Binary Numbers
Binary numbers are numbers with only two possible values for each digit: 0 and 1.
What is a Binary Number?
A binary number can only have digits with values 0 or 1.
Press the buttons below to see how counting in binary numbers works:
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It is important to understand binary numbers because they are the basis of all digital data, since computers can only store data in binary form, using bits and bytes.
The binary number 01000001 for example, stored in the computer, could be either the letter A or the decimal number 65 depending on the data type, how the computer interprets the data.
The term decimal comes from the Latin 'decem', meaning 'ten', because this number system (our normal everyday numbers) is based on ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, to represent values.
In a similar way, the term binary comes from the Latin 'bi', meaning 'two', because this number system uses only two digits: 0 and 1, to represent values.
Counting in Decimal Numbers
To better understand counting with binary numbers, it's a good idea to first understand the numbers we are used to: decimal numbers.
The decimal system has 10 different digits to choose from (0,..,9).
We start counting at the lowest value: 0.
Counting upwards from 0 looks like this: 1, 2, 3, 4, 5, 6, 7, 8, 9.
After counting up to 9, we have used up all the different digits available to us in the decimal system, so we need to add a new digit 1 to the left, and we reset the rightmost digit to 0, we get 10.
A similar thing happens at 99. To count further, we need to add a new digit 1 to the left, and we reset the existing digits to 0, we get 100.
Counting upwards, every time all possible combinations of digits have been used, we must add a new digit to continue counting. This is also true for counting using binary numbers.
Counting in Binary
Counting in binary is very similar to counting in decimal, but instead of using 10 different digits, we only have two possible digits: 0 and 1.
We start counting in binary:
0
The next number is:
1
So far, so good, right?
But now we have already used up all the different digits available to us in the binary system, so we need to add a new digit 1 to the left, and we reset the rightmost digit to 0, we get 10.
We continue counting:
10
11
It happened again! We have used up all the possible combinations of values, so we need to add another new digit 1 to the left, and reset the existing digits to 0, we get 100.
This is similar to what happens in decimal when we count from 99 to 100.
Using a third digit, we continue:
100
101
110
111
And now we have used up all the different digits again, so we need to add yet another digit 1 to the left, and reset the existing digits to 0, we get 1000.
Using the new fourth digit, we can continue counting:
1000
1001
...
..
And so on.
Understanding binary numbers becomes a lot easier if you're able to see the similarities between counting in binary and counting in decimal.
Converting Decimal to Decimal
To understand how binary numbers are converted to decimal numbers, it's a good idea to first see how decimal numbers get their value in the base 10 decimal system.
The decimal number 374 has 3 hundreds, 7 tens, and 4 ones, right?
We can write this as:
\[ \begin{equation} \begin{aligned} 374 {} & = 3 \cdot \underline{10^2} + 7 \cdot \underline{10^1} + 4 \cdot \underline{10^0} \\[8pt] & = 3 \cdot \underline{100} + 7 \cdot \underline{10} + 4 \cdot \underline{1} \\[8pt] & = 300 + 70 + 4 \\[8pt] & = 374 \end{aligned} \end{equation} \]
The math above helps us better understand how binary numbers are converted to decimal numbers.
Notice how \(10\) appears three times in the first line of calculation?
\[ 374 = 3 \cdot \underline{10}^2 + 7 \cdot \underline{10}^1 + 4 \cdot \underline{10}^0 \]
That is because \(10\) is the basis of the decimal number system. Each decimal digit is a multiple of \(10\), and that is why it is called a base 10 number system.
Converting Binary to Decimal
When converting from binary to decimal, we multiply the digits by powers of 2 (instead of powers of 10).
Let's convert the binary number 101 to decimal:
\[ \begin{equation} \begin{aligned} 101 {} & = 1 \cdot \underline{2^2} + 0 \cdot \underline{2^1} + 1 \cdot \underline{2^0} \\[8pt] & = 1 \cdot \underline{4} + 0 \cdot \underline{2} + 1 \cdot \underline{1} \\[8pt] & = 4 + 0 + 1 \\[8pt] & = 5 \end{aligned} \end{equation} \]
In the first line of calculation, each binary digit gets multiplied by 2 in the power of the digit's position.
The first position is 0, starting from the rightmost digit. So for example, the leftmost digit is multiplied by \(2^2\) since the leftmost digit's position is 2.
The fact that each binary digit is a multiple of 2 is why it is called a base 2 number system.
The calculation above shows that the binary number 101 is equal to the decimal number 5.
Click the individual binary digits below to see how other binary numbers are converted to decimal numbers:
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The further a binary digit is to the left, the more it is multiplied by, and that is why the leftmost binary digit is called the most significant bit.
Similarly, the rightmost digit is called the least significant bit, because it is just multiplied by \(2^0 = 1\).
Let's convert another binary number 110101 to decimal, just to get the hang of it:
\[ \begin{equation} \begin{aligned} 110101 {} & = 1 \cdot 2^5 + 1 \cdot 2^4 + 0 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0 \\[8pt] & = 32 + 16 + 0 + 4 + 0 + 1 \\[8pt] & = 53 \end{aligned} \end{equation} \]
As you can see, each binary digit is a multiple of 2, 2 in the power of the digit's position.
Converting Decimal to Binary
To convert a decimal number to a binary number, we can divide by 2, repeatedly, while keeping track of the remainders.
Let's convert 13 to binary:
\[ \begin{aligned} 13 \div 2 &= 6,\ \text{remainder } \underline{1} \\[8pt] 6 \div 2 &= 3,\ \text{remainder } \underline{0} \\[8pt] 3 \div 2 &= 1,\ \text{remainder } \underline{1} \\[8pt] 1 \div 2 &= 0,\ \text{remainder } \underline{1} \end{aligned} \]
Reading the remainders from bottom to top, we get 1101, which is the binary representation of 13.
Click the individual decimal digits below to see how a decimal number is converted to a binary number:
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Binary result:
Negative Binary Numbers
Binary numbers can represent negative values as well, by interpreting the leftmost bit as the sign.
A signed binary number is negative if the leftmost bit is 1, and positive if the leftmost bit is 0.
Using the leftmost bit as the sign bit, the highest positive number that can be represented with 8 bits is 01111111, which is 127.
Signed binary numbers that are positive (the leftmost bit being 0), are calculated into decimal numbers just like described above on this page.
But when the sign bit is 1, the number is negative, and something called the two's complement method is used to calculate the decimal value.
The two's complement method flips all the bits of the binary number (exept the sign bit) and adds 1 to the result.
Let's take the binary number 11111110 as an example.
The computer notices that the sign bit is 1, so it knows the number is negative, and uses the two's complement method to calculate the decimal value.
Removing the sign bit, we get 1111110.
Flipping all the bits, we get 0000001.
Adding 1 to the result, we get 0000010 which is 2.
Since the original number was negative, the result of converting the signed binary number 11111110 into decimal is -2.
Other Binary Operations
Binary numbers can be added, subtracted, multiplied and divided just like decimal numbers.
There are actually many more operations the computer can perform on binary numbers, but this page is just aimed at giving a basic understanding of binary numbers.
