division and rational number

Division and rational number. Any two integers have

such that xq = y. The process for finding q when three

such an integer is called division, and q is called the

quotient. Since division is not always possible in the

set of integers, it is advantageous to create a larger

set of number which division is possible in more

casess. The number which are introduced for this

prpose are the fractions, and the union of the set of

fractions and the set of integers is the set of rational

number

Division is splitting into

equal parts or groups.

It is the result of "fair

sharing".

12 Chocolates 12 Chocolates Divided by 3

Answer: 12 divided by 3 is 4: they get 4 each.

We use the ÷ symbol,

or sometimes

the / symbol to

mean divide:

12 ÷ 3 = 4

12 / 3 = 4

÷ /

Division is the opposite of multiplying. When we know a multiplication fact we can find a division fact:

Example: 3 × 5 =

15, so 15 / 5 = 3.

Also 15 / 3 = 5.

Multiplication... ...Division

3 groups of 5 make 15... so 15 divided by 3 is 5

and also:

5 groups of 3 make 15... so 15 divided by 5 is 3.

So there are four related facts:

3 × 5 = 15

5 × 3 = 15

1a5 / 3 = 5

15 / 5 = 3

Answer: 28 ÷ 7 = 4

Searching around

the multiplication

table we find that

28 is 4 × 7, so 28

divided by 7 must

be 4.?

There are special

names for each

number in a

division:

dividend ÷ divisor

= quotient

Example: in 12 ÷ 3 = 4:

12 is the dividend

3 is the divisor

4 is the quotient

Example: There are 7 bones

to share with 2 pups.

But 7 cannot be divided exactly

into 2 groups,

so each pup gets 3 bones,

but there will be 1 left over:

Hahahaha...

:D

A Rational Number is a real number that can be written

as a simple fraction (i.e. as

a ratio).

Example:

1.5 is a rational number because 1.5 = 3/2 (it can be written as a fraction)

Number As a Fraction Rational?

5 5/1 Yes

1.75 7/4 Yes

.001 1/1000 Yes

0.111... 1/9 Yes

√2(square root of 2) ? NO !

More formally we would say:

A rational number is a

number that can be in the

form p/q

where p and q are integers a

nd q is not equal to zero.

So, a rational number can be:

p

q

P q p / q =

1 1 1/1 1

1 2 ½ 0.5

55 100 55/100 0.55

1 1000 1/1000 0.001

253 10 253/10 25.3

7 0 7/0No! "q"

can't be zero!

If a rational number is still in the

form "p/q" it can be a little

difficult to use, so I have a

special page on how to:

Using Rational Numbers

Add, Subtract, Multiply and Divide

Rational Numbers

The ancient greek mathematician Pythagoras believed that all numbers were rational (could be written as a fraction), but

one of his students Hippasus proved (using geometry, it is

thought) that you could not represent the square root of 2 as

a fraction, and so it was irrational.

However Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers had

perfect values. But he could not disprove Hippasus' "irrational

numbers" and so Hippasus was thrown overboard and

drowned!