Explaining Addition

Date: 01/25/97 at 12:31:34
From: Tim Wu
Subject: A Question

Dear Dr. Math:

I have a question for you.  What is the answer to 6 + 2?  Can you show 
me how to come up with your answer?  Thank you very much.

Best regards,
Jennifer Woo

Date: 01/25/97 at 15:35:36
From: Doctor Ceeks
Subject: Re: A Question

Hi,

A number is something which tells you how many things there are.
For instance, look at the next line:

X X X X

There are some X's there.  How many do you see?  If you want to be 
able to express the answer to this, you have to learn about numbers.

When you see this:

X

People say there is "one" X.  They also write "1" for "one". 

People say "two X's" when they see: X X.  They also write "2" for 
"two".

People say "three X's" when they see: X X X.  They also write "3" for
"three".

People say "four X's" when they see: X X X X, and sometimes write "4"
instead of "four".

People say "five X's" when they see: X X X X X, and sometimes write 
"5" instead of "five".

People say "six X's" when they see: X X X X X X, and sometimes write 
"6" instead of "six".

People say "seven X's" when they see: X X X X X X X, and sometimes 
write "7" instead of "seven".

People say "eight X's" when they see: X X X X X X X X, and sometimes 
write "8" instead of "eight".

So, how many A's do you see below?

A A A A

The answer is: four A's.  When I get lazy or have to write quickly,
I'll write "4 A's" instead of "four A's".

How many A's are there below?

A
 A
  A
   A

The answer is: 4 A's.

How many C's are there below?

C C C
C C C

The answer is: 6 C's.  (There are two lines of 3 C's each.)

How many letters do you see on the next line?

ajbkKl

The answer is: 6 letters.

I've been using various letters to explain numbers, but you don't
have to count just letters.  You can count just about anything!  You 
can count people, cows, bottles, stars...

Now, 6 + 2 is the number which answers: How many X's do you see if you 
put 6 X's together with 2 X's?  Another way of saying it is: how many 
X's do you see if you add 2 X's to 6 X's?

X X X X X X   <== six X's over here.
  X  X        <== two X's over here.

How many X's do you see total?

The answer is 8 X's.

Therefore, 6 + 2 = 8.

Here's another example.  What is 2 + 3?

i j      <==== two letters here
a m o    <==== three letters here

...makes a total of 5 letters, so 2 + 3 = 5.

Look what happens if you put the line with "a m o" before the line 
with "i j" like this:

a m o    <==== three letters here
i j      <==== two letters here

See how you haven't changed the total number of letters?  So the total 
number of letters is still going to be 5.  But this is what you'd do 
if you tried to figure out 3 + 2 instead of 2 + 3. In other words,
2 + 3 = 3 + 2.  In fact, 2 + 6 = 6 + 2, and 1 + 4 = 4 + 1.  Do you see 
what I'm trying to say?  The order you add two numbers doesn't matter.

Now tell me, what is 8 + 1?  You know it's the number of X's you see
below:

X X X X X X X X   <== eight X's here
   X              <== one X here.

The number of X's you see above is equal to 8 + 1 (or 1 + 8).  That's
the answer, but if you want to say it, you have to say "nine", because
that's the word used to communicate how many X's there are there.  Did 
you already know that?  It's wonderful to know that people have worked 
out a way of expressing any number of things.

Starting with 1 X, if you keep adding X's one by one, you will see
respectively one, two, three, four, five, six, seven, eight, nine,
ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen,
eighteen, nineteen, twenty, twenty one, twenty two, twenty three, ...

Perhaps you can ask your mom or dad to explain how to go further.

When people write "1" instead of "one" or "2" instead of "two", they
are writing "numerals" down for the numbers.  Translating the above
sequence of numbers from "one" to "twenty three" into numerals looks 
like this: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 
18, 19, 20, 21, 22, 23, ...

It takes a lot less space, but it is another thing you have to learn
how to do.

Write back with any other questions!

-Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/