# The Magic of Math | Arthur T. Benjamin

Summary of: The Magic of Math: Solving for X and Figuring Out Why

By: Arthur T. Benjamin

## Introduction

Welcome to the magical world of mathematics, where numbers hold delightful secrets and astounding patterns. In the book summary of ‘The Magic of Math: Solving for X and Figuring Out Why’ by Arthur T. Benjamin, you’ll discover the enchanting properties of numerical patterns and the power of mental arithmetic. You’ll learn how math can be both beautiful and highly practical in real life situations. The book summary demonstrates a party trick based on algebra and unveils the wonders of Fibonacci Numbers. Brace yourself for a journey that explores the mysterious numbers π, i, and e, the peculiar aspects of infinity, and the sheer beauty of mathematical proofs.

## The Magic of Numerical Patterns

Mathematics is more than just tedious textbooks and laborious calculations. The author of this book discovered the beauty of numerical patterns since he was a child. These patterns, made of numbers, are not only magical but also useful in the real world of mental arithmetic.

The author employs the example of pairs of numbers adding up to 20, such as 9 and 11 or 10 and 10, to explain how these numerical patterns work. By examining the product of these pairs, certain patterns emerge, which apply no matter what number the pairs add up to. The numbers follow the sequence of 1², 2², 3², and so on.

Understanding these patterns can enhance our mental arithmetic and make our calculations easier, such as squaring numbers. Instead of calculating 13×13, we can perform the easier calculation 10*16, where both numbers add up to 26, just like 13 and 13. By following the pattern, we can add 3² to the result to get 13²=(10*16)+3²=160+9=169.

In conclusion, numerical patterns are not only fascinating but can also serve as practical tools in the real world.

## Mathemagics – The Power of Algebra

Learn how to amaze your friends with a party trick that involves the power of algebra. By following five simple steps, you can instantly guess two numbers your friend is thinking of and leave them amazed. This party trick is not just about impressing friends, but it also demonstrates the wonder of algebra and mathemagics. By breaking down the steps and using basic algebra, you can see how to arrive at the original numbers that your friend was thinking of. Demonstrating how letters in algebra are used in place of numbers, this trick is an excellent way to show how fun and useful math can be. By using algebra to explain the steps, it is easy to see how the sum of two numbers can lead to finding the original numbers being added.

## The Magic of Number 9

The number 9 holds two fascinating properties. First, the digits of multiples of 9 always add up to 9. Second, the product of an “abcde” number and 9 always equals 9.

The number 9 has mystical properties that have intrigued the author since childhood. One unique feature of 9 is that the digits in multiples of 9 always add up to 9. Even 99, which seems like an exception, adds up to 9 when its digits are summed.

Additionally, any number whose digits add up to a multiple of 9 is itself a multiple of 9. For instance, the total of the digits in 123,456,789 equals 45, which is a multiple of 9, and so 123,456,789 is also a multiple of 9.

Another fascinating aspect of 9 is related to certain numbers written in ascending order, such as 12345 or 135789. When one of these “abcde” numbers is multiplied by 9 and summed, the result is always 9.

These charming properties reveal the magic associated with the number 9.

## The Fascinating World of Fibonacci Numbers

In his book Liber Abaci, Fibonacci introduced a problem involving rabbits that led to the discovery of a sequence of numbers that has remarkable properties. Each number in the Fibonacci sequence is the sum of its two predecessors. Adding the squares of Fibonacci numbers leads to a new sequence, where every second number is left out. The sum of the squares of Fibonacci numbers is the product of two consecutive Fibonacci numbers. The discovery of the Fibonacci sequence has led to various applications in different fields, from nature to finance.

## The Power of Mathematical Proofs

Mathematics is the only science where propositions can be proven to be absolutely true. This is possible through the use of mathematical proofs – series of equations that hold true under all circumstances. For example, the proof that if m and n are even numbers, then m+n is an even number can be written as m=2k and n=2j, with k and j being integers. It follows that m+n=2k+2j=2(j+k), which means that m+n is a multiple of 2 and therefore even. The power of proofs lies in their ability to establish certainty without resorting to endless calculations. Another powerful method used in mathematics is induction, where the truth of a statement for a particular number is used to prove its truth for all subsequent numbers. Mathematicians find induction particularly elegant. Mathematical proofs are the foundation of mathematical truth and their use enables mathematicians to establish irrefutable evidence of the correctness of their propositions.