Answer :
- Perform binary multiplication for each pair of binary numbers.
- a) $101 \_2 \times 11 \_2 = 1111 \_2$
- b) $111 \_2 \times 10 \_2 = 1110 \_2$
- c) $1011 \_2 \times 101 \_2 = 110111 \_2$
- d) $1101 \_2 \times 10 \_2 = 11010 \_2$
- e) $1101 \_2 \times 101 \_2 = 1000001 \_2$
- f) $11010 \_2 \times 100 \_2 = 1101000 \_2$
- g) $10110 \_2 \times 110 \_2 = 10000100 \_2$
- h) $10111 \_2 \times 101 \_2 = 1110011 \_2$
- The final results are: $\boxed{a) 1111, b) 1110, c) 110111, d) 11010, e) 1000001, f) 1101000, g) 10000100, h) 1110011}$
### Explanation
1. Understanding Binary Multiplication
We are asked to perform binary multiplication for several pairs of binary numbers. Binary multiplication is similar to decimal multiplication, but we only use 0 and 1.
2. Binary Multiplication of 101 and 11
a) $101 \_2 \\times 11 \_2$:
$\ \ 101$
$\times 11$
$\overline{\ \ 101}$
$\ + 101\ \ $
$\overline{1111}$
So, $101 \_2 \times 11 \_2 = 1111 \_2$
3. Binary Multiplication of 111 and 10
b) $111 \_2 \times 10 \_2$:
$\ \ 111$
$\times 10$
$\overline{\ \ 000}$
$\ + 111\ \ $
$\overline{1110}$
So, $111 \_2 \times 10 \_2 = 1110 \_2$
4. Binary Multiplication of 1011 and 101
c) $1011 \_2 \times 101 \_2$:
$\ \ 1011$
$\times 101$
$\overline{\ \ 1011}$
$\ + 0000\ \ $
$\ + 1011\ \ \ $
$\overline{110111}$
So, $1011 \_2 \times 101 \_2 = 110111 \_2$
5. Binary Multiplication of 1101 and 10
d) $1101 \_2 \times 10 \_2$:
$\ \ 1101$
$\times 10$
$\overline{\ \ 0000}$
$\ + 1101\ \ $
$\overline{11010}$
So, $1101 \_2 \times 10 \_2 = 11010 \_2$
6. Binary Multiplication of 1101 and 101
e) $1101 \_2 \times 101 \_2$:
$\ \ 1101$
$\times 101$
$\overline{\ \ 1101}$
$\ + 0000\ \ $
$\ + 1101\ \ \ $
$\overline{1000001}$
So, $1101 \_2 \times 101 \_2 = 1000001 \_2$
7. Binary Multiplication of 11010 and 100
f) $11010 \_2 \times 100 \_2$:
$\ \ 11010$
$\times 100$
$\overline{\ \ 00000}$
$\ + 00000\ \ $
$\ + 11010\ \ \ $
$\overline{1101000}$
So, $11010 \_2 \times 100 \_2 = 1101000 \_2$
8. Binary Multiplication of 10110 and 110
g) $10110 \_2 \times 110 \_2$:
$\ \ 10110$
$\times 110$
$\overline{\ \ 00000}$
$\ + 10110\ \ $
$\ + 10110\ \ \ $
$\overline{10000100}$
So, $10110 \_2 \times 110 \_2 = 10000100 \_2$
9. Binary Multiplication of 10111 and 101
h) $10111 \_2 \times 101 \_2$:
$\ \ 10111$
$\times 101$
$\overline{\ \ 10111}$
$\ + 00000\ \ $
$\ + 10111\ \ \ $
$\overline{1110011}$
So, $10111 \_2 \times 101 \_2 = 1110011 \_2$
10. Final Results
The results of the binary multiplications are:
a) $1111 \_2$
b) $1110 \_2$
c) $110111 \_2$
d) $11010 \_2$
e) $1000001 \_2$
f) $1101000 \_2$
g) $10000100 \_2$
h) $1110011 \_2$
### Examples
Binary multiplication is used extensively in computer science for designing digital circuits and performing arithmetic operations within computer systems. For example, when a computer adds or multiplies two numbers, it does so using binary arithmetic. Understanding binary multiplication helps in designing efficient algorithms for computers.
- a) $101 \_2 \times 11 \_2 = 1111 \_2$
- b) $111 \_2 \times 10 \_2 = 1110 \_2$
- c) $1011 \_2 \times 101 \_2 = 110111 \_2$
- d) $1101 \_2 \times 10 \_2 = 11010 \_2$
- e) $1101 \_2 \times 101 \_2 = 1000001 \_2$
- f) $11010 \_2 \times 100 \_2 = 1101000 \_2$
- g) $10110 \_2 \times 110 \_2 = 10000100 \_2$
- h) $10111 \_2 \times 101 \_2 = 1110011 \_2$
- The final results are: $\boxed{a) 1111, b) 1110, c) 110111, d) 11010, e) 1000001, f) 1101000, g) 10000100, h) 1110011}$
### Explanation
1. Understanding Binary Multiplication
We are asked to perform binary multiplication for several pairs of binary numbers. Binary multiplication is similar to decimal multiplication, but we only use 0 and 1.
2. Binary Multiplication of 101 and 11
a) $101 \_2 \\times 11 \_2$:
$\ \ 101$
$\times 11$
$\overline{\ \ 101}$
$\ + 101\ \ $
$\overline{1111}$
So, $101 \_2 \times 11 \_2 = 1111 \_2$
3. Binary Multiplication of 111 and 10
b) $111 \_2 \times 10 \_2$:
$\ \ 111$
$\times 10$
$\overline{\ \ 000}$
$\ + 111\ \ $
$\overline{1110}$
So, $111 \_2 \times 10 \_2 = 1110 \_2$
4. Binary Multiplication of 1011 and 101
c) $1011 \_2 \times 101 \_2$:
$\ \ 1011$
$\times 101$
$\overline{\ \ 1011}$
$\ + 0000\ \ $
$\ + 1011\ \ \ $
$\overline{110111}$
So, $1011 \_2 \times 101 \_2 = 110111 \_2$
5. Binary Multiplication of 1101 and 10
d) $1101 \_2 \times 10 \_2$:
$\ \ 1101$
$\times 10$
$\overline{\ \ 0000}$
$\ + 1101\ \ $
$\overline{11010}$
So, $1101 \_2 \times 10 \_2 = 11010 \_2$
6. Binary Multiplication of 1101 and 101
e) $1101 \_2 \times 101 \_2$:
$\ \ 1101$
$\times 101$
$\overline{\ \ 1101}$
$\ + 0000\ \ $
$\ + 1101\ \ \ $
$\overline{1000001}$
So, $1101 \_2 \times 101 \_2 = 1000001 \_2$
7. Binary Multiplication of 11010 and 100
f) $11010 \_2 \times 100 \_2$:
$\ \ 11010$
$\times 100$
$\overline{\ \ 00000}$
$\ + 00000\ \ $
$\ + 11010\ \ \ $
$\overline{1101000}$
So, $11010 \_2 \times 100 \_2 = 1101000 \_2$
8. Binary Multiplication of 10110 and 110
g) $10110 \_2 \times 110 \_2$:
$\ \ 10110$
$\times 110$
$\overline{\ \ 00000}$
$\ + 10110\ \ $
$\ + 10110\ \ \ $
$\overline{10000100}$
So, $10110 \_2 \times 110 \_2 = 10000100 \_2$
9. Binary Multiplication of 10111 and 101
h) $10111 \_2 \times 101 \_2$:
$\ \ 10111$
$\times 101$
$\overline{\ \ 10111}$
$\ + 00000\ \ $
$\ + 10111\ \ \ $
$\overline{1110011}$
So, $10111 \_2 \times 101 \_2 = 1110011 \_2$
10. Final Results
The results of the binary multiplications are:
a) $1111 \_2$
b) $1110 \_2$
c) $110111 \_2$
d) $11010 \_2$
e) $1000001 \_2$
f) $1101000 \_2$
g) $10000100 \_2$
h) $1110011 \_2$
### Examples
Binary multiplication is used extensively in computer science for designing digital circuits and performing arithmetic operations within computer systems. For example, when a computer adds or multiplies two numbers, it does so using binary arithmetic. Understanding binary multiplication helps in designing efficient algorithms for computers.
