Chapter 86, Problem 1A

To determine

(a)

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

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

Hexadecimal	0	1	2	3	4	5	6	7
Decimal	0	1	2	3	4	5	6	7
Binary	0000	0001	0010	0011	0100	0101	0110	0111

Hexadecimal	8	9	A	B	C	D	E	F
Decimal	8	9	10	11	12	13	14	15
Binary	1000	1001	1010	1011	1100	1101	1110	1111

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

$hexadecimalnumber=B70905.F{8}_{16}$

$Nowconsideringindividualdigits$

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

$hexadecimaldigit=8+0+2+1$

$hexadecimaldigit=1\times 8+0\times 4+1\times 2+1\times 1$

$hexadecimaldigit=\overline{)1}\times 2^3+\overline{)0}\times 2^2+\overline{)1}\times 2^1+\overline{)1}\times 2^0$

$Sobinarynumber=\boxed{1011}$

$hexadecimaldigit=7=0+2^2+2^1+2^0$

$hexadecimaldigit=0+4+2+1$

$hexadecimaldigit=0\times 8+1\times 4+1\times 2+1\times 1$

$hexadecimaldigit=\overline{)0}\times 2^3+\overline{)1}\times 2^2+\overline{)1}\times 2^1+\overline{)1}\times 2^0$

$Sobinarynumber=\boxed{0111}$

$hexadecimaldigit=0=0+0+0+0$

$hexadecimaldigit=0+0+0+0$

$hexadecimaldigit=0\times 8+0\times 4+0\times 2+0\times 1$

$hexadecimaldigit=\overline{)0}\times 2^3+\overline{)0}\times 2^2+\overline{)0}\times 2^1+\overline{)0}\times 2^0$

$Sobinarynumber=\boxed{0000}$

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

$hexadecimaldigit=8+0+0+1$

$hexadecimaldigit=1\times 8+0\times 4+0\times 2+1\times 1$

$hexadecimaldigit=\overline{)1}\times 2^3+\overline{)0}\times 2^2+\overline{)0}\times 2^1+\overline{)1}\times 2^0$

$Sobinarynumber=\boxed{1001}$

$hexadecimaldigit=0=0+0+0+0$

$hexadecimaldigit=0+0+0+0$

$hexadecimaldigit=0\times 8+0\times 4+0\times 2+0\times 1$

$hexadecimaldigit=\overline{)0}\times 2^3+\overline{)0}\times 2^2+\overline{)0}\times 2^1+\overline{)0}\times 2^0$

$Sobinarynumber=\boxed{0000}$

$hexadecimaldigit=5=0+2^2+0+2^0$

$hexadecimaldigit=0+4+0+1$

$hexadecimaldigit=0\times 8+1\times 4+0\times 2+1\times 1$

$hexadecimaldigit=\overline{)0}\times 2^3+\overline{)1}\times 2^2+\overline{)0}\times 2^1+\overline{)1}\times 2^0$

$Sobinarynumber=\boxed{0101}$

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

$hexadecimaldigit=8+4+2+1$

$hexadecimaldigit=1\times 8+1\times 4+1\times 2+1\times 1$

$hexadecimaldigit=\overline{)1}\times 2^3+\overline{)1}\times 2^2+\overline{)1}\times 2^1+\overline{)1}\times 2^0$

$Sobinarynumber=\boxed{1111}$

$hexadecimaldigit=8=2^3+0+0+0$

$hexadecimaldigit=8+0+0+0$

$hexadecimaldigit=1\times 8+0\times 4+0\times 2+0\times 1$

$hexadecimaldigit=\overline{)1}\times 2^3+\overline{)0}\times 2^2+\overline{)0}\times 2^1+\overline{)0}\times 2^0$

$Sobinarynumber=\boxed{1000}$

$combineallthebinarydigitstogether={\left(101101110000100100000101.11111000\right)}_2$

To determine

(b)

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

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

Hexadecimal	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
Decimal	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15

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

Place	One lakh	Ten thousands	Thousands	Hundreds	Tens	Ones	Decimal point	Tens	Hundreds
Hexadecimal number	B	7	0	9	0	5	.	F	8
Multiplier	165	164	163	162	161	160	.	16-1	16-2
Decimal number	B×165	7×164	0×163	9×162	0×161	5×160		F×16-1	8×16-2

So decimal number is

  $\begin{array}{l}=B\times {16}^5+7\times {16}^4+0\times {16}^3+9\times {16}^2+0\times {16}^1+5\times {16}^0+F\times {16}^{-1}+8\times {16}^{-2}\\ =11\times 1048576+7\times 65536+0\times 4096+9\times 256+0\times 16+5\times 1+15\times 0.0625+8\times 0.00390625\\ 11534336+458752+0+2304+0+5+0.9375+0.03125\\ {11995397.96875}_{10}\end{array}$