Mathematics For Machine Technology
Mathematics For Machine Technology
8th Edition
ISBN: 9781337798310
Author: Peterson, John.
Publisher: Cengage Learning,
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Chapter 86, Problem 1A
To determine

(a)

Conversion of hexadecimal numbers B70905.F816 to a binary numbers.

Expert Solution
Check Mark

Answer to Problem 1A

Binary numbers is 101101110000100100000101.111112.

Explanation of Solution

Given information:

A hexadecimal numbers B70905.F816.

Calculation:

Binary number system uses the number 2 as its base. Therefore, it has 2 symbols: The numbers are 0 and 1.

And a hexadecimal number system uses the number 16 as its base i.e. it has 16 symbols, hexadecimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A,B,C,D,E and F.

Binary numbers are represented as from hexadecimal number

    Hexadecimal01234567
    Decimal01234567
    Binary00000001001000110100010101100111
    Hexadecimal89ABCDEF
    Decimal89101112131415
    Binary10001001101010111100110111101111

Each hexadecimal digit consists of 4 binary digits.

For example hexadecimal number 9 is equal to binary number 1001.

For converting hexadecimal numbers into binary numbers write down the hexadecimal numbers and represent each hexadecimal digit by its binary digit from the table.

Then combine all the digits together.

Same process follows for integer as well as fractional part.

Hexadecimal digits are equal to the summation of 2n where n = 0, 1, 2 and 3 (position from right)

For example 9 = 23+20; in this example 21 and 22 is not exist so at position 1 and 2 binary digit is zero and at position 0 and 3 binary digit is one; so binary of this digit "9" is

1 0 0 1

↓ ↓ ↓ ↓

23 22 21 20

hexadecimalnumber=B70905.F 8 16

Nowconsideringindividualdigits

hexadecimaldigit=B=11= 2 3 +0+ 2 1 + 2 0

hexadecimaldigit=8+0+2+1

hexadecimaldigit=1×8+0×4+1×2+1×1

hexadecimaldigit= 1 × 2 3 + 0 × 2 2 + 1 × 2 1 + 1 × 2 0

Sobinarynumber= 1011

hexadecimaldigit=7=0+ 2 2 + 2 1 + 2 0

hexadecimaldigit=0+4+2+1

hexadecimaldigit=0×8+1×4+1×2+1×1

hexadecimaldigit= 0 × 2 3 + 1 × 2 2 + 1 × 2 1 + 1 × 2 0

Sobinarynumber= 0111

hexadecimaldigit=0=0+0+0+0

hexadecimaldigit=0+0+0+0

hexadecimaldigit=0×8+0×4+0×2+0×1

hexadecimaldigit= 0 × 2 3 + 0 × 2 2 + 0 × 2 1 + 0 × 2 0

Sobinarynumber= 0000

hexadecimaldigit=9= 2 3 +0+0+1

hexadecimaldigit=8+0+0+1

hexadecimaldigit=1×8+0×4+0×2+1×1

hexadecimaldigit= 1 × 2 3 + 0 × 2 2 + 0 × 2 1 + 1 × 2 0

Sobinarynumber= 1001

hexadecimaldigit=0=0+0+0+0

hexadecimaldigit=0+0+0+0

hexadecimaldigit=0×8+0×4+0×2+0×1

hexadecimaldigit= 0 × 2 3 + 0 × 2 2 + 0 × 2 1 + 0 × 2 0

Sobinarynumber= 0000

hexadecimaldigit=5=0+ 2 2 +0+ 2 0

hexadecimaldigit=0+4+0+1

hexadecimaldigit=0×8+1×4+0×2+1×1

hexadecimaldigit= 0 × 2 3 + 1 × 2 2 + 0 × 2 1 + 1 × 2 0

Sobinarynumber= 0101

hexadecimaldigit=F=15= 2 3 + 2 2 + 2 1 + 2 0

hexadecimaldigit=8+4+2+1

hexadecimaldigit=1×8+1×4+1×2+1×1

hexadecimaldigit= 1 × 2 3 + 1 × 2 2 + 1 × 2 1 + 1 × 2 0

Sobinarynumber= 1111

hexadecimaldigit=8= 2 3 +0+0+0

hexadecimaldigit=8+0+0+0

hexadecimaldigit=1×8+0×4+0×2+0×1

hexadecimaldigit= 1 × 2 3 + 0 × 2 2 + 0 × 2 1 + 0 × 2 0

Sobinarynumber= 1000

combineallthebinarydigitstogether= ( 101101110000100100000101.11111000 ) 2

To determine

(b)

Conversion of hexadecimal numbers B70905.F816 to a decimal numbers.

Expert Solution
Check Mark

Answer to Problem 1A

Decimal numbers is 11995397.9687510.

Explanation of Solution

Given information:

A hexadecimal numbers B70905.F816.

Calculation:

Decimal number system uses the number 10 as its base. Therefore, it has 10 symbols: The numbers from 0 to 9; namely 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

And a hexadecimal number system uses the number 16 as its base i.e. it has 16 symbols, 10 as decimal symbol, the extra needed 6 digits are represented by the first 6 letters of english alphabet. Hence, hexadecimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 A,B,C,D,E and F.

Hexadecimal numbers are represented as

    Hexadecimal0123456789ABCDEF
    Decimal0123456789101112131415

Conversion of hexadecimal number into decimal number the following steps are used

For integer part

Ones place is multiply with 160

Tens place is multiply with 161

hundreds place is multiply with 162

and so on...

and for fractional part

Tenths place is multiply by 16-1

hundredths is multiply place by 16-2

and so on...

Now converting hexadecimal number into decimal number in tabular form

    PlaceOne lakhTen thousandsThousandsHundredsTensOnesDecimal pointTensHundreds
    Hexadecimal numberB70905.F8
    Multiplier165164163162161160.16-116-2
    Decimal numberB×1657×1640×1639×1620×1615×160F×16-18×16-2

So decimal number is

  =B×165+7×164+0×163+9×162+0×161+5×160+F×161+8×162=11×1048576+7×65536+0×4096+9×256+0×16+5×1+15×0.0625+8×0.0039062511534336+458752+0+2304+0+5+0.9375+0.0312511995397.9687510

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