Cube Root Notation

April 9, 2019

Friday May 3 sprint.

Today's sprint had two radical simplification problems on it. Most everyone did the arithmetic correctly. A few of you, however, wrote the wrong answer.

For example:

The answer shown, `5\sqrt(2)^(1/3)`, is marked as wrong. The incorrect answer is the same as writing `((2)^(1/2))^(1/3)` or `(2)^(1/6)`

There are two correct ways of writing the answer. The first is shown underneath the mistake as `5(2)^(1/3)`.

The second way of correctly answering the question is to write `5\root{3}2`. Notice the small 3 in the radical pocket. Without the small 3, it would read `5\sqrt2` which would be wrong. The small 3 is vital to communicate cube roots.

Music Theory and roots

When the modern piano was first being built in the early 1700's, composers such as J.S. Bach were looking for a way to arrange the notes so that they could build multiple harmonies. Two notes harmonize if their frequencies share a common factor. For example the key A above middle C is usually tuned to 440 or 432 cycles/second. If we tune middle A above C to 440 then, twelve notes up the keyboard, or one octave higher, the A note is tuned to 880 cycles/second or twice 440. Another octave higher and the next A is tuned to twice 880 or 1760 cycles/second. Go down to A below middle C and the frequency is 220 or 1/2 of 440. Go down another octave and A is tuned to 110.

There are 12 notes between 440 and 880. To make each note have the same doubling relationship between the octaves, they had to be evenly spaced. That is, every note on the keyboard is tuned to twice the frequency of same note one octave lower.

The problem was how do you arrange 12 numbers so that each number is twice its lower octave neighbor and do that with 12 keys? The answer was to multiply each note's frequency by `root[12]2`

When you mulitply any number by `root[12]2` twelve times, you get a number that's twice as large.