Chapter 85, Problem 30A

To determine

To express:

A hexadecimal number into a binary number.

Answer to Problem 30A

Binary number is 111100001110.100111010101₂.

Explanation of Solution

Given information:

A hexadecimal numbers F0E.9D5₁₆.

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. The 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 number into binary number, write down the hexadecimal number 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 2²are not there. So, at position 1 and 2, binary digit is zero, and at position 0 and 3, binary digit is one. Therefore, hexadecimal of binary 1001 is

1 0 0 1

↓ ↓ ↓ ↓

2³2²2¹2⁰

  $hexadecimalnumber=F0E.9D{5}_{16}$

  $Nowconsideringindividualdigits$

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

  $hexadecimaldigit=8+4+2+1$

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

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

  $Sobinarynumber=\boxed{1111}$

  $hexadecimaldigit=0=0+0+0+0$

  $hexadecimaldigit=0+0+0+0$

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

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

  $Sobinarynumber=\boxed{0000}$

  $hexadecimaldigit=E=14=2^3+2^2+2^1+0$

  $hexadecimaldigit=8+4+2+0$

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

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

  $hexadecimaldigit=9=2^3+0+0+2^0$

  $hexadecimaldigit=8+0+0+1$

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

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

  $Sobinarynumber=\boxed{1001}$

  $hexadecimaldigit=D=13=2^3+2^2+0+2^0$

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

  $hexadecimaldigit=8+4+0+1$

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

  $Sobinarynumber=\boxed{1101}$

  $hexadecimaldigit=5=0+2^2+0+2^0$

  $hexadecimaldigit=0+4+0+1$

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

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

  $Sobinarynumber=\boxed{0101}$

  $combineallthebinarydigitstogether={\left(111100001110.100111100101\right)}_2$