## Binary Numeral System

A numeral system is a way of representing numbers, not to be confused with number systems which classify numbers based on properties.

The binary numeral system has only two digits, 0 and 1. Every decimal number is represented with some combination of 0 and 1 in binary.

Suppose we had a binary number like 10110 that we wish to represent in decimal. How are we going to convert 10110 to decimal?

A numbers *place* in the decimal numeral system is based on some power of 10; each digit in a decimal number is multiplied by some power of ten to get its value. In binary, the base number is 2; so every digit is multiplied by some power of two. The exponent is determined by the number of digits to the left or right of the decimal.

In the decimal numeral system, the *ones* place is held by $ 10^0 = 1$. There is also a ones place in binary: $ 2^0 $ which is also equal to 1. In our number, 10110 there is a 0 in the ones place. So the ones place holds no value. $ 0 \times 10^1 = 0 $

The next place in the decimal numeral system is the *tens* place because $ 10^1 $ is ten. We don't have a tens place in binary. We have a *twos* place because $2^1$ is 2. In our number that we wish to convert, 10110, We have a 1 in the twos place. $1 \times 2^1 = 2. $

So far we have examined two places and our tally is 2 = 2 + 0. The next digit is the *fours* place because $ 2^2 $ is four. We have a 1 in the fours place. $1 \times 2^2=4 $. So our tally is 4 + 2 + 0 = 6.

The next digit is the *eights* place. We have a zero there so we don't need to tally anything.

The last place to examine in 10110 is the *sixteens* place, and we have a 1 there. $1 \cdot 2^4 = 16 $.

So our final tally is 16 + 0 + 4 + 2 + 0 = 22. So $ 10110_2=22_{10} $ . The subscripted numbers tell us the numeral system of that particular number.