How does one “invent new maths”? Like Isaac Newton inventing Calculus, or John Napier logs. How does one answer a mathematical question that’s never been answered?

Question

How does one “invent new maths”? Like Isaac Newton inventing Calculus, or John Napier logs. How does one answer a mathematical question that’s never been answered?

In: Mathematics

Answer 1

Here’s one way you might discover logarithms on your own. (From now on, log = log base 10.)

Suppose you’re back in the olden days before calculators and computers and your job sometimes involves multiplying numbers with a lot of digits. One day your boss asks you to find the area of a rectangle with dimensions 15432.54 by 827361.37. The area is 15432.54 * 827361.37. Calculators haven’t been invented yet, and doing this with pencil and paper would take ages. Is there a way to simplify this?

You could make a multiplication table, like the one you memorized in grade school. Then you could just look up the product in the table! Except that such a multiplication table would be HUGE if you wanted one which would be actually useful. Is there a way to shrink the amount of information required to make this reasonable?

Then you remember a neat fact about exponents: b^(x+y) = b^x b^y. In a way, exponents turn addition (an easy operation) into multiplication (a very hard operation). What if we could do the opposite? What if we could turn multiplication into addition?

Enter logarithms, which are just reverse exponents. These have the neat property that log(xy) = log(x) + log(y). i.e. they turn multiplication into addition.

So let’s use this to find 15432.54 * 827361.37.

First, we will find log(15432.54 * 827361.37). Then at the end we can take 10 to the power of whatever the answer is to get the final result.

In scientific notation, this becomes log(1.543254 * 10^(4) * 8.2736137 * 10^(5)).

Using what we found earlier, we know that this is equal to log(1.543254) + log(10^5) + log(8.2736137) + log(10^5).

This is log base 10, so we have now log(1.543254) + 4 + log(8.2736137) + 5.

We can look up those logarithms in a logarithm look-up table, which is what people used back in the days before computers. By using logs and scientific notation, we only need to know the values of logs for numbers between 1 and 10. Much more feasible than a giant multiplication table.

If you calculate this, you would find that log(15432.54 * 827361.37) = 10.10613. Therefore, 15432.54 * 827361.37 = 10^(10.10613265) = (10^(0.10613265)) * 10^(10) which you can solve using a different lookup table.