr, jiḏr, ǧaḏr or ǧiḏr, “root”), placed in its initial form (ﺟ) over a number to indicate its square root. The letter jīm resembles the present square root shape. Its usage goes as far as the end of the twelfth century in the works of the Moroccan mathematician Ibn al-Yasamin.[10]
The symbol '√' for the square root was first used in print in 1525 in Christoph Rudolff's Coss.[11]
Properties and uses[edit]
The graph of the function f(x) = √x, made up of half a parabola with a vertical directrix
The principal square root function f(x) = √x (usually just referred to as the "square root function") is a function that maps the set of nonnegative real numbers onto itself. In geometrical terms, the square root function maps the area of a square to its side length.
The square root of x is rational if and only if x is a rational number that can be represented as a ratio of two perfect squares. (See square root of 2 for proofs that this is an irrational number, and quadratic irrational for a proof for all non-square natural numbers.) The square root function maps rational numbers into algebraic numbers (a superset of the rational numbers).
For all real numbers x
x 2 = | x | = { x , if x ≥ 0 − x , if x < 0. {\displaystyle {\sqrt {x^{2}}}=\left|x\right|={\begin{cases}x,&{\mbox{if }}x\geq 0\\-x,&{\mbox{if }}x<0.\end{cases}}} (see absolute value)
For all nonnegative real numbers x and y,
x y = x y {\displaystyle {\sqrt {xy}}={\sqrt {x}}{\sqrt {y}}}
and
x = x 1 / 2 . {\displaystyle {\sqrt {x}}=x^{1/2}.}
The square root function is continuous for all nonnegative x and differentiable for all positive x. If f denotes the square-root function, its derivative is given by:
f ′ ( x ) = 1 2 x . {\displaystyle f'(x)={\frac {1}{2{\sqrt {x}}}}.}
The Taylor series of √1 + x about x = 0 converges for |x| ≤ 1 and is given by
1 + x = ∑ n = 0 ∞ ( − 1 ) n ( 2 n ) ! ( 1 − 2 n ) ( n ! ) 2 ( 4 n ) x n = 1 + 1 2 x − 1 8 x 2 + 1 16 x 3 − 5 128 x 4 + … , {\displaystyle {\sqrt {1+x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{(1-2n)(n!)^{2}(4^{n})}}x^{n}=1+\textstyle {\frac {1}{2}}x-{\frac {1}{8}}x^{2}+{\frac {1}{16}}x^{3}-{\frac {5}{128}}x^{4}+\dots ,}
The square root of a nonnegative number is used in the definition of Euclidean norm (and distance), as well as in generalizations such as Hilbert spaces. It defines an important concept of standard deviation used in probability theory and statistics. It has a major use in the formula for roots of a quadratic equation; quadratic fields and rings of quadratic integers, which are based on square roots, are important in algebra and have uses in geometry. Square roots frequently appear in mathematical formulas elsewhere, as well as in many physical laws.
Computation[edit]
Main article: Methods of computing square roots
Most pocket calculators have a square root key. Computer spreadsheets and other software are also frequently used to calculate square roots. Pocket calculators typically implement efficient routines, such as the Newton's method (frequently with an initial guess of 1), to compute the square root of a positive real number.[12][13] When computing square roots with logarithm tables or slide rules, one can exploit the identities
a = e ( ln  a ) / 2 = 10 ( log 10  a ) / 2 , {\displaystyle {\sqrt {a}}=e^{(\ln a)/2}=10^{(\log _{10}a)/2},}
where ln and log10 are the natural and base-10 logarithms.
By trial-and-error,[14] one can square an estimate for √a and raise or lower the estimate until it agrees to sufficient accuracy. For this technique it's prudent to use the identity
( x + c ) 2 = x 2 + 2 x c + c 2 , {\displaystyle (x+c)^{2}=x^{2}+2xc+c^{2},}
as it allows one to adjust the estimate x by some amount c and measure the square of the adjustment in terms of the original estimate and its square. Furthermore, (x + c)2 ≈ x2 + 2xc when c is close to 0, because the tangent line to the graph of x2 + 2xc + c2 at c=0, as a function of c alone, is y = 2xc + x2. Thus, small adjustments to x can be planned out by setting 2xc to a, or c=a/(2x).
The most common iterative method of square root calculation by hand is known as the "Babylonian method" or "Heron's method" after the first-century Greek philosopher Heron of Alexandria, who first described it.[15] The method uses the same iterative scheme as the Newton–Raphson method yields when applied to the function y = f(x) = x2 − a, using the fact that its slope at any point is dy/dx = f'(x) = 2x, but predates it by many centuries.[16] The algorithm is to repeat a simple calculation that results in a number closer to the actual square root each time it is repeated with its result as the new input. The motivation is that if x is an overestimate to the square root of a nonnegative real number a then a/x will be an underestimate and so the average of these two numbers is a better approximation than either of them. However, the inequality of arithmetic and geometric means shows this average is always an overestimate of the square root (as noted below), and so it can serve as a new overestimate with which to repeat the process, which converges as a consequence of the successive overestimates and underestimates being closer to each other after each iteration. To find x:
Start with an arbitrary positive start value x. The closer to the square root of a, the fewer the iterations that will be needed to achieve the desired precision.
Replace x by the average (x + a/x) / 2 between x and a/x.
Repeat from step 2, using this average as the new value of x.
That is, if an arbitrary guess for √a is x0, and xn + 1 = (xn + a/xn) / 2, then each xn is an approximation of √a which is better for large n than for small n. If a is positive, the convergence is quadratic, which means that in approaching the limit, the number of correct digits roughly doubles in each next iteration. If a = 0, the convergence is only linear.
Using the identity
a = 2 − n 4 n a , {\displaystyle {\sqrt {a}}=2^{-n}{\sqrt {4^{n}a}},}
the computation of the square root of a positive number can be reduced to that of a number in the range [1,4). This simplifies finding a start value for the iterative method that is close to the square root, for which a polynomial or piecewise-linear approximation can be used.
The time complexity for computing a square root with n digits of precision is equivalent to that of multiplying two n-digit numbers.
Another useful method for calculating the square root is the shifting nth root algorithm, applied for n = 2.
The name of the square root function varies from programming language to programming language, with sqrt[17] (often pronounced "squirt" [18]) being common, used in C, C++, and derived languages like JavaScript, PHP, and Python.
Square roots of negative and complex numbers[edit]
First leaf of the complex square root
Second leaf of the complex square root
Using the Riemann surface of the square root, it is shown how the two leaves fit together
The square of any positive or negative number is positive, and the square of 0 is 0. Therefore, no negative number can have a real square root. However, it is possible to work with a more inclusive set of numbers, called the complex numbers, that does contain solutions to the square root of a negative number. This is done by introducing a new number, denoted by i (sometimes j, especially in the context of electricity where "i" traditionally represents electric current) and called the imaginary unit, which is defined such that i2 = −1. Using this notation, we can think of i as the square root of −1, but notice that we also have (−i)2 = i2 = −1 and so −i is also a square root of −1. By convention, the principal square root of −1 is i, or more generally, if x is any nonnegative number, then the principal square root of −x is
− x = i x . {\displaystyle {\sqrt {-x}}=i{\sqrt {x}}.}
The right side (as well as its negative) is indeed a square root of −x, since
( i x ) 2 = i 2 ( x ) 2 = ( − 1 ) x = − x . {\displaystyle (i{\sqrt {x}})^{2}=i^{2}({\sqrt {x}})^{2}=(-1)x=-x.}
For every non-zero complex number z there exist precisely two numbers w such that w2 = z: the principal square root of z (defined below), and its negative.
Square root of an imaginary number[edit]
The square roots of i in the complex plane
The square root of i is given by
i = 1 2 2 + i 1 2 2 = 2 2 ( 1 + i ) . {\displaystyle {\sqrt {i}}={\frac {1}{2}}{\sqrt {2}}+i{\frac {1}{2}}{\sqrt {2}}={\frac {\sqrt {2}}{2}}(1+i).}
This result can be obtained algebraically by finding a and b such that
i = ( a + b i ) 2 {\displaystyle i=(a+bi)^{2}}
or equivalently
i = a 2 + 2 a b i − b 2 . {\displaystyle i=a^{2}+2abi-b^{2}.}
This gives the two simultaneous equations
{ 2 a b = 1 a 2 − b 2 = 0 {\displaystyle {\begin{cases}2ab=1\!\\a^{2}-b^{2}=0\!\end{cases}}}
with solutions
a = b = ± 1 2 . {\displaystyle a=b=\pm {\frac {1}{\sqrt {2}}}.}
The choice of the principal root then gives
a = b = 1 2 . {\displaystyle a=b={\frac {1}{\sqrt {2}}}.}
The result can also be obtained by using de Moivre's formula and setting
i = cos  ( π 2 ) + i sin  ( π 2 ) {\displaystyle i=\cos \left({\frac {\pi }{2}}\right)+i\sin \left({\frac {\pi }{2}}\right)}
which produces
i = ( cos  ( π 2 ) + i sin  ( π 2 ) ) 1 2 = cos  ( π 4 ) + i sin  ( π 4 ) = 1 2 + i ( 1 2 ) = 1 2 ( 1 + i ) . {\displaystyle {\begin{aligned}{\sqrt {i}}&=\left(\cos \left({\frac {\pi }{2}}\right)+i\sin \left({\frac {\pi }{2}}\right)\right)^{\frac {1}{2}}\\&=\cos \left({\frac {\pi }{4}}\right)+i\sin \left({\frac {\pi }{4}}\right)\\&={\frac {1}{\sqrt {2}}}+i\left({\frac {1}{\sqrt {2}}}\right)={\frac {1}{\sqrt {2}}}(1+i).\\\end{aligned}}}
Principal square root of a complex number[edit]
Visualisation of the square to sixth roots of a complex number z, in polar form reiφ where φ = arg z and r = |z | – if z is real, φ = 0 or π. Principal roots are in black.
To find a definition for the square root that allows us to consistently choose a single value, called the principal value, we start by observing that any complex number x + iy can be viewed as a point in the plane, (x, y), expressed using Cartesian coordinates. The same point may be reinterpreted using polar coordinates as the pair (r, φ), where r ≥ 0 is the distance of the point from the origin, and φ is the angle that the line from the origin to the point makes with the positive real (x) axis. In complex analysis, this value is conventionally written r eiφ. If
z = r e i φ with − π < φ ≤ π , {\displaystyle z=re^{i\varphi }{\text{ with }}-\pi <\varphi \leq \pi ,}
then we define the principal square root of z as follows:
z = r e i φ / 2 . {\displaystyle {\sqrt {z}}={\sqrt {r}}\,e^{i\varphi /2}.}
The principal square root function is thus defined using the nonpositive real axis as a branch cut. The principal square root function is holomorphic everywhere except on the set of non-positive real numbers (on strictly negative reals it isn't even continuous). The above Taylor series for √1 + x remains valid for complex numbers x with |x| < 1.
The above can also be expressed in terms of trigonometric functions:
r ( cos  φ + i sin  φ ) = r [ cos  φ 2 + i sin  φ 2 ] . {\displaystyle {\sqrt {r\left(\cos \varphi +i\,\sin \varphi \right)}}={\sqrt {r}}\left[\cos {\frac {\varphi }{2}}+i\sin {\frac {\varphi }{2}}\right].}
Algebraic formula[edit]
When the number is expressed using Cartesian coordinates the following formula can be used for the principal square root:[19][20]
z = | z | + Re  ( z ) 2 ± i | z | − Re  ( z ) 2 , {\displaystyle {\sqrt {z}}={\sqrt {\frac {|z|+\operatorname {Re} (z)}{2}}}\pm i\ {\sqrt {\frac {|z|-\operatorname {Re} (z)}{2}}},}
where the sign of the imaginary part of the root is taken to be the same as the sign of the imaginary part of the original number, or positive when zero. The real part of the principal value is always nonnegative.
Notes[edit]
Because of the discontinuous nature of the square root function in the complex plane, the following laws are not true in general:
z w = z w {\displaystyle {\sqrt {zw}}={\sqrt {z}}{\sqrt {w}}} (counterexample for the principal square root: z = −1 and w = −1)
1 z = 1 z {\displaystyle {\frac {1}{\sqrt {z}}}={\sqrt {\frac {1}{z}}}} (counterexample for the principal square root: z = −1)
z ∗ = ( z ) ∗ {\displaystyle {\sqrt {z^{*}}}=\left({\sqrt {z}}\right)^{*}} (counterexample for the principal square root: z = −1)
A similar problem appears with other complex functions with branch cuts, e.g., the complex logarithm and the relations log z + log w = log(zw) or log (z*) = log (z)* which are not true in general.
Wrongly assuming one of these laws underlies several faulty "proofs", for instance the following one showing that −1 = 1:
− 1 = i ⋅ i = − 1 ⋅ − 1 = ( − 1 ) ⋅ ( − 1 ) = 1 = 1 {\displaystyle {\begin{aligned}-1&=i\cdot i\\&={\sqrt {-1}}\cdot {\sqrt {-1}}\\&={\sqrt {\left(-1\right)\cdot \left(-1\right)}}\\&={\sqrt {1}}\\&=1\end{aligned}}}
The third equality cannot be justified (see invalid proof). It can be made to hold by changing the meaning of √ so that this no longer represents the principal square root (see above) but selects a branch for the square root that contains (√−1)·(√−1). The left-hand side becomes either
− 1 ⋅ − 1 = i ⋅ i = − 1 {\displaystyle {\sqrt {-1}}\cdot {\sqrt {-1}}=i\cdot i=-1}
if the branch includes +i or
− 1 ⋅ − 1 = ( − i ) ⋅ ( − i ) = − 1 {\displaystyle {\sqrt {-1}}\cdot {\sqrt {-1}}=(-i)\cdot (-i)=-1}
if the branch includes −i, while the right-hand side becomes
( − 1 ) ⋅ ( − 1 ) = 1 = − 1 , {\displaystyle {\sqrt {\left(-1\right)\cdot \left(-1\right)}}={\sqrt {1}}=-1,}
where the last equality, √1 = −1, is a consequence of the choice of branch in the redefinition of √.
Square roots of matrices and operators[edit]
Main article: Square root of a matrix
If A is a positive-definite matrix or operator, then there exists precisely one positive definite matrix or operator B with B2 = A; we then define A1/2 = B. In general matrices may have multiple square roots or even an infinitude of them. For example, the 2 × 2 identity matrix has an infinity of square roots,[21] though only one of them is positive definite.
In integral domains, including fields[edit]
Each element of an integral domain has no more than 2 square roots. The difference of two squares identity u2 − v2 = (u − v)(u + v) is proved using the commutativity of multiplication. If u and v are square roots of the same element, then u2 − v2 = 0. Because there are no zero divisors this implies u = v or u + v = 0, where the latter means that two roots are additive inverses of each other. In other words if an element a square root u of an element a exists, then the only square roots of a are u and -u. The only square root of 0 in an integral domain is 0 itself.
In a field of characteristic 2, an element either has one square root or does not have any at all, because each element is its own additive inverse, so that -u = u. If the field is finite of characteristic 2 then every element has a unique square root. In a field of any other characteristic, any non-zero element either has two square roots, as explained above, or does not have any.
Given an odd prime number p, let q = pe for some positive integer e. A non-zero element of the field Fq with q elements is a quadratic residue if it has a square root in Fq. Otherwise, it is a quadratic non-residue. There are (q − 1)/2 quadratic residues and (q − 1)/2 quadratic non-residues; zero is not counted in either class. The quadratic residues form a group under multiplication. The properties of quadratic residues are widely used in number theory.
In rings in general[edit]
In a ring we call an element b a square root of a iff b2 = a. To see that the square root need not be unique up to sign in a general ring, consider the ring Z / 8 Z {\displaystyle \mathbb {Z} /8\mathbb {Z} } from modular arithmetic. Here, the element 1 has four distinct square roots, namely ±1 and ±3. On the other hand, the element 2 has no square root. See also the article quadratic residue for details.
Another example is provided by the quaternions H {\displaystyle \mathbb {H} } in which the element −1 has an infinitude of square roots including ±i, ±j, and ±k.
In fact, the set of square roots of −1 is exactly
{ a i + b j + c k ∣ a 2 + b 2 + c 2 = 1 } . {\displaystyle \{ai+bj+ck\mid a^{2}+b^{2}+c^{2}=1\}.}
Hence this set is exactly the same size and shape as the unit sphere in 3-space.
The square root of 0 is by definition either 0 or a zero divisor, and where zero divisors do not exist (such as in quaternions and, generally, in division algebras), it is uniquely 0. It is not necessarily true in general rings, where Z/n2Z for any natural n provides an easy counterexample.
Principal square roots of the positive integers[edit]
As decimal expansions[edit]
The square roots of the perfect squares (1, 4, 9, 16, etc.) are integers. In all other cases, the square roots of positive integers are irrational numbers, and therefore their decimal representations are non-repeating decimals.
√0 = 0
√1 = 1
√2 ≈ 1.414213562373095048801688724209698078569671875376948073176679737990732478462 (article) 1 million digits, 2 million, 5 million, 10 million
√3 ≈ 1.732050807568877293527446341505872366942805253810380628055806979451933016909 (article) 1 million digits, 2 million
√4 = 2
√5 ≈ 2.236067977499789696409173668731276235440618359611525724270897245410520925638 (article) 1 million digits
√6 ≈ 2.449489742783178098197284074705891391965947480656670128432692567250960377457 1 million digits
√7 ≈ 2.645751311064590590501615753639260425710259183082450180368334459201068823230 1 million digits
√8 ≈ 2.828427124746190097603377448419396157139343750753896146353359475981464956924 1 million digits
√9 = 3
√10 ≈ 3.162277660168379331998893544432718533719555139325216826857504852792594438639 1 million digits
√11 ≈ 3.316624790355399849114932736670686683927088545589353597058682146116484642609 10 million digits (verified)
√12 ≈ 3.464101615137754587054892683011744733885610507620761256111613958903866033818 500,000 digits
√13 ≈ 3.605551275463989293119221267470495946251296573845246212710453056227166948293 200,000 digits
√14 ≈ 3.741657386773941385583748732316549301756019807778726946303745467320035156307
√15 ≈ 3.872983346207416885179265399782399610832921705291590826587573766113483091937
√16 = 4
√17 ≈ 4.123105625617660549821409855974077025147199225373620434398633573094954346338 100,000 digits
√18 ≈ 4.242640687119285146405066172629094235709015626130844219530039213972197435386
√19 ≈ 4.358898943540673552236981983859615659137003925232444936890344138159557328203 100,000 digits
√20 ≈ 4.472135954999579392818347337462552470881236719223051448541794490821041851276
√21 ≈ 4.582575694955840006588047193728008488984456576767971902607242123906868425547 100,000 digits
Note that if the radicand is not square-free, then one can factorize, for example
8 = 4 2 = 2 2 {\displaystyle {\sqrt {8}}\ =\ {\sqrt {4}}{\sqrt {2}}\ =\ 2{\sqrt {2}}}
12 = 4 3 = 2 3 {\displaystyle {\sqrt {12}}\ =\ {\sqrt {4}}{\sqrt {3}}\ =\ 2{\sqrt {3}}}
18 = 9 2 = 3 2 {\displaystyle {\sqrt {18}}\ =\ {\sqrt {9}}{\sqrt {2}}\ =\ 3{\sqrt {2}}}
20 = 4 5 = 2 5 {\displaystyle {\sqrt {20}}\ =\ {\sqrt {4}}{\sqrt {5}}\ =\ 2{\sqrt {5}}} .
As expansions in other numeral systems[edit]
The square roots of the perfect squares (1, 4, 9, 16, etc.) are integers. In all other cases, the square roots of positive integers are irrational numbers, and therefore their representations in any standard positional notation system are non-repeating.
The square roots of small integers are used in both the SHA-1 and SHA-2 hash function designs to provide nothing up my sleeve numbers.
As periodic continued fractions[edit]
One of the most intriguing results from the study of irrational numbers as continued fractions was obtained by Joseph Louis Lagrange c. 1780. Lagrange found that the representation of the square root of any non-square positive integer as a continued fraction is periodic. That is, a certain pattern of partial denominators repeats indefinitely in the continued fraction. In a sense these square roots are the very simplest irrational numbers, because they can be represented with a simple repeating pattern of integers.
√2 = [1; 2, 2, ...]
√3 = [1; 1, 2, 1, 2, ...]
√4 = [2]
√5 = [2; 4, 4, ...]
√6 = [2; 2, 4, 2, 4, ...]
√7 = [2; 1, 1, 1, 4, 1, 1, 1, 4, ...]
√8 = [2; 1, 4, 1, 4, ...]
√9 = [3]
√10 = [3; 6, 6, ...]
√11 = [3; 3, 6, 3, 6, ...]
√12 = [3; 2, 6, 2, 6, ...]
√13 = [3; 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, ...]
√14 = [3; 1, 2, 1, 6, 1, 2, 1, 6, ...]
√15 = [3; 1, 6, 1, 6, ...]
√16 = [4]
√17 = [4; 8, 8, ...]
√18 = [4; 4, 8, 4, 8, ...]
√19 = [4; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, ...]
√20 = [4; 2, 8, 2, 8, ...]
The square bracket notation used above is a sort of mathematical shorthand to conserve space. Written in more traditional notation the simple continued fraction for the square root of 11, [3; 3, 6, 3, 6, ...], looks like this:
11 = 3 + 1 3 + 1 6 + 1 3 + 1 6 + 1 3 + ⋱ {\displaystyle {\sqrt {11}}=3+{\cfrac {1}{3+{\cfrac {1}{6+{\cfrac {1}{3+{\cfrac {1}{6+{\cfrac {1}{3+\ddots }}}}}}}}}}}
where the two-digit pattern {3, 6} repeats over and over again in the partial denominators. Since 11 = 32 + 2, the above is also identical to the following generalized continued fractions:
11 = 3 + 2 6 + 2 6 + 2 6 + 2 6 + 2 6 + ⋱ = 3 + 6 ⋅ 1 20 − 1 − 1 20 − 1 20 − 1 20 − ⋱ . {\displaystyle {\sqrt {11}}=3+{\cfrac {2}{6+{\cfrac {2}{6+{\cfrac {2}{6+{\cfrac {2}{6+{\cfrac {2}{6+\ddots }}}}}}}}}}=3+{\cfrac {6\cdot 1}{20-1-{\cfrac {1}{20-{\cfrac {1}{20-{\cfrac {1}{20-\ddots }}}}}}}}.}
Geometric construction of the square root[edit]
The square root of a positive number is usually defined as the side length of a square with the area equal to the given number. But the square shape is not necessary for it: if one of two similar planar Euclidean objects has the area a times greater than another, then the ratio of their linear sizes is √a.
A square root can be constructed with a compass and straightedge. In his Elements, Euclid (fl. 300 BC) gave the construction of the geometric mean of two quantities in two different places: Proposition II.14 and Proposition VI.13. Since the geometric mean of a and b is a b {\displaystyle {\sqrt {ab}}} , one can construct a {\displaystyle {\sqrt {a}}} simply by taking b = 1.
The construction is also given by Descartes in his La Géométrie, see figure 2 on page 2. However, Descartes made no claim to originality and his audience would have been quite familiar with Euclid.
Euclid's second proof in Book VI depends on the theory of similar triangles. Let AHB be a line segment of length a + b with AH = a and HB = b. Construct the circle with AB as diameter and let C be one of the two intersections of the perpendicular chord at H with the circle and denote the length CH as h. Then, using Thales' theorem and, as in the proof of Pythagoras' theorem by similar triangles, triangle AHC is similar to triangle CHB (as indeed both are to triangle ACB, though we don't need that, but it is the essence of the proof of Pythagoras' theorem) so that AH:CH is as HC:HB, i.e. a / h = h / b , {\displaystyle \ a/h=h/b,} from which we conclude by cross-multiplication that h 2 = a b , {\displaystyle \ h^{2}=ab,} and finally that h = a b {\displaystyle h={\sqrt {ab}}} . Note further that if you were to mark the midpoint O of the line segment AB and draw the radius OC of length ( a + b ) / 2 {\displaystyle (a+b)/2} then clearly OC > CH, i.e. ( a + b ) / 2 ≥ a b {\displaystyle (a+b)/2\geq {\sqrt {ab}}} (with equality if and only if a = b), which is the arithmetic–geometric mean inequality for two variables and, as noted above, is the basis of the Ancient Greek understanding of "Heron's method".
Another method of geometric construction uses right triangles and induction: √1 can, of course, be constructed, and once √x has been constructed, the right triangle with 1 and √x for its legs has a hypotenuse of √x + 1. The Spiral of Theodorus is constructed using successive square roots in this manner.
See also[edit]
Apotome (mathematics)
Cube root
Integer square root
List of square roots
Methods of computing square roots
Nested radical
Nth root
Quadratic irrational
Root of unity
Solving quadratic equations with continued fractions
Square root principle
The square root of NOT gate (√NOT), one of the logic gates used in quantum computers (doesn't exist for non-quantum where NOT gates are used)
Notes[edit]
Jump up ^ Gel'fand, p. 120
Jump up ^ Zill, Dennis G.; Shanahan, Patrick (2008). A First Course in Complex Analysis With Applications (2nd ed.). Jones & Bartlett Learning. p. 78. ISBN 0-7637-5772-1. Extract of page 78
Jump up ^ "Analysis of YBC 7289". ubc.ca. Retrieved 19 January 2015.
Jump up ^ Anglin, W.S. (1994). Mathematics: A Concise History and Philosophy. New York: Springer-Verlag.
Jump up ^ Joseph, ch.8.
Jump up ^ Heath, Sir Thomas L. (1908). The Thirteen Books of The Elements, Vol. 3. Cambridge University Press. p. 3.
Jump up ^ Dauben (2007), p. 210.
Jump up ^ Selin 2008, p. 1268.
Jump up ^ "The Development of Algebra - 2". maths.org. Retrieved 19 January 2015.
Jump up ^ * Oaks, Jeffrey A. (2012). Algebraic Symbolism in Medieval Arabic Algebra (PDF) (Thesis). Philosophica. p. 36.
Jump up ^ Manguel, Alberto (2006). "Done on paper: the dual nature of numbers and the page". The Life of Numbers. ISBN 84-86882-14-1.
Jump up ^ Parkhurst, David F. (2006). Introduction to Applied Mathematics for Environmental Science. Springer. p. 241. ISBN 9780387342283.
Jump up ^ Solow, Anita E. (1993). Learning by Discovery: A Lab Manual for Calculus. Cambridge University Press. p. 48. ISBN 9780883850831.
Jump up ^ Aitken, Mike; Broadhurst, Bill; Hladky, Stephen (2009). Mathematics for Biological Scientists. Garland Science. p. 41. ISBN 978-1-136-84393-8. Extract of page 41
Jump up ^ Heath, Sir Thomas L. (1921). A History of Greek Mathematics, Vol. 2. Oxford: Clarendon Press. pp. 323–324.
Jump up ^ Muller, Jean-Mic (2006). Elementary functions: algorithms and implementation. Springer. pp. 92–93. ISBN 0-8176-4372-9. , Chapter 5, p 92
Jump up ^ "Function sqrt". CPlusPlus.com. The C++ Resources Network. 2016. Retrieved June 24, 2016.
Jump up ^ Overland, Brian (2013). C++ for the Impatient. Addison-Wesley. p. 338. ISBN 9780133257120. OCLC 850705706. Retrieved June 24, 2016.
Jump up ^ Abramowitz, Milton; Stegun, Irene A. (1964). Handbook of mathematical functions with formulas, graphs, and mathematical tables. Courier Dover Publications. p. 17. ISBN 0-486-61272-4. , Section 3.7.27, p. 17
Jump up ^ Cooke, Roger (2008). Classical algebra: its nature, origins, and uses. John Wiley and Sons. p. 59. ISBN 0-470-25952-3.
Jump up ^ Mitchell, Douglas W., "Using Pythagorean triples to generate square roots of I2", Mathematical Gazette 87, November 2003, 499–500.
References[edit]
Dauben, Joseph W. (2007). "Chinese Mathematics I". In Katz, Victor J. The Mathematics of Egypt, Mesopotamia, China, India, and Islam. Princeton: Princeton University Press. ISBN 0-691-11485-4.
Gel'fand, Izrael M.; Shen, Alexander (1993). Algebra (3rd ed.). Birkhäuser. p. 120. ISBN 0-8176-3677-3.
Joseph, George (2000). The Crest of the Peacock. Princeton: Princeton University Press. ISBN 0-691-00659-8.
Smith, David (1958). History of Mathematics. 2. New York: Dover Publications. ISBN 978-0-486-20430-7.
Selin, Helaine (2008), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, ISBN 978-1-4020-4559-2 .
External links[edit]
Wikimedia Commons has media related to Square root.
Algorithms, implementations, and more – Paul Hsieh's square roots webpage
How to manually find a square root
AMS Featured Column, Galileo's Arithmetic by Tony Philips – includes a section on how Galileo found square roots
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This page was last edited on 26 October 2017, at 15:16.
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