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