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Mathematics for Computer Scientists | Integers


Defendit numerus: There is safety in numbers

We begin by talking about numbers. This may seen rather elementary but is does
set the scene and introduce a lot of notation. In addition much of what follows is
important in computing.


I will assume you are familiar with the integers.
1,2,3,4,. . .,101,102, . . . , n, . . . , 2^32582657 − 1, . . .
sometime called the whole numbers. These are just the numbers we use for count-
ing. To these integers we add the zero, 0, defined as:
0 + any integer n = 0 + n = n + 0 = n
Once we have the integers and zero mathematicians create negative integers by
defining (−n) as:
the number which when added to n gives zero, so n + (−n) = (−n) + n = 0.
Eventually we get fed up with writing n+(−n) = 0 and write this as n−n = 0.
We have now got the positive and negative integers {. . . , −3, −2, −1, 0, 1, 2, 3, 4, . . .}
You are probably used to arithmetic with integers which follows simple rules.
To be on the safe side we itemize them, so for integers a and b

1.  a + b = b + a
2.  a × b = b × a or ab = ba
3.  −a × b = −ab
4.  (−a) × (−b) = ab
5.  To save space we write a k as a shorthand for a multiplied by itself k times.
So 3^4 = 3 × 3 × 3 × 3 and 2^10 = 1024. Note a^n × a^m = a^(n+m)
6.  Do note that n^0 =1.

Peace Out


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