# How To Count In Binary

### 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.

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

### 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