Multiplying by 1

Date: 09/01/98 at 13:04:40
From: Rizwan Abbasi
Subject: Multiplying by 1

Dear Dr. Math,

My question is as follows:

Multiply means increase in number. When 1 is multiplied by 1, the 
answer is 1. The answer is 1. Why is it? Each one independent unit is 
being multiplied but the number is not increased. Looks erratic to me. 
Please define.

Yours truly,
Ehsanullah Abbasi

Date: 09/01/98 at 17:35:56
From: Doctor Rick
Subject: Re: Multiplying by 1

Hello, Rizwan. This is an interesting question, and I can make it seem 
even stranger. Not only can you multiply by 1 and the result does not 
increase, but you can also multiply by 1/2 and the result is smaller.

If you look at the original meanings of words, the same problem arises 
with the word "add". It comes from the Latin "addere" meaning "to give 
to." Yet I can add a negative number, with the result that something 
is actually taken away.

I think the same sorts of problems will arise in any language, and in 
other disciplines besides math. A word that means one thing in everyday 
language will have a somewhat different meaning, or a very specific and 
specialized meaning, in math or physics or economics or another 
specialized field of study. When people have a new idea or invent a new 
product, sometimes they invent an entirely new word to identify it. But 
sometimes they just use an existing word that has a similar meaning. 
For instance, an electrical current is like a current in a river, but 
it is not exactly the same.

The basic words of math like "multiply" and "add" were adapted from 
everyday life long ago. Back then, concepts like negative numbers and 
even zero had not been developed. People would really only think in 
terms of multiplying by positive whole numbers. And why bother to 
multiply by 1? It doesn't do anything. So the use of the words made 
sense. 

But mathematicians gradually extended the meanings of the words. Not 
only can you multiply fractions or negative numbers, you can multiply 
matrices or numbers in modular arithmetic where the idea of one number 
being greater than another is meaningless.

The things that we call "multiplication" today have a lot in common 
with simple multiplication by an integer greater than 1, so it makes 
sense to use the same word for them. Why invent a new word just because 
the original narrow meaning of the word doesn't fit any more?

In short, the problem that you have raised is a reason for the 
existence of specialized dictionaries of science and technology. If 
you look up the meaning of a word in a dictionary of everyday language 
and try to apply the definition to the way the word is used in a 
specialized field like math, you will often only get confused. Just 
use a word the way it is defined in the field you are working in, and 
don't worry about what it means in everyday life.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/