Most people probably learned in school the miracle of the nine times table. No matter how far up you go (within reason), if you add the digits of the answer, they’ll always equal nine. Did you ever wonder why?

2 X 9 = 18, 1 + 8 = 9

5 X 9 = 45, 4 + 5 = 9

8 X 9 = 72, 7 + 2 = 9

12 X 9 = 108, 1 + 0 + 8 = 9

25 X 9 = 225, 2 + 2 + 5 = 9

And on, and on, at least up into the 400s. There are a few holes if you want to look for them, but it’s pretty impressive magic, nonetheless. How can this possibly be?

I was wide awake at 3 o’clock one morning, and having nothing better to do, I figured it out. It’s because…

The Four Basic Operations Aren’t.

Stay with me, here. We were always taught that there are four basic operations: add, subtract, multiply, and divide. Right?

Wrong. There are only two basic operations: adding and subtracting. The other two are mathematical concepts. They are shorthand ways to add and subtract.

So 9 X 3 is just a quick way of doing 9+9+9.

So look at mathematical operations as a series of levels. Add and subtract are on the bottom. Multiply and divide are quick ways of adding and subtracting. Exponents are a way of simplifying multiplication.

10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 = 10 X 3 = 10^{3}

But back to our original problem. Don’t look at it as if it is the nine times table. Look at it as a series of additions. 9+9+9 etc. Now look at 9 as if it’s 10-1, and 10 is the point where you move over into the next column.

So when you’re adding 9+9, think of it as adding 9 to 10 – 1 instead. 9 + 10 – 1 = 18. Drawn out, it looks like this:

Tens | Ones | Adds up to | |

0 | 9 | 9 | |

+ | 1 | -1 | 9 |

1 | 8 | 9 |

Showing that: 9 + 9 = 18. (Arithmetic)

And that: 1 + 8 = 9 (interesting observation that has nothing to do with arithmetic).

You add down the Tens column and you add down the ones column, but you don’t add the “Adds up to” column, because that isn’t part of the arithmetic. It’s just a description.

Larger Numbers

Every time you add a 10, it goes in the tens column. You already have 9 in the ones column, but the number you’re adding is one less, so you have to take 1 away. Do you begin to get the picture? Every time you add a 1 to the tens column, you take a 1 away from the ones column. Add 1, subtract 1, you haven’t changed anything. You always have 9.

Tens | Ones | Adds up to | |

0 | 9 | 9 | |

+ | 1 | -1 | 9 |

1 | -1 | 9 | |

1 | -1 | 9 | |

3 | 6 | 9 |

So this is a pictorial way of looking at 9 X 4 = 36. The one below is a shorter version. It has exactly the same amount of mathematical meaning. As in “absolutely none.”

Tens | Ones | Adds up to | |

0 | 9 | 9 | |

+ | 3 | – 3 | 30 – 3 = 27, 2 + 7 = 9 |

1 | 8 | 9 |

Don’t worry.

This doesn’t mean that mathematics is meaningless. When we add the ones column to the tens column we’re not doing mathematics. It just imaginary arithmetic, like when a politician quotes statistics. We’re not performing any function that comes out with a useful number. When we take 18 and add the 1 to the 8 and get 9, that 9 doesn’t really mean anything.

Because in arithmetic, addition can only work with equals. You can’t add apples to pears and get a functional answer. 1 apple plus 1 pear isn’t 2 of anything. It’s fruit salad. Unless you change the name to fruits. Under the heading “fruits,” apples and pears are equal, so 1 fruit plus 1 fruit makes 2 fruits, and we’re doing arithmetic again.

So, since a number in the tens column is ten times bigger than a number in the ones column, we can’t add them together and get anything more meaningful than salad. Unless for some reason the answer is interesting, as it is in this case. Or unless it proves something that we want to sell to people, but have no data to back up our sales pitch.

Taking it Farther

So, if we think of 8 as being 10 – 2, what would you expect to observe when you add the digits of the answers to the 8 times table? Interesting, but not so magical as the nines.

Which is why mathematics is so much fun.

And now I can go back to bed and get some sleep.