Binary Number System

In a binary number (or base two) system, we use only two digits: 0 and 1. The place value for a base two number is shown as follows:

Convert From Binary To Decimal

Like base ten numbers, we can determine the value of a base two number by placing each digit in its respective place value, then writing its expanded notation.

Example:
Find the value of each of the following binary numbers, giving the values in decimal numbers:
a)         101₂                 b)         1011₂               c)         10101₂           

Solution:
a)

                        101₂     = 1 × 2² + 0 × 2¹ + 1 × 2⁰
                                                = 4 + 0 + 1
                                                = 5₁₀ 

 b)

                        1011₂   = 1 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰
                                                = 8 + 0 + 2 + 1
                                                = 11₁₀  

c)

                        10101₂   = 1 × 2⁴ + 0 × 2³ + 1 × 2² + 0 × 2¹ + 1 × 2⁰
                                       = 16 + 0 + 4 + 0 + 1
                                       = 21₁₀ 

Convert From Decimal To Binary

We can convert a decimal number into a binary number by repeatedly dividing the base ten number by two. Then, write the remainders from the bottom to the top as the answer:

Example:
Write each of the following base ten numbers as a binary number:
a)         5₁₀                                b)         78₁₀                

Solution:

a)

5₁₀  = 101₂ 

b)

            78₁₀      = 1001110₂ 

The following video gives another example of decimal to binary conversion.

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