Sharp EL-501X: Some Complex Powers

 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

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Retro Review: TI-36 Solar

Retro Review:  TI-36 Solar 

Just the Facts:

Model:  TI-36 Solar

Company: Texas Instruments

Type: Scientific

Display:  10 digit with 2-digit exponents

Battery:  Solar

Logic: AOS

Memory Registers: 1

Years: 1984-1990

Features:

*  Complex Number Arithmetic

*  Rounding Numbers (to fixed number setting)

*  Base Conversions

*  Normal Distribution

*  Single-Variable Statistics

Examples with Complex Numbers

MODE 5:  CPLX

There are two separate registers to hold parts of complex numbers:

[ a ]:  Real part (rectangular),  Radius (polar)

[ b ]:  Imaginary part (rectangular), Angle (polar)

Example 1:  (4 + 3i) * (11 - 2i)

4 [ a ] 3 [ b ] [ × ] 11 [ a ] 2 [ +/- ] [ b ] [ = ]

Display:  50  

Press [ b ], Display:  25

Result:  50 + 25i

Example 2:  Convert 3 + 2.4i to polar form, degrees

3 [ a ] 2.4 [ b ] [ INV ] (R>P) 

Display:  3.841874542

Press [ b ],  Display:  38.65980826

3 + 2.4i = 3.841874542 ∠ 38.65980826°

Examples with Normal Distribution

MODE 6:  STAT

There are three functions that determine the area under the normal distribution curve:

P(t) from - ∞ to t,  lower tail curve

R(t) from 0 to t

Q(t) from t to ∞, upper tail curve

Note that for any t, P(t) + Q(t) = 1.

Also note that P(t), R(t), and Q(t) will operate on the standard normal curve, where μ = 0 and σ = 1, regardless of the amount of data points entered in Statistics mode through Σ+.

For z = t =2:

2 [ INV ] ( P(t) ) returns 0.97725

2 [ INV ] ( R(t) ) returns 0.47725

2 [ INV ] ( Q(t) ) returns 0.02275

Comparison:  TI-36 Solar vs. TI-35 Plus

The TI-36 Solar and TI-35 Plus have the same set of functions and features.  You can see my retro review from 2017 of the TI-35 Plus here:  http://edspi31415.blogspot.com/2017/09/retro-review-texas-instruments-ti-35.html

There are several keyboard differences:

Shift key is marked [ INV ] (with 2nd above it) for the TI-36 Solar, and the shift key is marked [ 2nd ] for the TI-35 Plus.

The equals key is twice in vertical height, covering the spans of the bottom two rows on the TI-35 Plus.

The top three rows* (all of the shift markings remain intact):

TI-36 Solar:

[CE/C] [ 1/x ] [ a ] [ b ] [ AC ]

[ hyp ] [ sin ] [ cos ] [ tan ] [ DRG ]

[ INV ] [ x^2 ] [ log ] [ ln x ] [ y^x ]

TI-35 Plus:

[ 2nd ] [ x^2 ] [ log ] [ ln x ] [ OFF ]

[ hyp ] [ sin ] [ cos ] [ tan ] [ DRG ]

[ y^x ] [ 1/x ] [ a ] [ b ] [ ÷ ]

The [ hyp ] key has the hyp^-1 label over it on the TI-35 Plus.  

*not counting the [ON/C] key above all the rows on the TI-35 Plus

Comparison:  TI-36 Solar vs. the original TI-36X Solar 

Here is my review from 2018 for the TI-36X Solar:  http://edspi31415.blogspot.com/2018/09/retro-review-texas-instruments-ti-36x.html

The TI-36X Solar:  

*  does not have the complex arithmetic mode of the TI-36 Solar

*  adds the Boolean functions AND, OR, XOR, XNOR, and NOT to the BIN, OCT, and HEX mode

*  adds linear regression

*  adds a second shift key, [ 3rd ]

*  there is no [ MODE ] key, every mode is selected through [ 3rd ] (key) combos

*  adds a fraction/decimal conversions

*  adds 8 scientific constants and 10 metric/US conversions

Verdict

Like the TI-35 Plus, the TI-36 Solar is a step up from the TI-30 series (1980s versions of TI-30).   Again, the TI-34 (1987) has the Boolean functions and fraction functions that the TI-36 Solar doesn't.  You don't have to worry about batteries at all since the TI-36 Solar runs entirely on solar/light power.

Eddie

All original content copyright, © 2011-2020.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Retro Review: Texas Instruments TI-35 PLUS

Retro Review:  Texas Instruments TI-35 PLUS

Essentials

Company:  Texas Instruments

Type:  Scientific

Year: 1986

Battery:  A76 x 2

Digits: 10

Memory Registers: 3, 2 temporary for certain functions and 1 permanent. Storage arithmetic commands SUM and EXC are included. 

Thank you for Bob Patton.  I won this calculator as one of the door prizes on last week’s HHC 2017.

Like the last retro review, I am going to describe the features by the modes available on the calculator.

Mode 1: Decimal Mode (Normal)

This is the normal mode where most of the mathematical operations are available.

The [ a ] and [ b ] keys are temporary registers for various functions, such as:

Number of Combinations:  n [ a ], r [ b ], [ 2nd ] [ ÷ ] (nCr)

Number of Permutations:  n [ a ], r [ b ], [ 2nd ] [ * ] (nPr)

Rectangular to Polar Conversions:  x [ a ], y [ b ], [ 2nd ] [ b ] (R>P); r stored in [ a ], θ stored in [ b ]

Polar to Rectangular Conversions: r [ a ], θ [ b ], [ 2nd ] [ a ] (P>R); x stored in [ a ], y stored in [ b ]

Mode 2:  Binary Mode

Entering Binary Mode converts the number into a binary integer.  Arithmetic operations are available.  The maximum binary number is 511 (2^9 – 1), and binary numbers are 10 bits including a signed bit (leftmost). 

Mode 3:  Octal Mode

Entering Octal Mode converts the number into an octal integer.  Arithmetic operations are available.  

Mode 4:  Hexadecimal Mode

Entering Hexadecimal Mode converts the number into a hexadecimal integer.  Arithmetic operations are available.  In this mode, the [ sin ], [ cos ], [ tan ], [ 1/x ], [ a ], and [ b ] are remapped to the hexadecimal digits A, B, C, D, E, and F, respectively.

Bob Patton gives this amazing demonstration of the Hexadecimal mode: 

[MODE] 4 [ tan ] [ 0 ] [ b ] [ b ] [ a ] [ a ] [STO]  (stores COFFEE_16 in memory)

[ tan ] [ 0 ] [ tan ] [ 0 ] [ sin ]  (inputs COCOA_16)

Repeat:

[ SUM ] [ 0 ] [ RCL ] [ = ] [ SUM ] [ +/- ]

(Note what happens while repeating this loop.  You can try a similar key stroke loop on similar calculators.  Thank you, Bob!)

Mode 5:  Complex Number Mode

Store the real part in temporary register [ a ], the imaginary part in temporary register  [  b ].  Arithmetic operations and polar/rectangular conversions are available.  Other math functions work on the components only.

Mode 6:  Statistics Mode

The TI-35 Plus offers 1-variable statistics with the standard measurements of mean, standard deviation (σn-1), population deviation (σn), and sums. The only way to clear the stat data is to exit stat mode, then enter it again.

An added feature is the three normal distribution probability functions.  Strangely, these functions do not rely on the data entered in the stat registers, and assume that the standard parameters apply (mean is zero, variance is one).

P(t):  lower tail probability

Q(t):  upper tail probability

R(t):  probability from 0 to z

Keyboard and Display

The keys are nice and responsive.  Over time the key markings wearing off.  I like the white font on the dark gray keys.  I wish there was a little more contrast for the secondary functions, which are black on dark gray. 

The display is nice and crisp. 

Final Verdict

I like the TI-35 Plus, it is a step up from the TI-30 series by adding complex number arithmetic and integer conversions.  However, the TI-35 Plus lacks the Boolean functions found on the TI-34.  It is a matter of what features are desired. 

Eddie

This blog is property of Edward Shore, 2017.

Sharp EL-501X (today) vs TI-35 Plus (1989)

TI-35 Plus Notes vs Sharp EL-501X (and almost all of its clones)

Ever wanted a TI-35 Plus? First you could go to eBay or other online stores and try to hunt one down or invest up to $10 for one of the many close-version that are still manufactured and sold today. There is even a version sold at the Dollar Store! (The keyboard of that $1 Store I bought is crappy.) Today I will be comparing the Sharp EL-501X against the 1989 TI-35 Plus (and its solar sister, TI-36 Solar (no X)).

Since I never actually owned a TI-35 Plus, all the information will be from its manual. I did buy a TI-36 Solar guidebook, 20 years ago or so, from a clearance bin at Toys R Us of all places. The Sharp EL-501X I bought this week primarily for this blog entry, and to replace a blue cased EL-501X I gave to my cousin's daughter.

Source of the TI-35 Plus information: www.datamath.org

TI-35 Plus Page: http://www.datamath.org/Sci/Modern/TI-35-PLUS.htm

Side plug: Datamath is an excellent web page for all things Texas Instruments calculators, past and preset.

Commonalities

Internal Digits: 12

Modes:
Computational
Complex Mode
Base Display Modes: Binary, Decimal, Octal, Hexadecimal
One Variable Statistics

Complex Mode is limited to arithmetic. The real part is stored and recalled using the [ a ] key and the [ b ] key is for the imaginary part. ( a + bi )

Base Display Modes (except for Decimal) is limited to arithmetic. To convert numbers, just call their respective mode.

Number of memory registers: 1

Factorial Function: n!. Only takes positive integers or zero unless an error occurs.

Polar/Rectangular Conversions:
[ a ] is used for x and r
[ b ] is used for y and θ

Basic Scientific Function set: hyperbolic functions, trigonometric functions and inverses, exponential functions and inverses, reciprocal, square root, cube root, power, arithmetic, π, 3-decimal random number (usually labeled RND)

Differences

Display:
TI-35 Plus: 10 digits plus 2 digit exponent
EL-501X:10 digits or 8 digit plus 2 digit exponent

Exchange Key:
TI-35 Plus has two: EXC to change the number in the display with the number in the memory register. x<>y exchanges operands in power, root, division, and subtraction calculations.
EL-501X: Only the latter is present, often symbolized by a vertical line with up and down arrows. ( [2ndF] [ ( ] )

Decimal Display Settings:
TI-35 Plus: Four permanent modes: floating, fixed, scientific, engineering
EL-501X: Two permanent modes: floating and fixed. Numbers can be toggled between floating/fixed and scientific notation by pressing [ F ←→ E ]

Things the TI-35 Plus had that the EL-501X and most of the clones do not:

Permutations and Combinations. The TI-35 Plus facilitated the use of the [ a ] and [ b ] keys.

Normal Cumulative Distribution Functions (mean = 0, variance = 1, given z):
R(z): area between 0 and z
Q(z): area from z to positive infinity (upper tail)
P(z): area from negative infinity to z (lower tail)

In more advanced calculators and mathematical software, the normalcdf function calculates P(z) (lower tail).

One variate of the EL-501X, the Canon F-604 has permutations, combinations, memory exchange, and fractions. Unfortunately the one I bought not to long ago, the zero key reseted the calculator, so I had to return it. Hopefully it is not the case with all F-604s.

---

Yes - entry level scientific calculators still exist in the market.


Take care everyone, thanks for comments and questions. Much appreciated. Eddie


This blog is property of Edward Shore. 2014