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manual:chapter3:complex [2017/06/17 13:52]
smartin [Commands for complex numbers]
manual:chapter3:complex [2017/07/05 19:30] (current)
smartin [Entering complex numbers]
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 ==== Entering complex numbers ==== ==== Entering complex numbers ====
  
-A complex number can be represented mathematically in rectangular form as $z = x + iy$ or in polar form as $z = re^{i\theta}$. ​ A complex number is entered as a pair of numbers comma or space separated and enclosed in parenthesis. ​ For example, the complex number $5+3i$ would be entered (in rectangular form) as ''​(5,​ 3)''​ or ''​(5 3)''​ or in polar form ($r,​\theta$) as ''​(5.83,​ ∡30.96)''​((The keyboard shortcut for the angle symbol (∡) is **AL-RS-6**.)). ​ Alternatively,​ a complex number can be created from two numbers ​on the stack using the command ''​R→C'':​+A complex number can be represented mathematically in rectangular form as $z = x + iy$ or in polar form as $z = re^{i\theta}$. ​ A complex number is entered as a pair of numbers comma or space separated and enclosed in parenthesis. ​ For example, the complex number $5+3i$ would be entered (in rectangular form) as ''​(5,​ 3)''​ or ''​(5 3)''​ or in polar form ($r,​\theta$) as ''​(5.83,​ ∡30.96)''​((The keyboard shortcut for the angle symbol (∡) is **AL-RS-6**.  In this case the calculator is set to ''​DEG''​ mode, angles can be expressed in radians by setting ''​RAD''​.)).  Alternatively,​ a complex number can be created from two numbers using the command ''​R→C'':​
  
 ''​5 3 R→C''​ ''​5 3 R→C''​
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 leads to ''​(#​101b,#​11b)''​ on the stack, but taking the complex conjugate (''​CONJ''​) results in ''​(5,​-3)''​. leads to ''​(#​101b,#​11b)''​ on the stack, but taking the complex conjugate (''​CONJ''​) results in ''​(5,​-3)''​.
  
-System flag -103 formally sets the calculator mode to complex: ''​-103 SF''​.+System flag -103 formally sets the calculator mode to complex: ''​-103 SF''​. ​ This is important to ensure all input to functions return complex results. ​ For example, without setting flag -103, taking the square root of -1 results an an error (Argument outside domain). ​ Whereas in complex mode, the square root of -1 returns the correct result ''​(0,​1)''​.
  
  
 ---- ----
 ==== Complex number arithmetic ==== ==== Complex number arithmetic ====
-Many of the functions that operate on regular (non-complex) numbers also operate on complex numbers. ​ These functions include the basic arithmetic functions ($+,​-,​\times,​\div$) along with many of the trigonometry ($\sin, \cos, \tan$) and power functions ($x^2, e^x, 10^x$) ​and their inverses.+Many of the functions that operate on regular (non-complex) numbers also operate on complex numbers. ​ These functions include the basic arithmetic functions ($+,​-,​\times,​\div$) along with many of the trigonometry ($\sin, \operatorname{asin}, \cos, \operatorname{acos}, \tan, \operatorname{atan}$) and power functions ($x^2, \sqrt x , e^x, \ln x, 10^x, \log x$).
 ---- ----
 ==== Commands for complex numbers ==== ==== Commands for complex numbers ====
-The following table summarizes commands applicable to complex numbers.+The following table summarizes commands applicable to complex numbers, $z=x+iy=re^{i\theta}$.
  
-| Command | Purpose | Example |+| Command | Purpose | Example ​($\theta$ in degrees) ​|
 | ''​C→R''​ | Break down complex number to two reals | ''​(5,​3) C→R''​ yields ''​(5,​3)''​ ''​5''​ ''​3''​| | ''​C→R''​ | Break down complex number to two reals | ''​(5,​3) C→R''​ yields ''​(5,​3)''​ ''​5''​ ''​3''​|
 | ''​R→C''​ | Combine two numbers to a complex | ''​5 3 R→C''​ yields ''​(5,​3)''​| | ''​R→C''​ | Combine two numbers to a complex | ''​5 3 R→C''​ yields ''​(5,​3)''​|
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 | ''​→RECT''​ | Convert from polar to rectangular | ''​(5.831.,​∡30.964.°) →RECT''​ yields ''​(5.,​3.)''​| | ''​→RECT''​ | Convert from polar to rectangular | ''​(5.831.,​∡30.964.°) →RECT''​ yields ''​(5.,​3.)''​|
 | ''​ABS''​ | $|z|=r=\sqrt{x^2+y^2}$ | ''​(5,​3) ABS''​ yields ''​5.831.''​| | ''​ABS''​ | $|z|=r=\sqrt{x^2+y^2}$ | ''​(5,​3) ABS''​ yields ''​5.831.''​|
-| ''​ARG''​ | $\theta = atan(y/x)$ | ''​(5,​3) ARG''​ yields ''​∡30.964.°''​|+| ''​ARG''​ | $\theta = \operatorname{atan}(y/x)$ | ''​(5,​3) ARG''​ yields ''​∡30.964.°''​|
 | ''​SIGN''​ | $z/|z|$ | ''​(5,​3) SIGN''​ yields ''​(0.857.,​0.514.)''​| | ''​SIGN''​ | $z/|z|$ | ''​(5,​3) SIGN''​ yields ''​(0.857.,​0.514.)''​|
 | ''​NEG''​ | $-z=-x-iy$ | ''​(5,​3) NEG''​ yields ''​(-5,​ -3)''​| | ''​NEG''​ | $-z=-x-iy$ | ''​(5,​3) NEG''​ yields ''​(-5,​ -3)''​|
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