The document discusses the concept of division and rational numbers, explaining how division involves finding quotients of integers and how it leads to the introduction of rational numbers, which can be represented as fractions. It provides examples of division and describes the relationship between division and multiplication through examples and definitions. Additionally, it touches on the historical context regarding rational and irrational numbers, referencing Pythagoras and Hippasus.

2. 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

3. Division is splitting into
equal parts or groups.
It is the result of "fair
sharing".

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

5. We use the ÷ symbol,
or sometimes
the / symbol to
mean divide:
12 ÷ 3 = 4
12 / 3 = 4
÷/

6. 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.

7. 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

8. Searching around
the multiplication
table we find that
28 is 4 × 7, so 28
divided by 7 must
be 4.?
Answer: 28 ÷ 7 = 4

9. 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

10. Example: There are 7 bones But 7 cannot be divided exactly
into 2 groups,
to share with 2 pups.
so each pup gets 3 bones,
but there will be 1 left over:
Hahahaha...
:D

11. 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)

12. 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 !

14. 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

15. Using Rational Numbers
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
No! "q"
can't be
zero!
7
0
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:
Add, Subtract, Multiply and Divide
Rational Numbers

16. 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!