Similarly, when you say you have $162 in the bank, that number is a way of expressing how many dollars you have in the bank (in Base10, otherwise known as "decimal numbers"). You would not say "one hundred and sixty two" if you are speaking Swedish, or Spanish, or some other language. It would sound different  but regardless of how you say it, you still have $162 dollars in the bank.
Humans think in decimal numbers largely because the vast majority of human hands have ten fingers, and that is how humans learned to count. Thus, counting in "base 10" is a natural extention of the human body. But base10 is not the only "language" of numbers. You could count in Base6, or Base13, if you wanted to, but hey, we have enough to worry about without changing the whole number structure of the human race.
So, then, why do we want to learn binary numbers (Base2)? Because that's the best way for a computer to think (and even though we modelled computers in our own image, they are decidedly not human). Using binary representations of numbers is the best way for a computer to tranfer information throughout its circuits with minimal error.
An indepth discussion about computer electronics and logic isn't necessary at this time, so just think of it this way. Base 2 means that there are only two possible digits, 0 and 1. Compare that to Base10 which has ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. If a computer is going to transmit a digit, it's much easier for the computer to check for errors in transmission if it has only two options. Electronically, it's much easier to distinguish between two states rather than a range of possiblities because it either transmits a voltage (1  On) or it doesn't (0  Off). None of this inbetween stuff. It's cut and dried. Nice and simple.
To describe a quantity of things, and represent it with digits, you
can do it in Base2, Base20, Base16 (also called hexidecimal), etc.
However, whatever "language" (number base) that you use to describe
this quantity, you will still have the same number of things. Your
number of it will just look different in Base2 than it does in Base10.
4275
it means that you sum the following to get a concept of the total
quantity of things represented by that number:








which is the same as:




(Remember that in mathematics, any number raised to the zero power equals 1)
The number that is the on the far right is in the "ones" place holder
(called the least significant digit). To the left of that is the
"tens" place holder. To the left of that is the "hundreds" place
holder, and so on. They are all powers of 10. Whatever digit
is sitting in a particular place holder, you multiply that place holder
amount ( power of 10) by the digit, and then add them all up.
^{(most significant digit)} 


^{(least significant digit)} 












= 4 thousand, 2 hundred, and seventyfive
= 4275
^{(most significatnt digit)} 


^{(least significant digit)} 




The far right colum is the "ones" place holder. To the left of that is the "twos" place holder. To the left of that is the "fours" place holder (2^{2} = 4). To the left of that is the "eights" place holder (2^{3} = 8). They are all powers of 2. Whatever digit is sitting in a particular place holder, you multiply that place holder amount ( power of 2) by the digit, and then add them all up. However, remember that a binary digit has only two options, 0 or 1, so it's very easy.
For example, let's look at the binary number:
You would this interpret in decimal as:1011




















The total is: 11 (in decimal) which equals 8 + 0 + 2 + 1
In other words,
(Note: the subscripted number indicates the number base)1011_{2} = 11_{10}
equal in decimal?0111 (also written as 0111_{2})
Answer:












Total = 7 (also written as 7_{10})
equal in decimal?1010 (also written as 1010_{2})
Answer:












Total = 10 (also written as 10_{10})
Binary Decimal
0000 = 00_{10}
0001 = 01_{10}
0010 = 02_{10}
0011 = 03_{10}
0100 = 04_{10}
0101 = 05_{10}
0110 = 06_{10}
0111 = 07_{10}
1000 = 08_{10}
1001 = 09_{10}
1010 = 10_{10}
1011 = 11_{10}
1100 = 12_{10}
1101 = 13_{10}
1110 = 14_{10}
1111 = 15_{10}
Can you see the pattern? Just like decimal counting, you start
with zero, add 1 to the least significant digit to count up by 1 each time.
When you've exhausted the total number of possibilities in that place holder
(which in binary is only two), the next time you add 1, you have to carry
it over to the next highest significant place holder, reset the less significant
place holder to 0, and repeat the process. Just like decimal numbers,
as you count up, the digits on the right hand side change more rapidly
than those further to the left.
















Check the following binary to decimal conversions:












You might notice something interesting about the number 11111111_{2}.
256K is a number often used in the computer world. So, why does 11111111_{2}
convert to 255_{10} and not 256_{10}? That's because,
one of the possibilities is zero! Any 8digit number that might have
a 1 in it (255_{10} total), plus one more possible state:
00000000_{2}.