Happy Numbers Essay

911 words - 4 pages

I chose this topic because it sounded like an interesting topic and it was a part of math I didn’t know about. Happy numbers are a part of recreational maths.
In this report I will be talking about happy numbers, these are numbers that when you square the digits of the number and add the squares together repeatedly, in the end you will end up with one. I will also talk about happy cube numbers, they are the same as happy numbers, but instead of squaring the digits of the original number, the digits are cubed. Furthermore, I will also explore happy prime numbers, these are prime numbers which are also happy. Additionally, I will mention unhappy numbers, these are numbers that do not end in ...view middle of the document...

1 is a special number because once it reach 1 it will remain there, because 1²=1, so it will always remain the same.`
Happy numbers are part of a set of numbers. If the number is seen as N and the digits M , then M²=N₁, M₁²=N₂…….until …..Mx²= 1. N≠0.
If a number is a happy all numbers in the sequence are also happy. So since 19 is happy 82, 68, 100 and 1 are also happy numbers.
Any permutation of the digits of a happy number also leads to a happy number. An example is 13 is a happy number ¹P₁=1, ³P₃=6. 1+6=7, 7 is a happy number.
All numbers that are 10ˣ are also happy numbers, this is because their digits will always be one 1 and one or more 0’s. an example would be 10²=100 and 100 is a happy number.
Furthermore if a happy number N has one more zero added to the end of it, it will still be a happy number this is because zero does not affect the sum of the squares. An example is 13 is a happy number and so is 130.
The first few happy numbers are 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100…..

Unhappy numbers
Unhappy numbers are numbers that after having their digits squared and added together repeatedly the number always leads to 4. This is how you find out if a number is unhappy:
16:
1²+6²= 1+36= 37
3²+7²= 9+49= 58
5²+8²= 25+64= 89
8²+9²= 64+81= 145...

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