Types of Numbers
Figurate numbers
Some numbers can be represented in particular configurations of dots. These numbers represent a link between geometry and arithmetic.
Square Numbers
The first four square numbers are
Here is the pattern which generates square numbers
1=1
1 + 3 = 2 x 2 = 4
1 + 3 + 5 = 3 x 3 = 9
1 + 3 + 5 + 7 = 4 x 4 = 16
1 + 3 + 5 + 7 + 9 = 5 x 5 = 25
The sum of any number of consecutive odd numbers starting with 1 is a perfect square
What is the 7th and 10th square numbers?
Every square number ends in a 0, 1, 4, 5, 6 or 9
Every square number is divisible by 3 or is when it is decreased by 1
Every square number is divisible by 4 or is when it is decreased by 1
Every square number is divisible by 5 or is when it is increased or decreased by 1
Triangular Numbers
The first four triangular numbers are
Here is the pattern which generates triangular numbers
1=1
1 + 2 = 3
1 + 2 + 3 = 6
1 + 2 + 3 + 4 = 10
1 + 2 + 3 + 4 + 5 = 15
What is the 10th triangular number?
There is a link between Square and triangular numbers
1st triangular number + 2nd triangular number = 2nd Square number 1 + 3 = 4
2nd triangular number + 3rd triangular number = 3rd Square number 3 + 6 = 9
3rd triangular number + 4th triangular number = 4th Square number 6 + 10 = 16
Determine
a) the sum of the 4th and 5th triangular numbers
b) the sum of the 9th and 10th triangular numbers
c) a) the sum of the 49th and 50th triangular numbers
Any square number is the sum of two successive triangular numbers.
Fibonacci Numbers
1, 1, 2, 3, 5, ...........
Describe the pattern
Write down the first 10 Fibonacci numbers
Pick any Fibonacci number other than 1. Square that number. Find the product of the two numbers either side of the chosen Fibonnaci number. Repeat this a number of times.
What do you notice?
Palindromic Numbers
A palindrome reads the sum way when it is reversed
A palindromic number is a number which is not affected by reversing the digits
eg 313 and 15351
This procedure will generate a palindromic number
Choose a 2-digit number 48
Reverse the digits 84
Add them 132
Reverse the digits 231
Add them 363 This is a palindromic number
Sometimes to generate a palindromic number by this method my take a few more steps
Try this process with a 3 digit number.
Prime Numbers
A prime number's only factors are 1 and itself. It is not divisible by any other number.
Every number can be described by a unique product of prime numbers (prime fctorisation)
Goldbach Conjecture - Every even integer other than 2 is the sum of two prime numbers.
As early as about 300 BC Euclid provided the first proof that the number of prime numbers are infinite.
Every number can be described by a unique product of prime numbers (prime fctorisation)
Goldbach Conjecture - Every even integer other than 2 is the sum of two prime numbers.
As early as about 300 BC Euclid provided the first proof that the number of prime numbers are infinite.
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