1.2.3.A BinaryNumberSystem.pptx

The Binary Number System and

Conversions

© 2014 Project Lead The Way, Inc. Digital Electronics

Bridging the Digital Divide

0010

0

0101011

01

0

10010110₁

011011 1101 010 00101101

0010

0

0101011

0010

0

01

0

10010110₁

1101 010 00101101

01

01

01

1

Decimal ‒to‒ Binary Conversion

The Process : Successive Division

a) Divide the Decimal Number by 2; the remainder is the LSB of

Binary Number .

b) If the quotient is zero, the conversion is complete; else repeat step (a) using the quotient as the Decimal Number. The new remainder is the next most significant bit of the Binary Number.

Example:

Convert the decimal number 6₁₀ into its binary equivalent.

 6₁₀ = 110₂

3 Bit t Significan Most 1 r 0 1 2 1 r 1 3 2 Bit t Significan Least 0 r 3 6 2     

Dec → Binary : Example #1

Dec → Binary : Example #1

Convert the decimal number 26₁₀ into its binary equivalent.

Solution:

 26₁₀ = 11010₂

5 LSB 0 r 13 26

2  

MSB 1 r 0 1

2  

1 r 6 13 2  0 r 3 6 2  1 r 1 3 2 

Dec → Binary : Example #2

Dec → Binary : Example #2

Convert the decimal number 41₁₀ into its binary equivalent.

Solution:

 41₁₀ = 101001₂

7 LSB 1 r 20 41

2  

0 r 10 20 2  0 r 5 10 2  1 r 2 5 2  MSB 1 r 0 1

2  

0 r 1 2 2 

Dec → Binary : More Examples

a) 13

= ?

b) 22

= ?

c) 43

= ?

Dec → Binary : More Examples

a) 13

= ?

b) 22

= ?

c) 43

= ?

d) 158

= ?

1 1 0 1

1 0 1 1 0

1 0 1 0 1 1

1 0 0 1 1 1 1 0

Binary ‒to‒ Decimal Process

The Process : Weighted Multiplication

a) Multiply each bit of the Binary Number by it corresponding bit-weighting factor (i.e. Bit-0→20=1; Bit-1→21=2; Bit-2→22=4; etc).

b) Sum up all the products in step (a) to get the Decimal Number. Example:

Convert the decimal number 0110₂ into its decimal equivalent.

 0110 ₂ = 6 ₁₀

0

1

1

0

23 ₂2 ₂1 ₂0

8 4 2 1

0 + 4 + 2 + 0 =

6

Bit-Weighting Factors

Binary → Dec : Example #1

Convert the binary number 10010₂ into its decimal equivalent.

Binary → Dec : Example #1

Convert the binary number 10010₂ into its decimal equivalent.

1

0

0

1

0

24 ₂3 ₂2 ₂1 ₂0

16 8 4 2 1

16 + 0 + 0 + 2 + 0 =

18

Binary → Dec : Example #2

Convert the binary number 0110101₂ into its decimal equivalent.

Binary → Dec : Example #2

Convert the binary number 0110101₂ into its decimal equivalent.

0

1

1

0

1

0

1

26 ₂5 ₂4 ₂3 ₂2 ₂1 ₂0

64 32 16 8 4 2 1

0 + 32 + 16 + 0 + 4 + 0 + 1 =

53

Binary → Dec : More Examples

a) 0110

= ?

b) 11010

= ?

c) 0110101

= ?

d) 11010011

= ?

Binary → Dec : More Examples

a) 0110

= ?

b) 11010

= ?

c) 0110101

= ?

d) 11010011

= ?

6

26

53

Summary & Review

Base

₁₀

DECIMAL

Base

₂

BINARY

Division

a) Divide the Decimal Number by 2; the remainder is the LSB of Binary Number .

b) If the Quotient Zero, the conversion is complete; else repeat step (a) using the Quotient as the Decimal Number. The new remainder is the next most significant bit of the Binary Number.

a) Multiply each bit of the Binary Number by it corresponding bit-weighting factor (i.e. Bit-0→20_{=1; Bit-1→2}1_{=2; Bit-2→2}2_{=4; etc).}

b) Sum up all the products in step (a) to get the Decimal Number. Weighted

Multiplication

Base

10

DECIMAL

Base

₂

BINARY

Image Resources

• _{Microsoft, Inc. (2008). Clip Art. Retrieved March 15, 2008 from}