Page 45 - Computer Science V2.0 Class 11

Given two non-negative numbers n and m (m ≠ 0), using the division algorithm, we can write n = m.q + r, where
0<=r<m, for a suitable choice of q and r. Thus, q is the quotient obtained by dividing n by m, and r is the remainder.
Now, let us rewrite the above computations using the division algorithm, which reveals more clearly why the above
method of converting a number from base 10 to base 2 works.
70

= 2 × 35 + 0
0
= 2 × 35 + 2 × 0

0
= 2 × (2 × 17 + 1) + 2 × 0
= 2 × 17 + 2 × 1 + 2 × 0
2
0
1
0
1
2
= 2 × (2 × 8 + 1) + 2 × 1 +2 × 0

3
= 2 × 8 + 2 × 1 + 2 × 1 + 2 × 0
2
1
0

0
1
2
3
= 2 × (2 × 4 + 0) + 2 × 1 + 2 × 1 + 2 × 0
4
1
0
3
2
= 2 × 4 +2 × 0 + 2 × 1 + 2 × 1 + 2 × 0
1
0
3
2

4
= 2 × (2 × 2 +0) + 2 × 0 +2 × 1 + 2 × 1 + 2 × 0
2
1
4
0
3
= 2 × 2 + 2 × 0 + 2 × 0 + 2 × 1 + 2 × 1 + 2 × 0
5
4
2
1
3
= 2 × (2 × 1 + 0) + 2 × 0 +2 × 0 + 2 × 1 + 2 × 1 + 2 × 0
5

0
2
0
3
= 2 × 1 +2 × 0 + 2 × 0 +2 × 0 +2 × 1 + 2 × 1 + 2 × 0
5
1
6
4
Thus, we have expressed 70 in terms of a sum of the powers of 2:
2
3
4
1
5
0
6
70 = 2 × 1 +2 × 0 +2 × 0 +2 × 0 +2 × 1 + 2 × 1 + 2 × 0
So, we can write binary representation of 70, as a sequence of coefficients of powers of 2 (beginning the coefficient of
the highest power of 2) in the above sum as follows:
(70 ) = (1000110) 2
10
Find binary equivalents of 5, 8, 16.
Binary to Decimal Conversion
Let us convert a binary number to decimal. For this, we use the place values with a base value of 2. The conversion
process involves the following steps:
1. Write the binary number
2. Below each digit, write the place value of the digit, represented as a power of 2.
3. Multiply the face value of each digit with its place value.
4. Add all values obtained in step 3 to get the decimal number.
In example 1, (1000110) is the binary representation of the decimal number (70) . So, if we express the binary
10
2
number (1000110) as a decimal number, we must get the original number (70) shown below:
10
2
Find decimal equivalents of 1001, 1100, and 110.
Number Systems and Encoding Schemes 31