Using Bases other than Base 10
We could pick any value to be the base of our number system, in fact many cultures have used other bases over time. For example, the Babylonian used base 60 and the Mayan used base 20. Even today we use other bases depending on the situation. Base 12 is one we use every day, look at the clock or inches & feet. However, base 2, 8, and 16 are very common when working with computers, they even have special names, binary, octal, and hexadecimal respectively.
Since our number system has special characters for 0 – 9, when our base is below 10, we don’t have to worry about a character to use for the values. However, for hexadecimal we need a single character to represent 10 – 15, usually we use the beginning of the alphabet, A – F. If we were to go larger than base 16, we could keep going with the letters.
Base 2 ~ Binary
When we are working with binary, we only have 2 digits, 0 & 1. Each place value in our chart can only hold one of those 2 digits. We get our place values for base 2, the same way we did for base 10, by looking at the powers of 2. Unlike with the powers of 10, we are not going to look at the fractional portions.
Binary Place Values |
||||
3 | 2 | 1 | 0 | |
Powers of 2 | 2^{3} | 2^{2} | 2^{1} | 2^{0} |
numbers | 1000 | 100 | 10 | 1 |
Value in base 10 | 8 | 4 | 2 | 1 |
So if we have the number 7_{10}, it would be 111_{2}. One way to work with our bases and to understand what is happening is to use an expanded form of our numbers.
Expanded form is when we take our digit and multiply it by the place value. For example, in base 10 we have 576, this is the same as 5 x 100 + 7 x 10 + 6 x 1 or 5 x 10^{2} + 7 x 10^{1} + 6 x 10^{1}. So if we have 7 in base 10 and want to change it to base 2, we look how it is added with powers of 2. 7 = 4 + 2 + 1, 4 = 2^{2}, so we have 7 = 2^{2} + 2^{1} + 2^{0}, but we still need the multiplier for each of those powers, since they are each in our expression, there are no 0s, only 1s. Now we have 1 x 2^{2} + 1 x 2^{1} + 1 x 2^{0}, so in base 2 7 is written as 111_{2}.