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Counting in different bases replaces our base ten with a different base. Time is an example of counting in a different base, for seconds it's 60; for minutes, 60; for hours, 24; for months, 12. And we have come to accept these things.

Expanded form

We know that 1234 does not equal to 1+2+3+4. The "1" in "1234" actually represents 1000, or 103. And we know that the "2" in "1234" is equal to 200, which is 2x102.

Therefore, we can expand any number to a certain form: the expanded form. The expanded form of 1234 would be 1x103 + 2x102 + 3x101 + 4x100.

Base

If we want to change the base of a number, we would want to change the base of each term in the expanded form of the number. Therefore, 1234 in base 8 would represent a value of 1x83 + 2x82 + 3x81 + 4x80, or 668 in base 10.

Symbols

In base 10 (i.e. the normal system), we have 10 symbols: 0,1,2,3,4,5,6,7,8,9. In base 8, we have only 8 symbols: 0,1,2,3,4,5,6,7. Therefore, in base n, we have n symbols.

In base 16, we should have 16 symbols. Therefore, we would need to invent new symbols for the numbers after 9 (i.e. 10,11,12,13,14,15 in base 10). Usually, we would use the letters to represent them. i.e., A=10, B=11, C=12, etc.

Base Equations

Base equations also follow the same rules.

6x9 in base 13.

1,2,3,4,5,6,7,8,9,10,1A,1B,1C,11,12,13,14,15,16,17,18,19,20,2A,2B,2C,21,22,23,24,25,26,27,28,29,30,3A,3B,3C,31,32,33,34,35,36,37,38,39,40,4A,4B,4C,41,42

What is happening here is that for every symbol, one is subtracted from 6x9 in base 10, which is 54, and continued that way until the number that is last is reached.

So, 6x9=42.

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