Understanding Binary Numbers

by Vanessa Else

Step 1: Relax

Don't be afraid of understanding binary numbers and how to convert them to our common decimal number system. It's easy. Before we delve into mathematics, though, let's consider a linguistic analogy. Consider that "love" and "amour" are two different ways of saying the same thing. One is English. One is French. However, they both represent the same concept. The abstract concept "love" is independent of how we choose to label it. It is a concept. We give it a name so we can communicate with each other when describing to someone that we feel love. The name for that feeling is different depending upon your frame of reference. If you say "love," you might say you are speaking in Base English. If you say "amour," you could say you are speaking in Base French.

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 Base-10, 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 base-10 is not the only "language" of numbers. You could count in Base-6, or Base-13, 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 (Base-2)? 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 in-depth 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 Base-10 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 in-between 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 Base-2, Base-20, Base-16 (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 Base-2 than it does in Base-10.

Decimal Numbers

Let's take a closer look at something most of us take for granted. Counting in decimal numbers. For example, when you see the decimal number:

4275

it means that you sum the following to get a concept of the total quantity of things represented by that number:

Thousands Hundreds Tens Ones

4 x 1000 plus 2 x 100 plus 7 x 10 plus 5 x 1

which is the same as:

4 x 10³ plus 2 x 10² plus 7 x 10¹ plus 5 x 10⁰

(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.

^{Thousands column}
^{(most significant digit)} ^{Hundreds column} ^{Tens column} ^{Ones column}
^{(least significant digit)}

10³ 10² 10¹ 10⁰

digit from 0-9 x 1000 digit from 0-9 x 100 digit from 0-9 x 10 digit from 0-9 x 1

4 2 7 5

= 4 thousand, 2 hundred, and seventy-five
= 4275

Binary Numbers

Similarly, in binary (Base-2), all the columns are powers of 2. Think of the following table as the four least significant digit place holders for the binary number system just like the tables above using the decimal number system:

^{Eights column} ^{(most significatnt digit)}	^{Fours column}	^{Twos column}	^{Ones column} ^{(least significant digit)}
2³	2²	2¹	2⁰

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 (2² = 4). To the left of that is the "eights" place holder (2³ = 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:

1011

You would this interpret in decimal as:

	^Eights	^Fours	^Twos	^Ones
	1 x 2³	plus 0 x 2²	plus 1 x 2¹	plus 1 x 2⁰
which equals	1 x 8	plus 0 x 4	plus 1 x 2	plus 1 x 1
which equals	8	plus 0	plus 2	plus 1

The total is: 11 (in decimal) which equals 8 + 0 + 2 + 1

In other words,

1011₂ = 11₁₀

(Note: the subscripted number indicates the number base)

Try It Yourself

Here are two more examples of binary numbers converted to decimal. Try the conversion yourself before peaking at the answer.

Example 1:

What does this binary number:

0111 (also written as 0111₂)

equal in decimal?

Answer:

0 x 2³ plus 1 x 2² plus 1 x 2¹ plus 1 x 2⁰

= 0 x 8 plus 1 x 4 plus 1 x 2 plus 1 x 1

= 0 + 4 + 2 + 1

Total = 7 (also written as 7₁₀)

Example 2:

What does this binary number:

1010 (also written as 1010₂)

equal in decimal?

Answer:

1 x 2³ plus 0 x 2² plus 1 x 2¹ plus 0 x 2⁰

= 1 x 8 plus 0 x 4 plus 1 x 2 plus 0 x 1

= 8 + 0 + 2 + 0

Total = 10 (also written as 10₁₀)

Trick Question:

If you were going to count from 0 to 15 (a total of 16 possibilites), what would it look like in binary?

Answer:

Binary    Decimal
0000       = 00₁₀
0001       = 01₁₀
0010       = 02₁₀
0011       = 03₁₀
0100       = 04₁₀
0101       = 05₁₀
0110       = 06₁₀
0111       = 07₁₀
1000       = 08₁₀
1001       = 09₁₀
1010       = 10₁₀
1011       = 11₁₀
1100       = 12₁₀
1101       = 13₁₀
1110       = 14₁₀
1111       = 15₁₀

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.

Let's Get Advanced

In computers, each "bit" of information is like a place holder digit in a binary number. An 8-bit piece of computer data is often called a "byte" and is represented by an 8-digit binary number. The values of these binary place holders are as follows (with the "most significant digit" on the left hand side, and the "least significant digit" on the right had side -- just like decimal numbers):

2⁷	2⁶	2⁵	2⁴	2³	2²	2¹	2⁰
= 128	= 64	= 32	= 16	= 8	= 4	= 2	= 1

Check the following binary to decimal conversions:

Binary number Decimal Number Summation of Decimal Numbers

1000 1111 143 (128 + 8 + 4 + 2 + 1)

0110 0110 102 (64 + 32 + 4 + 2)

1111 1111 255 (128 + 64 + 32 + 16 + 8 + 4 + 2 + 1)

You might notice something interesting about the number 11111111₂. 256K is a number often used in the computer world. So, why does 11111111₂ convert to 255₁₀ and not 256₁₀? That's because, one of the possibilities is zero! Any 8-digit number that might have a 1 in it (255₁₀ total), plus one more possible state: 00000000₂.

Thousands	Hundreds	Tens	Ones
4 x 1000	plus 2 x 100	plus 7 x 10	plus 5 x 1

^{Thousands column} ^{(most significant digit)}	^{Hundreds column}	^{Tens column}	^{Ones column} ^{(least significant digit)}
10³	10²	10¹	10⁰
digit from 0-9 x 1000	digit from 0-9 x 100	digit from 0-9 x 10	digit from 0-9 x 1
4	2	7	5

0 x 2³	plus 1 x 2²	plus 1 x 2¹	plus 1 x 2⁰
= 0 x 8	plus 1 x 4	plus 1 x 2	plus 1 x 1
= 0	+ 4	+ 2	+ 1

Binary number	Decimal Number	Summation of Decimal Numbers
1000 1111	143	(128 + 8 + 4 + 2 + 1)
0110 0110	102	(64 + 32 + 4 + 2)
1111 1111	255	(128 + 64 + 32 + 16 + 8 + 4 + 2 + 1)