History Continued

In today's world, almost every industrial culture uses the numeral 0 through 9 but these symbols weren't invented until the 3rd century BC in India and it took another 800 years for the idea of zero with place value such as tens, hundredths and thousandths to be constructed. This big idea dramatically changed the face of mathematics.

We humans have always shared with one another. When early cultures shared their food and water or wanted to divide their land in ways that were fair and equal, fractions gradually emerged as symbols for those fair share situations.

The ancient Egyptians used unit fractions, fractions where the numerator is 1 like one-half \((\frac{1}{2})\), one-third \((\frac{1}{3})\) and one-fifth \((\frac{1}{5})\) and would add and half these fractions. If they wanted to divide three loaves of bread equally among five family members, they would first divide the first and second loaves into thirds, then they'd divide the third loaf into fifths. Finally, they'd take the remaining one-third from the second loaf and divide that into five pieces. They wrote this as one-third \((\frac{1}{3})\), one-fifth \((\frac{1}{5})\) and one-fifteenth \((\frac{1}{15})\).

Today, we would represent this sharing with a fraction, three-fifths \(\frac{3}{5}\), three-fifth of a loaf for each person or three loaves divided by five people.

The Sumerians and the early Babylonians invented a number system of fractions based on 60 that we still use four thousand years later. Our days have 60-minute hours and 60-second minutes, and our circles encompass 360 degrees. Even, earlier clocks like the sundial (see image below) used this same concept before the advent of more modern clocks.

A sundial is an instrument showing the time by the shadow of a pointer cast by the sun onto a plate marked with the hours of the day. Image credit: public domain

Chinese societies used an abacus (see image below) with a system based on tens although it had no zero. An early form of decimal fractions came from the abacus. For example, three-fifths would be 6 out of 10 on an abacus. The Chinese lovingly named the numerator, the son and the denominator, the mother. It wasn't until the 12th century that common fractions with the bar notation that we use today were invented. Even then, these fractions weren't widely used until the Renaissance period only 500 years ago.

An abacus

Throughout history, every culture around the globe has created inventive ways to calculate. To solve a problem, say \(12 × 15\), early Russian peasants used a system of doubling and halving i.e. \(12 × 15\) \(= 6 × 30\) \(= 3 × 60\) \(= 1\frac{1}{2} × 120\). When an odd number had resulted in a fraction, they rounded down i.e. \(1\frac{1}{2} × 120\) \(= 1 × 120\), then they added the factors associated with the odd multipliers, in this case, 60, resulting in \(60 + 120 = 180\).

To solve a problem, say \(12 × 15\), early Russian peasants used a system of doubling and halving.

Early Egyptians relied on a doubling procedure until they produced enough groups, i.e. \(12 × 15 ⇒\) \(1 × 15 = 15;\) \(2 × 15 = 30;\) \(4 × 15 = 60;\) \(8 × 15 = 120.\) Then, they added these groups to find the answer. Therefore, since \(4 + 8 = 12,\) \(12 × 15\) \(= 60 + 120 = 180.\)

Across Europe and Asia, during the Middle Ages, the abacus was the handheld calculator at its day but only few people knew how to use it, usually wealthy merchants and moneylenders. By simply moving beads that each have a place value (the numerical value that a digit has by virtue of its position in a number), an abacus was a highly efficient way to compute.

Hindu-Arabic numerals, set of 10 symbols—1, 2, 3, 4, 5, 6, 7, 8, 9, 0—that represent numbers in the decimal number system. They originated in India in the 6th or 7th century and were introduced to Europe through the writings of Middle Eastern mathematicians, especially al-Khwarizmi and al-Kindi, about the 12th century. Image credit: https://www.britannica.com/topic/Hindu-Arabic-numerals

After a while, the great Arab mathematician, Al-Khwarizmi introduced the Hindu-Arabic numerals 0 through 9 (which are still in use today) into North America and Europe, and created new procedures for computation. These procedures or algorithms could now be written on paper and over the centuries, learning them became the hallmark of education, as students were taught to compute long columns of figures, borrow and carry, as well as do long division efficiently and reliably. They could now keep records of these procedures and check results.

Today, complex calculations are done with a hand-held calculator. The first electronic calculator was the Casio 14a, still so large that it required fitting into a desk. As technology advanced, the improved computing power of the time enabled calculators to add more strings to their bow. Now, we have the scientific calculator which can work with scientific notation, logarithmic and trigonometric calculations, exponential functions, roots and powers, and other long complicated mathematical terms that many people have never even heard of.

Casio 14a, the first electronic calculator. Image credit: http://www.casio-calculator.com

As we journey through the rich and vibrant history of mathematics, we can see how ideas and creations grew out of our very human need to solve the problems in our everyday lives. Through time, the mathematical explorations of men and women from around the globe have given us fascinating lenses that help us to mathematically view and make sense of our world.