I realized recently that I have three stories that I tell my students about operations, but I've never shared these stories broadly. None of these stories are true in a strict sense, but I think they help to convey important mathematical ideas. This is the first story.
Math seems like a set of ideas bestowed upon us from Mount Euclid, but in fact it has been slowly developed over many hundreds of years by people.
At first, people invented math systems so they could count sheep, loaves of bread, and other tradable goods. Babylonians used a system based on 60, but most societies eventually settled on a system based on 10.
But as people counted and counted and counted, they realized they could make a shortcut. Instead of counting one group of sheep and then another group, why not just add the two groups together? And so addition was born. Of course, shortly thereafter people realized they needed an operation that undid addition, and so subtraction was developed as a related operation. These two operations are known as inverses, which means they have opposite effects. For a long time, addition and subtraction were the only operations.
But after a while, people got sick of adding the same number over and over again. Eight people have how many fingers? Well 10 + 10 is 20 and then 20 + 10 is 30 and then 30 + 10 is 40...
So ancient mathematicians invented a beautiful shortcut for adding the same number repeatedly. They called it multiplication. And to undo multiplication, people invented division. Two more inverse operations.
Whenever people had a big long math problem, they would multiply and divide anywhere they could before moving to addition and subtraction. Multiplication and division were shortcuts, of course, so it made perfect sense to do them first.
And that's how it was for hundreds of years. People would multiply and divide, and then add and subtract. But over time, people got sick of repeatedly multiplying the same number over and over again.
So they invented yet another shortcut, known as the exponent. It's a quicker way of doing repeated multiplication, which made things much faster. And at the same time, people came up with its inverse, which is finding the root of a number.
At this point, people felt pretty good with their operations. You do exponents and roots first, then multiply and divide, and then add and subtract. First the shortest shortcut, then the next one, and finish up with old-fashioned adding and subtracting.
But what if you want to add first? How do you communicate that to other people? You need some sort of symbol that indicates a smaller, special group of numbers and operations. So people invented parentheses, brackets, fraction bars, and all sorts of other grouping symbols.
So you do the math inside grouping symbols first, then do exponents and roots, then multiply and divide, and finally add and subtract.
And that's how the order of operations was invented. Out of order.