How To Count In Binary

Posted by Jeff Schuler on Jul 27, 2009 in Uncategorized |

How do binary numbers work?

Binary numbers work exactly like our decimal system. It’s called Decimal because it has 10 total combinations per digit. For instance we count 0-1-2-3-4-5-6-7-8-9 which gives us 10 total numbers we can use in a single digit. When we run out of digits we add one to the leading digit and switch the current digit to the first combination. For instance when we count in decimal and we get to 19 we add 1 to the first digit making it a 2 and turn the current digit to our first combination,a zero, making a 20. Whenever there is no digit in front we assume its a zero so when we count to 10 and we’re at 9 we add one to the first digit turning it from a zero to a one. Then we turn our current digit, the nine, to a zero making 10. It works infinitely so we can be at 999,999 and our next number would be 1,000,000.

Binary counting is exactly the same only even simpler because it only has two combinations, 0-1. The only combinations in digits you have available to you is zero or one. So you start your count with 0. Then you add one to bring you to 1. Then when you add another, since you’re at your max combinations for that single digit you need to add one to the leading digit and turn your current digit to zero. So the decimal #2 becomes 10 in binary. That may look like a 10 in decimal but in binary since you only have two combinations per digit it translate to the number 2. You’re next number would be done the same way. You add one to your current digit which in the 10 would become 11. This becomes the number 3 in decimal. Once again to add one you need to add one to the leading digit which would be the 11 and turn your current digit to zero making it 100 for the count of four. Much like learning to count it gets much simpler the more you practice.

Try figuring out what the next number would be in this sequence:
0000=0
0001=1
0010=2
0011=3
0100=4
0101=5
0110=6
0111=7
1000=8
1001=9
1010=10
1011=11
____=12

Answer: 1100

Figuring Out Long Numbers In Binary

As mentioned binary works exactly like our decimal system only it has two combinations per digit instead of ten. When you’re brain is trying to figure out how large a decimal number is it looks at all the placements of the digits and their values. For instance the number 321 is 3 x 10^2 + 2 x 10^1 + 1 x 10^0 = 321 . The 10 base number is because there is 10 combinations per digit. The power is the placement of that digit and the number being multiplied by it is the number in that position. So to get the number 400 it is 4 x 10^2 . In reality this is done for every digit so 50 would actually be 5 x 10^1 + 0 x 10^0 . Since anything multiplied by zero is zero we tend to ignore those as to not confuse ourselves when doing large numbers. Binary is done the same way except since there is only ones and zeros you only need to concern yourself with the ones. So the binary number 0100 would be 1 x 2^2 = 4 Therefore 1010 would be 1 x 2^3 + 1 x 2^1 = 10 . This makes it easy to figure out really long binary numbers such as 00110111 because logically on paper that would be 1 x 2^5 + 1 x 2^4 + 1 x 2^2 + 1 x 2^1 + 1 x 2^0 for a decimal answer of 55.

Using What You’ve Learned Try To Figure Out The Decimal Equivilant of This Binary Number:
10110111

Answer: 183

Bits And Bytes

When people talk about bits and bytes of binary they’re referring to how many assumed leading zeros are in the number. So a 4 bit binary number of two would be 0010. An 8bit version of the number 2 would be 00000010. As with decimal the zeros are always there they just aren’t shown unless you need to specify a maximum amount that you want. So if you said you want a 8 bit binary representation of the number 85 you would get 01010101. A byte is simply 8 bits. So if you said you wanted a 2 byte representation of the number 85 you would get 0000000001010101 for a total of 16 bits.

Good luck and happy counting!

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