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### Multiplying by 1

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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.

Yours truly,
Ehsanullah Abbasi
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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/
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Associated Topics:
Elementary Multiplication