The binary system (BIN) is commonly used in computer science. Computers can handle only two different states: a de-energized state or an energized state. These states can be described with the numbers 0 and 1. The base number of the binary system is 2, and its only numbers are 0 and 1. In a binary system, each number can be represented as the power of number 2. The position of zero or one indicates how large the number is.

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

Note! Because the decimal system is the most common numeral system, its numbers do not normally require the use of a subscript.

Example 2

Convert the binary number 101101 to the decimal system.

​[[$ \begin{align} & 1 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0 \\ & = 1 \cdot 32 + 0 \cdot 16 + 1 \cdot 8 + 1\cdot4 + 0\cdot2 + 1\cdot1 \\ & =32 + 8 + 4+ 1 \\ & = 45 \end{align} $]]​

Answer: 101101₂ = 45₁₀

Example 3

Convert the number 83 from the decimal system into the binary system.

• Let us first take the largest power of two that is ​[[$ \leq 83. \text{ } 2^6 = 64 \leq 83 $]]​.
• Subtract this from the variable[[$ 83-64 = 19 $]]​.
• The maximum power of two, which is[[$ \leq 19 $]], is [[$ 2^4 = 16 $]]​​.
• Subtract this from the variable number[[$ 19-16 = 3 $]].​
• The maximum power of two, which is [[$ \leq 3 $]], is [[$ 2^1 = 2 $]]​​.
• [[$ 3-2 = 1 $]], so the last power of the two is [[$ 2^0 = 1 $]]​.
• Powers [[$ 2^6, 2^4, 2^1 $]] and [[$ 2^0 $]] ​​will be replaced by a number 1 in the binary format and the places between them will be filled with zeros.

• The binary number is 1010011

Answer: [[$ 83_{10} = 1010011_2 $]]​