Python – Earth’s Radius and Gravity in US Units

Python – Earth’s Radius and Gravity in US Units

Introduction

The following script, gravus2.py, estimates the Earth’s gravity in ft/sec^2 given:

1. The observer’s latitude on Earth (North or South; you don’t have to enter negative numbers for South)

2. The elevation where the observer is in feet.

The Python script is made with a TI-84 Evo, but it should work with all platforms with Python.

Equation Used: With Everything Converted to U.S. Units:

Calculating g_0 at sea level with latitude φ:

g_0 = 32.17192 – 0.08530 * cos(2 * φ°) = 32.17192 – 0.08530 * cos(2 * φ * π ÷ 180)

Calculating g_h with height h:

g_h = g_0 * (erad ÷ (erad + h ÷ 5280))^2

Where (Earth’s radius at latitude φ):

erad = √((a^4 + b^4 * tan(φ)^2) ÷ (a^2+ b^2 * tan(φ)^2)) 

a = 3963.1905919

b = 3949.90276423189

Code: gravus2.py

'''

Earth's Gravity in US Units

EWS 05/26/2026

'''

from math import *

# title screen

print("Earth's Gravity")

print("Given Latitude and Elevation")

print("U.S./Imperial Units")

print("-----------------")

# angles must be converted to radians

print("Latitude in DMS:")

l1=eval(input("Whole Degrees? "))

l2=eval(input("Whole Minutes? "))

l3=eval(input("Seconds? "))

langle=l1+l2/60+l3/3600

# convert to radians

langle*=pi/180

# elevation

elev=eval(input("Elevation in feet? "))

# calculate radius 

a=3963.1905919

b=3949.90276423189

erad=(a**4+b**4*(tan(langle))**2)

erad/=(a**2+b**2*(tan(langle))**2)

erad=sqrt(erad)

# calculation

g=32.17192-0.08530*cos(2*langle)

g*=(erad/(erad+elev/5280))**2

print("Earth's radius: {0:.5f} mi".format(erad))

print("Estimated Gravity: {0:.5f} ft/s**2".format(g))

Examples (Results are rounded to five decimal places)

Latitude (φ)	Height (h ft)	Earth’s Radius (mi)	g_h (ft/s^2)
34°03’00”	500	3959.04875	32.13857
64°11’27”	1650	3952.43868	32.21979
10°57’00”	395	3962.71501	32.09156

Source

Grainger Engineering Office of Marketing and Communications. (answer written by Rebecca H.) (2016, November 21). “How gravitational force varies at different locations on Earth.” Illinois. https://van.physics.illinois.edu/ask/listing/64061. Retrieved March 10, 2026.

Planetcalc. “Earth Radius by Latitude (WGS 84)” Timur. 2021. https://planetcalc.com/7721/ Retrieved March 22, 2026.

Eddie

All original content copyright, © 2011-2026. 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.

The MU Key on a Four Function Calculator and Programs for the DM42/HP 42S

 The MU Key on a Four Function Calculator and Programs for the DM42/HP 42S

Not too long ago, I purchased this very colorful, four function calculator from the University I work at, the Dexin BST DX-818.

The DX-818 is powered by a solar panel with an option to be run by an AAA battery when using the calculator in less than bright light conditions. The AAA battery is optional.

A Standard Calculator

This calculator is the standard four function key calculator, complete with the square root function, percent function, the standard three memory keys (M+, M-, and MRC), and a grand total (GT) key. A grand total is started when calculations are completed by pressing the equals key [ = ]. The grand total is both recalled and cleared when the [ GT ] is pressed.

The Markup Equation

Before we head to the calculators, the mark up is a calculation of between selling price (new), cost (old), and mark up. The markup is ratio between the selling price and cost over cost and as expressed as a percentage (%):

MU = (selling price – cost) ÷ cost = (selling price ÷ cost - 1)

Solving for the other variables yield:

selling price = cost * (1 + MU)

cost = selling price ÷ (1 + MU)

In these equations, MU is expressed as a decimal.

If let the selling price be the new value and cost be the old value, then we have the equation for percent change (Δ%):

Δ% = (new – old) ÷ old = (new ÷ old - 1)

Similarly:

new = old * (1 + Δ%)

old = new ÷ (1 + Δ%)

In these equations, Δ% is expressed as a decimal.

The Markup Key (MU)

On this particular calculator, there is an [ MU ] key. This is the markup key. Apparently, there are two types of [ MU ] key, one set of syntax for Casio calculators with the [ MU ] key. An example of a Casio calculator with an [ MU ] key is the MJ-120D plus. They way Casio calculators use the [ MU ] key is a calculation:

cost [ MU ] markup percentage [ % ] returns selling price.

Immediately pressing [ = ] returns selling price – cost.

However, some calculators, like the colorful calculator featured (I really wish I knew the model number) with the [ MU ] have another syntax, and that is what I will address from here on out.

On this particular calculator, the syntax, using one of the four arithmetic functions is:

y < +, -, ×, or ÷ > x [ MU ]

No use of the equals key is needed, unless you want to add the result to the grand total.

The four calculations with the [ MU ] key, on this particular calculator (and possibly the NewYes calculators from China) are:

Key Sequence	Formula Used	Notes
Y [ + ] X [ MU ]	100 * (Y ÷ X + 1)
Y [ - ] X [ MU ]	100 * (Y ÷ X - 1)	Calculates markup or percent change (Δ%). Y = selling price, new value X = cost, old value
Y [ × ] X [ MU ]	Y * (1 + X ÷ 100)	Adds X% to Y. Calculates the new value (selling price) given old value (cost) (Y) and markup (X).
Y [ ÷ ] X [ MU ]	Y ÷ (1 – X ÷ 100)	Finds the original retail price given selling price (Y) and discount rate (X). This can also be used for gross-up calculations.

Sample Calculations

Problem	Keystrokes	Result
Determine 100 * (Y ÷ X + 1) with Y = 420, X = 25	420 [ + ] 25 [ MU ]	1780
An item that cost the company $14.75 sells for $19.99. Find the markup.	19.99 [ - ] 14.75 [ MU ]	35.5254237288 (≈35.53%)
A company manufactures a product with a unit cost of $32.84 and wants a 20% markup. What is the selling price?	32.84 [ × ] 20 [ MU ]	39.408 (about $39.41)
A product is currently on sale for $39.99. If the discount is 35%, what is the original retail price.	39.99 [ ÷ ] 35 [ MU ]	61.523076923 (about $61.52)

RPN Programming with the Swiss Micros DM42

Here are four RPN programs, which are were programmed using a Swiss Micros DM42, which can be used on the HP 42S (and probably the HP 41C).

Program MU+: Y [ + ] X [ MU ]

00 {17-Byte Prgm}

01 LBL “MU+”
02 ÷

03 1

04 +

05 100

06 ×

07 RTN

08 END

Program MU-: Y [ - ] X [ MU ]

00 {17-Byte Prgm}

01 LBL “MU-”
02 ÷

03 1

04 -

05 100

06 ×

07 RTN

08 END

Program MUMLT: Y [ × ] X [ MU ]

00 {12-Byte Prgm}

01 LBL “MUMLT”
02 %

03 +

04 RTN

05 END

Program MUDIV: Y [ ÷ ] X [ MU ]

00 {21-Byte Prgm}

01 LBL “MUDIV”
02 ENTER

03 100

04 ÷

05 1

06 X<>Y

07 -

08 ÷

09 RTN

10 END

Sources

 Casio. “MJ-120D Plus” https://www.casio.com/intl/basic-calculators/product.MJ-120DPLUS/ Accessed May 23, 2026.

Eaton, Glenn. “The Calculator MU Key” Eaton Family Website. Posted on June 30, 2017. https://eatonfamily.au/calculator-mu-key/ Accessed May 23, 2026.

“How to use the MU button on a calculator? #1” YouTube Short: @newyes1162. Posted December 5, 2022. https://www.youtube.com/shorts/OiS8yFKKJSU Accessed May 23, 2026

Eddie

All original content copyright, © 2011-2026. 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.

Guest Post: Why Kids (and Adults) Are Rediscovering the Abacus in the Age of Calculators by The AbacusTool Team

Today’s post is a guest post by The AbacusTool Team, and it is about discovering mathematics by using the time-honored tool, the abacus. I have a small abacus collection, including a small eight rod abacus that sits at my desk at work. I have a working knowledge of adding, subtracting, and multiply, though I am no expert. I love how the abacus is set up to efficiently add and subtract. We can set the decimal point anywhere among the rods. I enjoy using the abacus and appreciate how it contributed to mathematics and education.

And now, here's the guest post from The AbacusTool Team, as a request. I thank The AbacusTool Team. Please visit their website at https://abacustool.xyz/.

Eddie

------------------------

From the AbacusTool Team: 

(pictures are from the AbacusTool Team)

A few months ago, I watched a nine-year-old solve a three-digit addition problem in her head faster than I could type it into my phone. No calculator, no scratch paper — just her fingers twitching slightly in the air as she "saw" the numbers. She'd been training on an abacus for about a year.

It's a strange thing to witness in 2026. We carry supercomputers in our pockets, yet one of the oldest counting tools on earth — a wooden frame with sliding beads — is having a real moment in classrooms and homeschool circles again. Here's why, and how you can actually try it yourself without buying anything.

The Tool Everyone Forgot About

The abacus isn't new. Counting boards go back to Mesopotamia around 2700 BCE, and versions of it evolved independently across Rome, Greece, China, and Japan over the following millennia. China's suanpan — with two beads on top and five below each rod — shows up in written records from the Han dynasty, roughly 200 BCE. Japan later adapted the design into the soroban, simplifying it down to one top bead and four bottom beads, which is now the standard taught in most abacus classes worldwide.

What's interesting isn't the history trivia, though — it's what happens in someone's brain after enough practice.

It's Not About the Beads, It's About What Happens After

The real payoff isn't learning to move beads quickly. It's what researchers call mental abacus — the ability to visualize the bead frame in your head and manipulate it without a physical tool in front of you. Once that clicks, people can add and subtract multi-digit numbers almost as fast as punching them into a calculator.

This isn't just anecdotal. Cognitive psychologist James Stigler's research in the 1980s, along with follow-up studies since, found that trained abacus users develop distinct visual-spatial strategies for arithmetic that regular calculator use never builds. It's less "memorizing math facts" and more "training a mental workspace."

That's also why abacus programs like UCMAS and Aloha have stuck around for decades — they're not really teaching a tool, they're teaching a cognitive habit.

Who Actually Benefits From This

Young kids (5–12) tend to take to it fastest, mostly because numbers stop being abstract. A kid who struggles to understand what "36" means can physically see it as three beads and one heaven bead. That concreteness seems to matter a lot for how quickly place value clicks.

Students get a more practical edge — faster mental math during exams, quicker estimation, less reliance on a calculator for basic arithmetic.

Adults are honestly an underrated group here. Abacus practice functions a bit like a puzzle app, except it's also sharpening real-world skills — splitting a bill, estimating a tip, doing quick math at work. It's turning into a fairly popular low-key brain-training habit for people who've aged out of "flashcards."

Teachers and parents use it as a physical teaching aid, especially for explaining carrying, place value, and basic operations in a way that clicks visually instead of just numerically. Some even print practice worksheets for offline drilling when screen time isn't the goal.

You Don't Need to Buy One to Try It

This is the part that surprised me most — you don't actually need a physical abacus to start. There are free interactive versions online now that replicate real bead logic (not a gimmicky animation), let you switch between Soroban and Suanpan styles, and throw practice problems at you so it doesn't feel like homework.

I've been using AbacusTool.xyz to get a feel for both styles before deciding whether to buy a physical one for my kid. It's free, doesn't ask for a signup, and has a built-in tutorial that starts from "here's what a bead is worth" and works up to actual multi-digit calculations. There's also a step-by-step solver that shows you exactly how a problem is worked out bead by bead, which honestly made more sense to me than most YouTube tutorials.

If you want to go further, it's worth trying flash anzan drills — numbers flash on screen and you add them mentally, visualizing the beads instead of looking at a physical frame. It's the same drill competitive mental-math kids use, and it's brutal at first, then oddly addictive. There's also a complement drills tool for practicing the 5- and 10-complement technique, which is the actual foundation that makes fast abacus calculation possible — most beginners skip it and then wonder why they're stuck.

If you're not sure where to start, there's a structured learning path that walks you from complete beginner to mental calculation in order, instead of randomly clicking around.

Abacus vs. Vedic Math vs. Just Using a Calculator

People sometimes ask whether abacus training is "better" than Vedic math tricks or just... learning to type fast on a calculator. They're not really competing with each other:

• Abacus builds visualization and number sense, especially in younger kids who need something concrete to hold onto.

• Vedic math teaches shortcut techniques and works better once someone already understands how arithmetic works.

• Calculators are just faster for anything complex — nobody's suggesting you long-divide a mortgage payment by hand.

The abacus isn't trying to replace calculators. It's training the part of your brain that calculators let go soft.

Worth Trying, Even If You Think You're "Not a Math Person"

The thing that convinced me this wasn't just a kids' toy was trying it myself. Watching numbers turn into physical bead positions — and slowly not needing to look at them anymore — is a genuinely different kind of mental exercise than anything else I've tried. It's slow at first, faster than expected after a week of casual practice, and free to test without committing to anything.

If you're curious, it takes about two minutes to open a browser tab and move your first few beads.

---

Have you tried abacus training, either as a kid or as an adult brain-training habit? I'd be curious to hear how it compares to other mental math methods you've used.

Python (TI-84 Evo) and HP 32S: Shannon-Wiener Diversity Index

Python (TI-84 Evo) and HP 32S: Shannon-Wiener Diversity Index

Introduction

The Shannon-Wiener Diversity Index measures a level of how diverse a group of populations are by the following formula:

H = -Σ( p_i * ln(p_i), i = 1 to n) = Σ( -p_i * ln(p_i), i = 1 to n)

where p_i = (population of segment i) ÷ (total population)

As stated by M. Robert F. (see Sources), there are three levels of diversity:

Low Diversity: H < 1.5

Medium Diversity: 1.5 ≤ H ≤ 2.5

High Diversity: H > 2.5

Python: diverse.py

This script was done with a TI-84 Evo. The script uses only the math module and should be able to use on any calculator with Python.

# Math Calculations

from math import *

print("Shannon-Wiener Diversity\nIndex")

print("Enter a list of populations.")

lpop=eval(input("list: "))

# sum

s=sum(lpop)

# proportions

lpro=[i/s for i in lpop]

# -p log p

llog=[-i*log(i) for i in lpro]

# diversity index

h=sum(llog)

# results

print("diversity index:\n{0:.5f}".format(h))

if h>2.5:

  print("high diversity")

elif h<1.5:

  print("low diversity")

else:

  print("medium diversity")

HP 32S Program: Shannon Wiener Diversity Index

This was programmed on an original HP 32S, and the code should be same for the later HP 32SII and Swiss Micros DM32.

Instructions:

If need be, clear the variables

Enter the population size in the variables A, B, C, D etc… up to 25.

Enter the number of populations, XEQ A

Four labels are used:

LBL A: program start, initialization

LBL Z: determine total population (pop = A + B + C + D….)

LBL Y: determine proportion of each population (A = A/pop, B = B/pop, C = C/pop, …)

LBL X: calculate the Shannon-Weiner Diversity Index: -Σ(p_i * ln(p_i) from i=1 to n)

Code:

A01 LBL A

A02 ENTER

A03 ENTER

A04 ENTER

A05 CL x

A06 STO Z

A07 R↓

A08 STO i

Z01 LBL Z

Z02 RCL(i)

Z03 STO+ Z

Z04 R↓

Z05 DSE i

Z06 GTO Z

Z07 STO i

Z08 RCL Z

Y01 LBL Y

Y02 STO÷ (i)

Y03 DSE i

Y04 GTO Y

Y05 R↓

Y06 STO i

Y07 CL x

Y08 STO Z

X01 LBL X

X02 RCL (i)

X03 ENTER

X04 LN

X05 ×

X06 +/-

X07 STO+ Z

X08 DSE i

X09 GTO X

X10 RCL Z

X11 RTN

Size:

LBL A: 12.0 bytes

LBL Z: 12.0 bytes

LBL Y: 12.0 bytes

LBL X: 16.5 bytes

Total: 52.5 bytes

This does not count the 9.5 bytes needed for each variable used (Z, i, A, B, C, etc.). This program takes up to 25 populations (A through Y).

Examples

Example 1:

4 populations (n = 4).

A = 40

B = 30

C = 70

D = 20

Index: 1.2820 (low diversity index)

Example 2:

9 populations (n = 9).

A = 41

B = 36

C = 40

D = 97

E = 64

F = 37

G = 16

H = 18

I = 36

Index: 2.06303 (medium diversity index)

Sources

M. Robert F. The Math Behind Biology Lightning Source4 LLC. LaVergne, TN. 2026. pp. 6-8

Nolan, K.A. and J.E. Callahan. 2006. Beachcomber biology: The Shannon-Weiner Species Diversity Index. Pages 334-338, in Tested Studies for Laboratory Teaching, Volume 27 (M.A. O’Donnell, Editor). Proceedings of the 27^th Workshop/Conference of the Association for Biology Laboratory Education (ABLE), 383 pages. 2006. ISBN 1-890444-09-X. Article: https://www.ableweb.org/biologylabs/wp-content/uploads/volumes/vol-27/22_Nolan.pdf. Retrieved May 6, 2026.

Eddie

All original content copyright, © 2011-2026. 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.