Changing to Different Number Bases
Numbers such as 130, 2010, and 79 can be represented in different ways. These different ways are called bases.
Any number written is in the standard base, base 10; however, these numbers can be changed to different bases,
such as base 2, 5, 8, 9, 12 and so on. The process to follow when changing any number to a different base is the
following:
1.
2.
3.
4.
Identify the base to which the number will be changed to. The base will become the divisor.
Divide the number by the divisor (base).
Take the result and divide it by the divisor (base).
Continue doing step 3 until the result is no longer divisible by the divisor (base).
Change 212010 to base 8.
Example:
Step 1: base 8, the divisor is
265
Step 2: 8 2120
33
Step 3: 8 265
4
Step 4: 8 33
0
8 4
-16
52
-48
40
-40
0
-24
25
-24
1
-32
1
-0
4
Stop
5. To write the changed number, take the last result then write all the remainders from the previous
divisions after it in reverse order.
6. The base is written in subscript.
212010 = 41108
Numbers can also be changed from any base to base 10. The process of converting a number from any base to
base 10 is the following:
1.
2.
3.
4.
Write down the position of every number starting from right to left and starting with position zero.
Take the first digit of the number and multiply it by the base raised to the power of its position.
Take the second digit and multiply it by the base raised to the power of its position.
Keep doing this process until you have multiplied all the digits by the base raised to the power of its
corresponding position.
5. Add all the results.
Example:
Convert 41108 to base 10.
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Step 1:
4 1 1 08
3 2 1 0
Number
Positions
Step 2:
4 x 83 = 4 x 512 = 2048
Step 3:
1 x 82 = 1 x 64 = 64
Step 4:
1 x 81 = 1 x 8 = 8
0 x 80 = 0 x 1 = 0
Step 5:
2048 + 64 + 8 + 0 = 212010
Similarly, numbers can be changed from any base to any other base. The process to convert a number of any
base to another base is the following:
1. Convert the number to base 10.
2. Convert the number from base 10 to new base.
Example:
Convert 1205 to base 9.
Step 1:
convert 1205 to base 10.
1 2 05
2 1 0
Number
Positions
1 x 52 + 2 x 51 + 0 x 50
1 x 25 + 2 x 5 + 0 x 1
25 + 10 + 0
= 3510
Step 2:
convert 3510 to base 9.
3
9 35
0
93
-27
8
-0
3
STOP
= 389
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