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Find the binary equivalent for the following decimal numbers. Find the round off error, if there is any:  a. (5.75)10 = b. (2.625)10 =

Find the binary equivalent for the following decimal numbers. Find the round off error, if there is any:  a. (5.75)10 = b. (2.625)10 =

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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 Find the binary equivalent for the following decimal numbers. Find the round off error, if there is any: 
a. (5.75)10 =
b. (2.625)10 =

Expert Solution
Step 1: Converting (5.75)10 to binary

Converting the integer part (5) to binary

To convert an integer to binary, we repeatedly divide the number by 2 and keep track of the remainders.

table row cell text Division by 2 end text end cell cell colon text Remainder end text end cell row cell 5 divided by 2 end cell cell equals 2 colon 1 end cell row cell 2 divided by 2 end cell cell equals 1 colon 0 end cell row cell 1 divided by 2 end cell cell equals 0 colon 1 end cell end table

Reading the remainders from bottom to top, the binary representation of 5 is .

Converting the fractional part (0.75) to binary

To convert a fraction to binary, we multiply the fraction by 2 and keep track of the integer parts.

table row cell text Multiplication by 2 end text end cell cell colon text Integer Part end text end cell row cell 0.75 cross times 2 end cell cell equals 1.5 colon 1 end cell row cell 0.5 cross times 2 end cell cell equals 1.0 colon 1 end cell end table

Reading the integer parts from top to bottom, the binary representation of 0.75 is .

Combining the results from Steps 1 and 2

Combining the results, left parenthesis 5.75 right parenthesis subscript 10 equals left parenthesis 101.11 right parenthesis subscript 2

There's no round-off error in this case because 5.75 can be exactly represented in binary.

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