Sharp EL-501X: Some Complex Powers
Introduction
This covers the class of calculators, including the Sharp EL-501X, TDE ATC-1335 Scientific Calculator, the 1980s calculator Texas Instruments TI-35 Plus and some scientific calculators (including Jot) that are purchased from dollar stores.
These set of calculators have a complex mode which allows with arithmetic calculations (addition, subtraction, multiplication, division) and conversions between polar and rectangular form. The keys [ a ] and [ b ] serve as two temporary registers in calculations.
However, since the complex mode does not allow for extended calculations or storing and recalling numbers in the memory register, I am not going to use the complex number mode for today's algorithms.
Let z = x + yi (i = √-1), we can calculate the following:
Square of the complex number: z^2
Square root of the complex number: √z
Cube Root of the complex number: z^(1/3)
Reciprocal of the complex number: 1/z
Recall the complex number can be taken by any power n, where n is a real number, can be found by:
(x + y*i)^n = (r * e^(i*θ))^n = r^n * e^(i*θ*n)
where
r = abs(x + yi) = x^2 + y^2
θ = arg(x + yi) = atan2(x,y)
e^(i*θ*n) = cos(θ*n) + i * sin(θ*n)
r^n * e^(i*θ*n) = r^n * cos(θ*n) + i * r^n * sin(θ*n)
We can use the memory register and the polar (→rθ) and rectangular (→xy) conversions.
I used and tested the following calculators: Sharp EL-501X and TDE ATC-1335 (the calculators pictured above).
Warning: Pressing the equals key [ = ] clears out the a and b registers. This limits the range of powers n can be take (without having to write immediate results).
The Algorithms
Note: Due to potential keyboard differences, I will leave out the shifted key. Please refer to your calculator manual for details.
Please note the sequence of the keystrokes. Recall that pressing the equals key clears out the a and b registers.
Also, since there are conversion to and from the polar form, the angle measure does not matter.
Finally, do not switch the calculator to complex number mode. Be in regular mode.
Square: (x + yi)^2
x [ a ] y [ b ] [ →rθ ] [ STO ]
[ b ] [ × ] 2 [ = ] [ b ]
[ RCL ] [ x^2 ] [ a ] [ →xy ]
(real part displayed)
[ b ] (imaginary part displayed)
Example: (6 - 3i)^2 = 27 - 36i
6 [ a ] 3 [+/-] [ b ] [ →rθ ] [ STO ]
[ b ] [ × ] 2 [ = ] [ b ]
[ RCL ] [ x^2 ] [ a ] [ →xy ]
27
[ b ] -36
Square Root: √(x + yi)
x [ a ] y [ b ] [ →rθ ] [ STO ]
[ b ] [ ÷ ] 2 [ = ] [ b ]
[ RCL ] [ √ ] [ a ] [ →xy ]
(real part displayed)
[ b ] (imaginary part displayed)
Example: √(6 - 3i) ≈ 2.52073441 - 0.595064674i
6 [ a ] 3 [+/-] [ b ] [ →rθ ] [ STO ]
[ b ] [ ÷ ] 2 [ = ] [ b ]
[ RCL ] [ √ ] [ a ] [ →xy ]
2.52073341
[ b ] -0.595064674
Cube Root: (x + yi)^(1/3)
x [ a ] y [ b ] [ →rθ ] [ STO ]
[ b ] [ ÷ ] 2 [ = ] [ b ]
[ RCL ] [ ³√ ] [ a ] [ →xy ]
(real part displayed)
[ b ] (imaginary part displayed)
Example: (6 - 3i)^(1/3) ≈ 1.863493911- 0.290316636i
6 [ a ] 3 [+/-] [ b ] [ →rθ ] [ STO ]
[ b ] [ ÷ ] 2 [ = ] [ b ]
[ RCL ] [ ³√ ] [ a ] [ →xy ]
(real part displayed)
[ b ] (imaginary part displayed)
Reciprocal: 1/(x + yi)
x [ a ] y [ b ] [ →rθ ] [ STO ]
[ b ] [ ÷ ] 1 [ +/- ] [ = ] [ b ]
[ RCL ] [ 1/x ] [ a ] [ →xy ]
(real part displayed)
[ b ] (imaginary part displayed)
Example: 1/(6 - 3i) ≈ 0.133333333 + 0.066666667i
6 [ a ] 3 [+/-] [ b ] [ →rθ ] [ STO ]
[ b ] [ ÷ ] 1 [ +/- ] [ = ] [ b ]
[ RCL ] [ 1/x ] [ a ] [ →xy ]
0.133333333
[ b ] 0.066666667
Eddie
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