Special Numbers
There are numbers that form special mathematical patterns. We call these numbers special numbers. In this chapter, we will go through a few different types of special numbers.
Palindromes
In maths, a palindrome is a number that can be read in the same order in either forward or reverse direction.
Examples
$232$232 is a palindrome because the $2$2 comes first, then the $3$3, then another $2$2, whether we start from the left or the right.
$561$561 is not a palindrome because if we read this from right to left, the number would be $165$165.
Just for fun: words (such as "racecar") and phrases (like "stack cats") are also palindromes.
Triangular Numbers
The picture below shows a number sequence that is made from a pattern of dots which form triangles increasing in size.
| 1st | 2nd | 3rd | 4th | 5th | 6th |
|---|---|---|---|---|---|
 |
 |
 |
 |
 |
 |
| 1 | 3 | 6 | 10 | 15 | 21 |
By adding another row of dots and counting all the dots we can find the next number of the sequence.
So the next number in the sequence (the 7th number) would have $7$7 numbers on the bottom and the total number of dots would be $7+6+5+4+3+2+1=28$7+6+5+4+3+2+1=28.
There is a rule we can use to calculate triangular numbers so we don't have to count from the beginning each time.
It is:
Tn = $\frac{n\times\left(n+1\right)}{2}$n×(n+1)2
Example
Evaluate: What is the tenth triangular number?
Think: If I want to find the 10th triangular number or T10, I can use this rule.
| $T_{10}$T10 | $=$= | $\frac{10\times\left(10+1\right)}{2}$10×(10+1)2 |
| $=$= | $\frac{10\times11}{2}$10×112 | |
| $=$= | $\frac{110}{2}$1102 | |
| $=$= | $55$55 |
Square Numbers
Square numbers are a bit like triangular numbers, except now we are making our pattern with squares instead of triangles!
The picture below shows a number sequence that is made from a pattern of dots which form squares increasing in size.
| $1^2$12 | $2^2$22 | $3^2$32 | $4^2$42 | $5^2$52 |
|---|---|---|---|---|
 |
 |
 |
 |
 |
| 1 | 4 | 9 | 16 | 25 |
Remember, squaring a number means multiplying the number by itself.
$1^2=1\times1$12=1×1 $=$= $1$1
$2^2=2\times2$22=2×2 $=$= $4$4
$3^2=3\times3$32=3×3 $=$= $9$9 and so on
Fibonacci Numbers
Fibonacci numbers are a special sequence of numbers named after Leonardo Fibonacci. This pattern is pretty cool as you can see examples of it in nature, such as branching in a tree, the pattern on the skin of a pineapple or in the centre of a sunflower.
The first two Fibonacci numbers are $0$0 and $1$1. The next term is made by adding the previous two so the next numbers would be:
$0+1=1$0+1=1
$1+1=2$1+1=2
$2+1=3$2+1=3
So the first ten Fibonacci numbers are: $0,1,1,2,3,5,8,13,21$0,1,1,2,3,5,8,13,21 and $34$34.
Completing patterns
You need to be like a detective and try to spot these patterns. So get familiar with each set of special numbers.
Examples
question 1
Evaluate: Write the next three numbers in the pattern: $36,49,64$36,49,64.
Think: This pattern is made of square numbers- $6^2$62 is $36$36, $7^2$72 is $49$49 and $8^2$82 is $64$64. We just need to keep going with this pattern.
Do: $81,100,121$81,100,121
question 2
Evaluate: Write the next three numbers in the pattern: $21,34,55$21,34,55
Think: This pattern is made of Fibonacci numbers: $21+34=55$21+34=55. So keep going with this pattern!
Do: $89,144,233$89,144,233
Question 3
Write down the next square number after $16$16.
Question 4
Write down the next $3$3 triangular numbers.
$1$1, $3$3, $6$6, $10$10, $15$15, $\editable{}$, $\editable{}$, $\editable{}$