 Mathematics

# Base Ten Numbers Base eight Numbers Tabitha Akery's image for:
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Image by: Figuring out how to understand numbers when they have a different base can be difficult. Changing the base of a number is usually indicated by a little number off set on the right hand side. Only the base ten system does not have the little number. The normal system is Base ten. This means that the numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are used. With the number 10 equaling 10.

Computers use the binary code which is a base two. Only the numbers 0 and 1 are used.
So in binary code:

Base 10 number Base 2 number
0 0
1 1
2 10
3 11
4 100
5 101
6 110
7 111
8 1000
9 1001
10 1011

This pattern repeats itself as the numbers go higher. The higher the numbers go the bigger the binary numbers become. This is why this system is only used on computers it can become impossible to keep track of really large numbers this way.

The base three system uses 0, 1, and 2. This is called tertiary.
So in tertiary code:

Base 10 number Base 3 number
0 0
1 1
2 2
3 10
4 11
5 12
6 100
7 101
8 102
9 110
10 111

The base eight system is the mostly commonly used system and is often used in place of the base ten. A little 8 will appear on the right side of the number. This number 8 is offset like a footnote. Only the numbers 0, 1, 2, 3, 4, 5, 6, 7 are used.
Base 10 number Base 8 number
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 10
9 11
10 12
11 13
12 14
13 15
14 16
15 17
16 20

At every multiple of 8, a pattern emerges. The number 10 gets multiplied by how ever many multiples of 8 there are. For example, at 24 in base ten, the number would be 1000. This makes things easier when trying to convert a base ten number to a base eight.

Take the base ten number and divide it by 8. The whole number will tell how tens need to be multiplied together. For example, take the number 72. 72/8 = 9. Therefore, multiply 10 together nine times to get the base eight number. 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 1,000,000,000.

If the number is not a multiple of 8, the base eight number can still be found. For example, take the number 41. 41/8 = 5.125. Take the whole number and multiply ten together five times. 10 x 10 x 10 x 10 x 10 = 10,000. Only the first decimal number is important. Do not round that number up. Add that number to the whole number answer. 10,000 + 1 = 10,001. The resulting number is the base eight number. So the number 41 in base eight is 10,001.

Remembering the rules for using a different base number will help in converting from one base to another. Making a list will help with smaller numbers and help identify patterns that will assist in dealing with the larger numbers. As with everything in math, practice makes perfect.

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