Arithmetic
Counting
In mathematical problems, there is a difference between the way you count numbers and values. In simple base 10 math, the methods appear to be the same; However, if you intend to use an alternative base for your mathematical problems, you can immediately see the difference.
A numerical representation for a value, is divided into exponents according to it's base. For example, in base 10 math, the digits representing values are read from right to left as ones, tens, hundreds, thousands, etc. - which are all multiples of 10.
In base two math, they read from right to left as ones, twos, fours, eights. Similarly, base three math reads ones, threes, nines, twenty-sevens, etc.
The following table illustrates the four basic digits that represent values for each base up to ten.
Base One |
D.N.E. |
D.N.E. |
D.N.E. |
D.N.E. |
|
Base Two |
1 = 1 |
10 = 2 |
100 = 4 |
1000 = 8 |
|
Base Three |
1 = 1 |
10 = 3 |
100 = 9 |
1000 = 27 |
|
Base Four |
1 = 1 |
10 = 4 |
100 = 16 |
1000 = 64 |
|
Base Five |
1 = 1 |
10 = 5 |
100 = 25 |
1000 = 125 |
|
Base Six |
1 = 1 |
10 = 6 |
100 = 36 |
1000 = 216 |
|
Base Seven |
1 = 1 |
10 = 7 |
100 = 49 |
1000 = 343 |
|
Base Eight |
1 = 1 |
10 = 8 |
100 = 64 |
1000 = 512 |
|
Base Nine |
1 = 1 |
10 = 9 |
100 = 81 |
1000 = 729 |
|
Base Ten |
1 = 1 |
10 = 10 |
100 = 100 |
1000 = 1000 |
|
Addition
Addition, like all mathematical operations, is not dependent on the base you are using. To perform addition, you simply count inside of a placeholder until it carries over into the next place.
For example:
Base 2:
101 + 1 = 110 (5 + 1 = 6)
Because base 2 math only allows 1's and zeros, you must carry over any remaining digits.
Base 8:
107 + 1 = 110 (71 + 1 = 72)
Because base 8 math only allows for numbers up to 8, you must carry over any remaining digits.
Subtraction
Subtraction is always the same despite the base that you are using. Subtraction is simple the inverse of addition. Rather than adding a positive number, you are adding a negative number - you are removing a number.
So, here is an example of how to achieve this in base 2, 8, and 10:
Multiplication
Division
Exponents
Roots
Irrational/Fractional Numbers