The remainder and quotient of the binary division (101110) 2by (110) 2are respectively
(100) 2and (111) 2
Understanding Binary Division
Binary division is similar to decimal division, but it involves only two digits: 0 and 1. We perform division using the long division method, following the same principles as decimal long division. In this problem, we need to divide the binary number (101110)2 by (110)2 to find the remainder and quotient.
Step-by-Step Binary Long Division
Let's perform the binary long division of (101110)2 by (110)2.
The dividend is 101110 and the divisor is 110.
| Step | Process | Result | Quotient Bit |
|---|---|---|---|
| 1 | Compare divisor (110) with the first few bits of the dividend. 101 is less than 110. Consider the first four bits: 1011. How many times does 110 go into 1011? It goes 1 time (since $110 \times 1 = 110$, and $110 \times 2 = 1100$ which is greater than 1011). | $1011 - 110 = 101$ | 1 |
| 2 | Bring down the next bit from the dividend (which is 1). The new number is 1011. Compare 110 with 1011. It goes 1 time. | $1011 - 110 = 101$ | 1 |
| 3 | Bring down the next bit from the dividend (which is 0). The new number is 1010. Compare 110 with 1010. It goes 1 time. | $1010 - 110 = 100$ | 1 |
Let's visualize the long division process:
111 <-- Quotient
_______
110 | 101110 <-- Dividend
- 110
-----
1011
- 110
-----
1010
- 110
-----
100 <-- Remainder
From the long division, we find:
- The quotient is (111)2.
- The remainder is (100)2.
To verify, we can convert the numbers to decimal:
- (101110)2 = $1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0$ = $32 + 0 + 8 + 4 + 2 + 0 = 46$
- (110)2 = $1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 4 + 2 + 0 = 6$
- (111)2 (Quotient) = $1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 4 + 2 + 1 = 7$
- (100)2 (Remainder) = $1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 = 4 + 0 + 0 = 4$
In decimal, $46 \div 6$ gives a quotient of 7 and a remainder of 4. This matches our binary results.
Binary Division Results
The remainder of the binary division (101110)2 by (110)2 is (100)2.
The quotient of the binary division (101110)2 by (110)2 is (111)2.
The question asks for the remainder and quotient respectively. Therefore, the answer is (100)2 and (111)2.
Revision Table: Binary Division
| Concept | Description | Example |
|---|---|---|
| Binary Number System | A base-2 number system using only digits 0 and 1. | (101)2 = 5 in decimal |
| Binary Division | Arithmetic operation to divide one binary number by another. | Similar to decimal long division. |
| Quotient | The result of the division. | (111)2 in this problem. |
| Remainder | The amount left over after the division. | (100)2 in this problem. |
Additional Information on Binary Arithmetic
Binary arithmetic is fundamental in digital electronics and computer science. Besides division, other binary operations include addition, subtraction, and multiplication. Understanding these operations is key to understanding how computers perform calculations.
- Binary Addition: $0+0=0$, $0+1=1$, $1+0=1$, $1+1=0$ (with a carry of 1).
- Binary Subtraction: $0-0=0$, $1-0=1$, $1-1=0$, $0-1=1$ (with a borrow from the next position).
- Binary Multiplication: $0 \times 0=0$, $0 \times 1=0$, $1 \times 0=0$, $1 \times 1=1$. This involves shifts and additions, similar to decimal multiplication.
Binary division uses binary subtraction at each step to determine how many times the divisor fits into the current part of the dividend.
Similar Questions
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Important Questions from Binary Operations
Multiplication of 111 2by 101 2is
Let * be binary operation defined on R by \(\rm p * q=\frac{p+q}{2}, \forall \) p, q ∈ R. The operation is:
Convert 29 into binary.
A. 10101
B. 11110
C. 11101
D. 11001
The product of the two binary numbers 011 and 110 is:
If G is the set of integer numbers, and a.b = a – b, ∀ a, b ∈ G, then G is
