Number Theory

Lesson

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.

In maths, a palindrome is a number that can be read in the same order in either forward or reverse direction.

$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.

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:

T_{n} = $\frac{n\times\left(n+1\right)}{2}$`n`×(`n`+1)2

**Evaluate:** What is the tenth triangular number?

**Think:** If I want to find the 10th triangular number or T_{10}, 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 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 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.

You need to be like a detective and try to spot these patterns. So get familiar with each set of special numbers.

**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

**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

Write down the next square number after $16$16.

Write down the next $3$3 triangular numbers.

$1$1, $3$3, $6$6, $10$10, $15$15, $\editable{}$, $\editable{}$, $\editable{}$