Binary Number System

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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:

Place Value
Sixty-fours	Thirty-twos	Sixteens	Eights	Fours	Twos	Units/ Ones
26	25	24	23	22	21	20

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)

24	23	22	21	20
		1	0	1

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

 b)

24	23	22	21	20
	1	0	1	1

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

c)

24	23	22	21	20
1	0	1	0	1

                        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 

Base 10 to Base 2 (Decimal to Binary) Conversion
A demonstration of the repeated divide by 2 method for converting numbers from base 10 (or decimal) into base 2 (or binary) form.