To think about a mathematical
problems that arise from curiosity about numbers.
The usual counting numbers
1, 2, 3, 4…
Are called the natural numbers (or positive integers)
Each natural number either
even or odd, the even natural numbers,
2, 4, 6, 8.
Are those that can be divided
by 2 exactly; that is, an even number can be divided evenly’ into two parts
8÷2=4
The odd natural numbers
1, 3, 5, 7……
Are those that cannot be
divided by 2 exactly; that is when an odd number is divided by 2, there is 1
left over
7÷2=3 remainder 1
It is not just the positive
integers that are either even or odd.
For example 0, -2 and -4 are
even numbers, and -1 and -3 are odd numbers
The square numbers
1, 4, 9, 16….
Are obtained by multiplying each
natural number by itself.
1=1×1
4=2×2
9=3×3
16=4×4
The square numbers can be
represented as patterns of dots arranged as squares.
You will see shortly that
this can be helpful in discovering properties of numbers
Activity 28 Types
of numbers.
(A)
The
sixth natural number
Answer: 1, 2, 3, 4, 5, 6 the answer is 6
(B)
The
sixth even number
Answer: 2, 4, 6, 8, 10,12 the answer is 12
(C)
The
sixth odd number
Answer: 1,3,5,7,9,11 the answer is 11
(D) The sixth square number
Answer: 1×1=1, 2×2=4, 3×3=9, 4×4=12, 5×5=25, 6×6=36 the answer
is 36
Activity 29
|
(A)
How
many odd numbers
|
Sum
|
|
1
|
1=1
|
|
2
|
1+3=4
|
|
3
|
1+3+5=9
|
|
4
|
1+3+5+7=16
|
|
5
|
1+3+5+7+9=25
|
|
6
|
1+3+5+7+9+11=36
|
(B)
All
sums are square numbers.
The sums look familiar – they
are all square numbers. Each sum is the square of the number of odd numbers
added. Odd numbers starting from 1 always results in the square of the number
of odd numbers that are added. This statement is a conjecture – an informed guess about what might be true, from
considering a few cases.
No matter how many odd
numbers are added.
If we use the letter n to
represent any natural number, then the conjecture can be expressed in the
following neat way
Conjecture
If you add up the
first n odd numbers, then the sum is always n²
When you add the first seven
odd numbers, the answer should be 7², which is 49.
But no amount of checking of
individual cases can prove that it is true for all natural numbers n. However,
it turns out that we can prove this by considering patterns of dots.
You can make larger and
larger squares of dots by adding larger and larger L shaped patterns of dots.
Where n is a natural number, then the results is a square of n² dots.
Because you can do this for
any natural number n, you can see that the conjecture is true.
A mathematical statement that
has been approved is called theorem
or a result
Result
If you add up the
first n odd numbers, then the sum is always n².
Activity 30
Use the results
above to find the sum of the first 100 odd numbers:
100²= 100×100=10000
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