Binary numbers

Binary numbers are a number system with base 2. In the binary system there are only two digits: 0 and 1.

 

Each digit in a binary number has a place value, which is a power of 2.

From right to left the place values are \(2^0, 2^1, 2^2, 2^3, \ldots\).

 

$$ \large 10101_2 = 1\cdot2^4 + 0\cdot2^3 + 1\cdot2^2 + 0\cdot2^1 + 1\cdot2^0 = 16 + 0 + 4 + 0 + 1 = 21_{10} $$

 

From binary to decimal

To convert a binary number to decimal, add the place values where there is a 1.

 

$$ \large 11001_2 = 1\cdot2^4 + 1\cdot2^3 + 0\cdot2^2 + 0\cdot2^1 + 1\cdot2^0 = 16 + 8 + 0 + 0 + 1 = 25_{10} $$

 

From decimal to binary

A quick method is to find the powers of 2 that add up to the number. Put 1 in the places you use and 0 in the others.

 

Example: 37 in decimal can be written as \(32 + 4 + 1\). This corresponds to \(2^5 + 2^2 + 2^0\).

 

$$ \large 37_{10} = 100101_2 $$

 

Here are the numbers from 0–15. We use 4 bits with leading zeros for clarity.

 

Decimal	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Binary (4 bits)	0000	0001	0010	0011	0100	0101	0110	0111	1000	1001	1010	1011	1100	1101	1110	1111

 

 

Bits and bytes

A bit is a single binary digit (0 or 1). A byte is 8 bits. The largest value on 8 bits is \(11111111_2\), which equals 255 in decimal.

 

$$ \large 11111111_2 = 2^7+2^6+2^5+2^4+2^3+2^2+2^1+2^0 = 128+64+32+16+8+4+2+1 = 255_{10} $$

 

 

Binary addition

The rules for one column are: \(0+0=0\), \(0+1=1\), \(1+0=1\), \(1+1=10\) (write 0 and carry 1 to the next column).

 

Example:

 

$$ \large 1011_2 + 110_2 = 10001_2 $$

 

 

Binary subtraction

When subtracting numbers in binary, use the same method as in decimal. If the digit above is too small, borrow from the next column.

 

Example: \(1010_2 - 11_2\)

 

$$ \large 1010_2 - 0011_2 = 0111_2 $$

 

Here we borrow from the third column, so \(0 - 1\) becomes \(10 - 1 = 1\), etc. The result 0111 is 7 in decimal.

 

 

Binary multiplication and division

Multiplication follows the same logic as in decimal, but you only multiply by 0 or 1. Division also follows the same rules as in decimal.

 

Example of multiplication:

 

$$ \large 101_2 \cdot 11_2 = 1111_2 $$

 

Explanation: \(101_2 = 5_{10}\), \(11_2 = 3_{10}\). So the result \(1111_2\) equals \(15_{10}\).

 

Example of division:

 

$$ \large 1100_2 \div 11_2 = 100_2 $$

 

Explanation: \(1100_2 = 12_{10}\), \(11_2 = 3_{10}\). The result \(100_2 = 4_{10}\).

 

 

Typical patterns

A binary number that consists only of 1’s corresponds in decimal to the sum of a series of powers of 2.

 

$$ \large 1111_2 = 2^3+2^2+2^1+2^0 = 8+4+2+1 = 15_{10} $$

$$ \large 1000_2 = 2^3 = 8_{10} $$

 

 

Summary

Binary numbers use only the digits 0 and 1, and each position carries the weight of a power of 2.

You convert by using the place values \(2^0, 2^1, 2^2, \ldots\), and you can go the other way by breaking the decimal number into powers of 2.