Math Without Numbers - a book review

In preparing and studying for this blog, I read as many math books as possible. I haven't posted book reviews in this blog before, but Math Without Numbers by Milo Beckman and illustrated by M Erazo deserves a very positive review. Beckman truly succeeds in walking the reader through many mathematical topics without using numbers. To do so, many real life examples are given to go with the concepts.

Several mathematical proofs are presented in a refreshing format- sometimes with easy-to-follow illustrations but never with numbers or formulas. 

One of my favorite sections gave a good introduction to topology. Another presents the incompleteness theorem of Gödel in a dialog format. A surprisingly clear analogy is offered on how the law of gravity is comparable to estimating the trade between two countries. As a fan of automata, I enjoyed an example demonstrating how the traffic phenomena of gapers' block can be modeled with simple rules. To cap off the tour through many subjects, Beckman delivers an understandable explanation of the Standard Model of particle physics.

milobeckman.com

Twitter milobela

See prior post for other math book and website recommendations.

Pi Day 3-14-2021

This blog has several posts about pi, including one that used random numbers to estimate pi.

Using Random to Estimate Pi

Matt Parker and Steve Mould posted a YouTube video demonstrating how to estimate pi using Avogadro's number (Avogadro's number is used in chemistry and is the number of particles in one mole of a substance).

Matt Parker and Steve Mould use Avogadro's Number to Estimate Pi

For pi day, WIRED posted this video about pi and keeping trains on the track:

https://www.wired.com/story/how-pi-keeps-train-wheels-on-track/

Other pi-related posts in this blog include:

Prime Number Spirals

One-Time Pad Coding

Find Your Birthday in the Digits of Pi

Pi - How Many Digits are Really Needed?

Additional Thoughts on the Pythagorean Triple Generator

A prior post list a formula for producing PythagoreanTriples. A Pythagorean Triple is a set of three integers (a, b, c) that will form a right angle triangle. The formula is:

Let n and m be any positive integers where n>m. The Pythagorean Triple generated is formed by the numbers:
2nm
n² - m²
n² +m²

This prior post did not state way this formula works. We can show why in just a few steps. The two requirements for a Pythagorean Triple (a, b, c) are that a, b, and c are integers and that a² + b² = c².

For the first requirement, we need to show that 2nm,  n² - m² , and  n² +m² are all integers. It is given that n and m are integers, Since both n and m are integers, a product of two integers is an integer and 2 times an integer is an integer, so 2nm is an integer. Likewise, for n² - m²
and n² +m², we are only dealing with squares of integers, their sums and their differences. All these operations will produce only integers, therefore n² - m² and n² +m² are also integers.

For the second requirement, we need to show that (2nm)² + (n² - m²)² = (n² +m²)²

Expanding the left-hand side,

 (2nm)² + (n² - m²)² = 4n²m² + n⁴ – 2n²m² + m⁴

Simplifying and reordering,       

                              = n⁴ + 2n²m² + m⁴

Compare this result to squaring the original right-hand side, we see the Pythagorean Triple generator formula will produce triples that meet the requirements of being proper Pythagorean Triples.

(n² +m²)² = n⁴ + 2n²m² + m⁴

^{Update 9/24/2022: A spreadsheet has been created to produce these triples.}

Ending Digits of Numbers Raised to Various Powers

 

A table of the numbers 1 through 10 each raised to the powers of 1 through 5. This table is shown below.

 

Table of numbers 1 to 10 raised to different powers

For this post, I would like to focus on just the ending digit of each result of the exponentiations. The following table lists the ending digit of each result and now we can explore some of the patterns that arise for the different powers. When we get the results for Power 5, we see why we only have to consider this explorations for just 5 powers as the pattern will repeat for powers higher than 5.

One additional note is that each row of this table applies not only to the numbers 1 through 10 but also to any number, higher than 10 that has the same last digit. For example, the results for raising the number 12 to various powers will have the same ending digit as 2 raised to the same powers. For this reason, the first column of the second table is named “Starting Number’s Last Digit” and the last row’s name has been changed from “10” to “0.”

Table of last digits resulting from exponentiation

Discussion of results: 

Power 0

The table doesn't list the results of raising numbers to the power of 0. All numbers raised to the power of 0 equal 1. One result that will likely become another post on its own is zero raised to the power of zero. There is disagreement on the result (see link). Some think this should equal 1; some believe it should undefined as with division by zero.

Power 1

These results are a bit trivia but are included for completeness of the exercise. Any number raised to the power of 1 is itself and therefore the last digit of the result will match the last digit of the starting number. A shorthand way of expressing this result is 1à1, 2à2, etc. For Power 1, this is the identity function, for all i, i à i.

Power 2

Squaring numbers ending in 1, 5, 6 and 0 give results that will end with 1, 5, 6, and 0, respectively. This pattern continues for all powers. Results of squaring numbers 2 and 8 will end in 4. For the numbers ending in 3 and 7 yield 9. Squaring numbers that end in 9 yields 1. Another observation is there are no numbers that when squared which will have an ending digit of 2, 3, 7 or 8. Summarized in shorthand expressions:

Identity function for 1, 5, 6, 0
2à4, 8à4, 3à9, 7à9
Result exclusions: 2, 3, 7, 8

Power 3

Unlike Power 2 in which it was impossible to obtain results with certain end digits (2, 3, 7 or 8), we observe cubes that end in all digits 0 through 9. Cubing numbers ending in 1, 4, 5, 6, 9 and 0 give results that will end with 1, 4, 5, 6, 9 and 0, respectively. For the other four digits: numbers ending in 2 yield 8 and those ending in 8 yield 2; numbers ending in 3 yield 7 and those ending in 7 yield 3.

Summary
Identity function for 1, 4, 5, 6, 9 and 0
Inverse function for 2, 3, 7, 8: 2à8, 8à2, 3à7, 7à3
No exclusions

Power 4

Reviewing Power 2 we observed that resultant ending digits excluded 2, 3, 7 and 8 and with Power 3 all digits were possible. With Power 4, there are more exclusions. All numbers raised to the 4^th power will only end in 0, 1, 5 or 6. There are no numbers when raised to the 4^th power that end in 2, 3, 4, 7, 8 or 9. As with the other powers, raising numbers ending in 1, 5, 6 and 0 will yield the same end digit as the starting number. Along with 6, numbers ending in 2, 4 or 8 will also yield a result having an end digit of 6. Similarly, numbers ending in 1, 3, 7, 9 yield 1.

Summary
Identity function for 1, 5, 6 and 0
(1, 3, 7, 9) à 1
(2, 4, 6, 8) à 6
Result Exclusions: 2, 3, 4, 7, 8 and 9

Power 5

Power 5 completes the full cycle of the power pattern. All numbers when raised to the 5^th power will have an ending digit that is the same as the starting number’s last digit. This is the same as Power 1 which resulted in the identity function for all ending digits. Another observation is that we could continue this exercise for powers higher than 5. The patterns and results we observed for Powers 2 to 5 will repeat for Powers 6 to 9, Powers 10 to 13 and higher.