This is Math Newsletter number 2; Wednesday, August 4, 2010.
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Extending the number system

The number 1 is the foundation number of all arithmetic.

In arithmetic and algebra, 1 is important as the unit of
counting. In geometry, 1 is important as the unit distance.
"1" is the distance between two arbitrarily chosen points on
a line. These two point are labeled "0" and "1". Note that
the "1" refers both to a point on the line, and also to the
distance of that point from the "0" point.


When we count pennies, pebbles, socks, or whatever,
we are repetitively adding 1.

Two socks: 1 sock plus 1 sock.
Three pennies: 1 penny plus 1 penny plus 1 penny.

All positive integers are reached by repetitively adding 1
sufficiently often.

Now we have in our repertoire all the positive integers,
1,2,3,4,5,6,7,8,9,10,11,12,...

If we count nickels instead of pennies, then we would want to
count by 5's instead of by 1's.

If we wish to know the value of the coins in our possession,
then we would need to sometimes add 1, sometimes add 5,
sometimes add 10, sometimes 25, and sometimes 50.

Now we have generalized counting to addition.

For any two positive integers we now know what it means to
add them together.

We invent the word "plus" for the process of addition,
and invent the symbol "+" for indicating addition.

75 + 20 is equal to 95.

The existence of the addition process soon invites the need
for the inverse of addition. What did I add to 75 to get 95?

This inverse process to addition, we call subtraction.

Now that we have the subtraction process, we use it to create
more numbers.

We invent the word "minus" for the process of subtraction,
and invent the symbol "-" for indicating subtraction.

1 - 1 is a new number that we call 0.

0 - 1 is a new number that we call -1.

-1 - 1 is a new number that we call -2.

etc.

Now we have all the negative integers,
and need to elaborate the rules for adding negative
integers to negative integers,
and for adding negative integers to positive integers.

The symbol "=" is used to mean "is equal to".

1 + (-1) = 1 + (0 - 1) = (1 + 0) - 1 = 1 - 1 = 0

-1 + -1 = (0 - 1) + (0 - 1) = 0 + 0 - 1 - 1 = 0 - 2 = -2

Another extension of addition is multiplication.

Suppse we have 19 nickels, and we want to know the value of
these 19 nickels. Each nickel is worth 5 pennies. How many
pennies is worth 19 nickels?

We wish to calculate: 5+5+5+5+5+5+5+5+5+5+5+5+5+5+5+5+5+5+5.

We simplify the expression of repetitively adding the same
number, and invent the process of multiplication.

5+5+5+5+5+5+5+5+5+5+5+5+5+5+5+5+5+5+5 is abbreviated 5 * 19.

Now we can define multiplication of any integer by another
positive integer as repetitively addition.

4+4+4 is represented by 4 * 3

It does not matter whether the integer being repetitively
added is positive or negative.

(-5)+(-5)+(-5)+(-5) is represented by (-5)*4

Now that we have defined the multiplication by positive
integers,it is natural to ask for the meaning of
multiplication by negative integers.

What would 5 * (-4) mean?

It would mean 5 added repetitively (-4) times.
Since the inverse of addition is subtraction,
this means 5 subtracted repetitively 4 times.

5 * (-4) means 0-5-5-5-5 = 0 -(5+5+5+5) = 0 - 5*4
= -(5*4)

and
(-5)*(-4)means 0-(-5)-(-5)-(-5)-(-5) = 0+5+5+5+5 = 5*4


Also, now we can ask, "what does it mean to multiply by 0?"


5 * 0 means to repetitively add 5 zero times.
5 * 0 = 0 because no 5's are added.

0 * 5 = 0+(0+0+0+0+0) = 0

0 times anything is 0.
anything times 0 is 0.

Next, we define the inverse of multiplication.
What did I multiply 9 by to get 36.

This inverse of multiplication is given the name "division".

36/9 = 4.

Now that we have the process of division, we create more new
numbers.

1/2 is the number we multiply 2 by to get 1.
1/3 is the number we multiply 3 by to get 1.
2/3 is the number we multiply 3 by to get 2.

-4/5 is the number we multiply 5 by to get -4.

22/7 is the number we multiply 7 by to get 22.

19/7 is the number we multiply 7 by to get 19.

For any two integers non-zero integers, we can create their
ratio, one of the non-zero integers divided by the other.

The new ratio numbers, we call the "rational numbers".

Note that all integers are also rational numbers.

7 = 7/1.
-5 = (-5)/1

0/7 is the number we multiply 7 by to get 0.

7 * 0 = 0.

0/7 = 0.

What about division by 0?

Is it possible to define a number 0/0?

If it were, we would say
0/0 is the number we multiply 0 by to get 0.

This means that 0/0 is every number.
It is not any particular number. It is every number.
This is why it is not possible to define a number
0/0.

What about 1/0?

1/0 is the number that we multiply 0 by to get 1.

We have found that 0 times any number is 0.

There is not any number that we can multiply 0 by to get 1.

1/0 is not a number.


This is why the standard arithmetic textbooks have stated the
rule that division by 0 is not permitted.


We elaborate the rules for division of and by negative
numbers.

(-5)/4 = -(5/4)
5/(-4) = -(5/4)
(-5)/(-4) = 5/4

Now that we have the multiplication process, consider the
special case of multiplying one integer by itself.

0 * 0 = 0
1 * 1 = 1
(-1) * (-1) = 1
2 * 2 = 4
(-2) * (-2) = 4

We call these integers that are an integer multiplied by
itself, square integers. You can guess why we call them
square integers.

Squaring an integer is a special case of the multiplication
process.

Square of 5 is 25.
square of 6 is 36.
etc

Now that we have the square process, we ask, "what is the
inverse of the square operation?"



What do we square to get 49?

The inverse process we call square root.

Since the square of 7 is 49,
7 is the square root of 49.
we write 7 = sqrt(49).

Even though the square of (-7) is also 49,
since we want sqrt(49) to be one number, not two numbers,
we make the convention that sqrt(49) is 7, not (-7).

sqrt(any positive number) is positive.

We apply the square root process to all our existing numbers,
the rational numbers, to make yet more numbers.

sqrt(1) = 1
sqrt(2) is a number whose square is 2.
It is between 1 and 2.

sqrt(3) is a number whose square is 3.
It is between sqrt(2) and 2.

sqrt(4) = 2.

sqrt(-1) is a number whose square is -1.

sqrt(-1) is not a negative number, because negative times
negative is positive.

sqrt(-1) is not a positive number, because positive times
positive is positive.

The sqrt(-1) is called the "imaginary unit."

It is given the name "i".

We create the "complex numbers" as follows:

The rational pure imaginary numbers consist of
rational numbers multiplied by i.

Examples of pure imaginary numbers are
-5i, 4i/5, 10i, ...

Now we need to distinguish the rational numbers
previously defined from the pure imaginary rational numbers.

We say real rational number for 1/1, 1/2, 2/1, -1/2, -2/1,
19/7, 65/128, etc.

Now we define a complex rational number as the sum of a real
rational number and a pure imaginary rational number.

1 + i, 1 + 2 i, 2 + i, 3 + 4 i, 3/5 + 4i/5, ...


We also may use the sqrt numbers as complex number
components.

sqrt(2)/2 + i sqrt(2)/2, 1/2 + i sqrt(3) /2, ...


Have fun playing with all these numbers.

In next newsletter, I'll describe even more number
extensions.

Kermit Rose