Base10: The Decimal System

Overview 

The numbers that we use every day for things like prices, math, banking, how many pizzas to get for a party, etc are in a system called base10, or decimal. We call it that because each 'column', or place, is associated with a power of 10. The system also has a radix of 10, meaning that each column can hold 1 of 10 possible digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9.

ex: 20310203_{10}

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203=(2102)+(0101)+(3100)203 = (2 * 10^2) + (0 * 10^1) + (3 * 10^0)
203=200+0+3203 = 200 + 0 + 3

The digit in each place is multiplied by a power of 10, and then when the products are added together, the result is the number written in base10. The 1's column is multiplied by 10010^0 , which is just 1 , the 10's column is multiplied by 101=1010^1 = 10 , the 100's column is multiplied by 102=10010^2 = 100 , and so on. It's a different way of looking at the numbers we use every day, but it hopefully gives you some insight into how and why they are constructed the way that they are.

Note: The subscript 10 is used to denote the base of the number. We don't include this in every day math because it's assumed that unless specified, numbers are already written in base10.

Numbers smaller than 1 

It's a pretty straight-forward process for getting bigger and bigger numbers- just increase the power of 10 of the multiplier. But what about numbers smaller than 1?

Let's look at 0.31100.31_{10} as an example. Thinking in the context of number bases, we can see that the decimal point actually specifies where our 10010^0 column is. When moving to the right of the decimal point, the powers of 10 simply continue to decrease:

ex. 0.31100.31_{10}

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0.31=(0100)+(3101)+(1102)0.31 = (0 * 10^0) + (3 * 10^{-1}) + (1 * 10^{-2})
0.31=0+0.3+.010.31 = 0 + 0.3 + .01

Recall that:

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101=1101=110=0.110^{-1} = \frac{1}{10^1} = \frac{1}{10} = 0.1
102=1102=1100=0.0110^{-2} = \frac{1}{10^2} = \frac{1}{100} = 0.01
103=1103=11000=0.00110^{-3} = \frac{1}{10^3} = \frac{1}{1000} = 0.001

If you've never experienced numbers in different bases before, hopefully this is some interesting insight into the number system we use every day. Now that we've figured out how our decimal system works as written, it should be an easy task to apply this to other number bases.