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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.



 

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