Page 21 - ComputerScience_Class_11

1.2.1 Decimal to Binary
Firstly, the integer part is converted into binary and then the fractional part is converted. The steps to be followed are:
1. For integer part, divide the number by 2 and store the remainder of the division separately.
2. Assign quotient as the new dividend.
3. Repeat steps 1 & 2 until new dividend = 0.
4. Arrange the remainders from bottom (LSB) to top (MSB) to get the result.
5. For decimal part, repeat step 6 until the fractional part becomes 0 or up to 4 decimal places.
6. Multiply fractional part by 2 and store the integer part of the product separately.
7. Arrange the binary digits of the integer part from top to bottom for the result of the fractional part.

We usually read data in decimal representation. However, digital systems internally represent and process numbers in
binary form. The circuits inside computing devices convert decimal numbers into binary for digital computation.
Let us understand with an example:
Example: Convert (77.45) to binary.
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1. On dividing 77 by 2, we get 38 as quotient and 2 77
1 as remainder. 38 (quotient) 1 (First remainder)
2. Now, 38 is the new dividend. On dividing it by 2 38
2, we get 19 as quotient and 0 as remainder. 19 (quotient) 0 (Second remainder)

3. On dividing 19 by 2, we get 9 as quotient and 2 19
1 as remainder. 9 (quotient) 1 (Third remainder)

4. Dividing 9 by 2 gives 4 as quotient and 1 as 2 9
remainder. 4 (quotient) 1 (Fourth remainder)

5. Dividing 4 by 2 gives 2 as quotient and 0 as 2 4
remainder. 2 (quotient) 0 (Fifth remainder)
6. Dividing 2 by 2 gives 1 as quotient and 0 as 2 2
remainder. 1 (quotient) 0 (Sixth remainder)
7. Dividing 1 by 2 gives 0 as quotient and 1 2 1
as remainder. As quotient 0 is reached, the 0 (quotient) 1 (Seventh remainder)
remainders are read upwards.
8. Multiplying fractional part 0.45 by 2 gives 0.90.
Store integral part 0. 0.45 × 2 = 0.90 Integral part = 0
9. Multiplying fractional part 0.90 by 2 gives 1.80.
Integral part 1 is separated. 0.90 × 2 = 1.80 Integral part = 1
10. Multiplying fractional part 0.80 by 2 gives 1.60.
Keep 1 separately. 0.80 × 2 = 1.60 Integral part = 1
11. Multiplying fractional part 0.60 by 2 gives 1.20.
Keep 1 separately. The process will continue 0.60 × 2 = 1.20 Integral part = 1
infinitely. Thus, we stop after four decimal
places to take the approximate answer by
arranging from top to bottom.
Corresponding binary number will be
(1001101.0111) .
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System of Numeration 19