Computer Math Part 2: Converting Decimal to/from Binary

Wednesday, I discussed the very basics of binary and how to count, or increment, in binary. Today I’ll be discussing converting decimal numbers to and from binary numbers.

Before I cover conversion, let’s talk for a second about bases. That’s what this series boils down to. I’m not talking about military bases, but number bases. Binary is base 2, meaning that there are 2 digits. Because decimal uses 10 digits, it is base 10.

When a number is of a certain base, you denote that by putting a subscripted 2, or 10, immediately after the number. So 1002 means 100 in binary, while 10010 means 100 in decimal. If you don’t fully grasp this, I’m sure you will once you finish the exercises for today.

So, we all know that 12 is 110, but what does 102 equal in base 10? It’s 210. Check out this chart:

BinaryDecimal
12110
102210
1002410
10002810
1000021610
10000023210
100000026410
10000000212810
100000000225610

For every extra zero on the binary side, the decimal side doubles. This is because binary is base 2. You can say that, counting from the right in binary, each digit placement is worth double the previous.

You can use this chart to convert a binary number to a decimal number. For every 1, add the decimal equivilent. For instance, the number 1112 is 110 + 210 + 410 which is 710. Another example is 10012, which is 810 + 110, or 9.

OK, converting binary to decimal is the easier part for today. On to converting decimal to binary. You do pretty much the same thing as converting binary to decimal, but in reverse. The first step is to grab a scrap piece of paper, or open notepad.

To walk through the process, I’ll convert 10510 to binary. The first step is to find the decimal number in the chart that is closest to 10510 without going over, which is 6410. 6410 is 10000002, so write that binary number on the first line. The second step is to subtract 6410 from 10510, which is 4110.

Then we repeat. So, 3210, or 1000002, is the next number. Write 1000002 on the second line, making sure to line up the numbers on the right side, same as you would for decimal. Then subtract 3210 from 4110, which is 910.

Then we repeat. So, 810, or 10002, is the next number. Write 10002 on the second line, making sure to line up the numbers on the right side, same as you would for decimal. Then subtract 810 from 910, which is 110.

Then we repeat. So, 110, or 12, is the next number. Write 12 on the second line, making sure to line up the numbers on the right side, same as you would for decimal. Then subtract 110 from 110, which is 010.
Because, we have reached zero, we can move on the last step. On your paper/notepad should be:

1000000
100000
1000
1

All you have to do is combine these. Starting on the left, any column that has a 1 in it, write a 1, or if the column only has zeros, write a 0. This yeilds: 11010012. Converting this back to decimal will let you know you did it right.

Homework:

Convert the following binary numbers to decimal:
1001
1101
10
10000011
10101010
10110010
100011

Convert the following decimal numbers to binary:
37
123
251
6
192
168
68

Remember to convert it back to check your work. Check back Monday when I’ll discuss adding binary numbers.