Part I: The Origins of Algebraic Thinking and the Modern Number System
Have you ever had to solve a word problem with an unknown quantity? For example, “There are 4 apples in a basket. The apples and the basket together weigh 16 ounces, but the basket weighs 4 ounces. How much does one apple weigh?” In A Grade Ahead math classes, we teach students to solve this problem by creating an algebra equation: 4x + 4 = 16. The “x” stands in for the weight of an individual apple, which means that we are using algebra to figure out this problem.
For most of us, algebra is the use of symbols or letters to stand for unknown quantities in equations and formulas. Mathematicians and scientists, though, more specifically use it to study the properties and relationships between abstract concepts. So, we use algebra to write computer programs and calculate the trajectory of rockets as well as to find the weight of an individual apple in math class.
A Grade Ahead students apply these concepts to real-world situations throughout the year in our enrichment classes. We also devote the entire month of May to the practical applications of math! Learn more on our website.
It is easy to take all these things for granted as we sit through math classes, but A Grade Ahead appreciates the complicated evolution of our understanding of math. In this first part of our new series on the history of math, we will explore the development of algebraic thinking.
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Origins of Algebraic Thinking
Scholars have found evidence that the first word problems emerged in ancient Mesopotamian classes. There, the students learned to apply concepts similar to algebra as we know it to real-world situations, like solving for unknown property measurements. Indeed, the earliest arithmetic problems appear in cuneiform tablets created by the Babylonians almost 4,000 years ago! Babylonian math, though, used a number system based on units of 60 rather than our modern system, which is based on units of 10. They also did not have a concept of 0 as a place holder as we do today.
The oldest known text using algebraic thinking is the Rhind papyrus, which was written in Egypt over 3,000 years ago. This papyrus demonstrates that the Egyptians already knew how to solve linear equations for one unknown variable, like the equation 4x + 4 = 16. Over 2,000 years ago, in 300 BC, Egyptian students were learning to solve systems of equations for two variable and quadratic equations. Quadratic equations involve variables taken to the second power. For example, x2 + 2x + 3 = 4 is a quadratic equation. These equations are useful for scientists studying curves, like the trajectory of an object through space.
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These ancient mathematicians, however, did not substitute symbols or letters for the unknown variables, and it is clear that they were meant to be read and answered aloud in math class. They were not meant to be written down and solved as we do today.
The Greeks’ Mind-Blowing Contributions
The Greeks elevated our understanding of math, moving it from everyday practical applications to a theoretical appreciation of the world around us.
One of the best examples of this is the discovery of irrational numbers, like √2. In 430 BC, Pythagorean mathematicians discovered that not all lengths can be measured by the same unit. They came to this important conclusion while exploring the relationship between the side of a square and its diagonal. They had already learned the Pythagorean Theorem from the Babylonians. This theorem tells us that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse, the third side. This is known to us as a2 + b2 = c2. Because that is true, though, a square with side lengths of 1 unit has a diagonal of √2 units, which is not a rational or whole number. It results in a decimal that continues with no end.
Greek scholars did not fully understand decimals, so this concept was challenging for them. It continues to challenge A Grade Ahead’s math students learning number systems, like they do in our high school math program.
Like the Egyptians and Mesopotamians, though, the Greek mathematicians did not use formal equations as we know them or use symbols as we do today. Then, around 250 AD, Diophantus of Alexandria began using a sort of shorthand to calculate something like an equation. He did not use equal signs and negative coefficients or even suggest manipulating the variables to solve these equations.
Did you know that A Grade Ahead math students begin studying algebra as early as 3rd grade?
Indian Revelations
Ancient Indian mathematicians further revolutionized our understanding of the number system by introducing the concepts of decimals, integers, and especially the importance of zero.
Today, zero serves as a placeholder, which allows us to use a base-10 number system. This means that the ones place goes from 0 to 9 before starting over. So, 11 follows 10, and 21 follows 20. This system allows for decimals, like 2.5. This revelation changed the way we think about numbers and means that we can calculate more complicated equations and formulas.
Positive and negative numbers, which we call integers, further expanded what we can do with math. They allow us to use numbers less than 0 or greater than 0. Business leaders use them to calculate profits and losses. We also use integers to calculate fluctuations in temperature and the sea levels, which allows us to keep track of the changing environment.
Scholars believe that Indian mathematicians developed these concepts and were using them over 1,000 years ago. Their revelations spread both east and west as Chinese and Islamic scholars began working with them as well.
Chinese Pioneers
At A Grade Ahead, we teach our algebra students to solve quadratic equations using radicals, just as Chinese scholars did centuries ago. Radicals are the roots of numbers, which appear under a root symbol (√). Thanks to these ancient mathematicians, we can find not only the square root of a number but also the cube root of a number and more!
To calculate radicals, Chinese mathematicians used an abacus, the world’s first calculator. An abacus uses rods and beads to complete calculations, like adding and subtracting as well as multiplication. They were using this invention over 2,000 years ago, and it helped them develop multiplication tables for decimals as well as whole numbers.
Algebra Gets Its Name: Islamic Mathematicians
In 825 AD, Muḥammad ibn Mūsā al-Khwārizmī wrote his most important work, al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa’l-muqābala. This work, which was translated to Algebra et Almucabal in Latin, forms the origins of the European understanding of “algebra.” Al-Khwārizmī was one of many Islamic mathematicians building on the discoveries of the entire ancient world, from the Chinese scholars to the Greek theoreticians. He particularly connected quadratic equations to geometry and the study of curves, including parabolas, hyperbolas, and circles. As we have seen, this connection is essential to modern scientists.
Islamic mathematicians began to further loosen some of the constraints that the Greek mathematicians had put on numbers and number theory, using more abstract ideas that allowed for the emergence of the modern equation. For example, Abu Kāmil, an Egyptian mathematician who lived over a 1,000 years ago, started treating the quadratic equation as a specific number instead of as a line segment or area as previous mathematicians had.
These ideas spread to Europe as Italian merchants increasingly established trading relationships with Islamic communities along the shores of the Mediterranean Sea.
What do you think? Are you interested in the history of math? What area of math are you most interested in and why? What topic would you like to see covered next? For more about the further advancements of math and algebra, check back for Part II of this blog series.
Don’t forget that summer is swiftly approaching! See if an A Grade Ahead near you is offering Mathventure Jr., Mathventure, or one of our other Enrichment Camps in your area! For more information, check out this blog post.
Author: Susanna Robbins, Teacher and Franchise Assistant at A Grade Ahead
