 Binary Numbers: Maths Notes for NDA, CDS, AFCAT & CAPF

In the decimal system, we use 10 digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 so its base or radix is 10. In binary system, we use 2 digits 0 and 1. So its base or radix is 2.

Decimal to binary conversion : - To convert integer to binary, start with the integer given in question & divide it by 2, keeping notice of the quotient & the remainders. Continue dividing the quotient by 2 until you get a quotient of zero. Then just write out the remainder in the reverse order.

E.g. — conversion of 180 into a binary number is : —
Ans.10110100

Conversion of fractional decimal into binary: → fractional numbers can be converted to binary form by successive multiplication by 2. In each step, the digit before the decimal point is being transferred to the binary record & process is repeated with the remaining fractional number.

E.g. —Conversion of 0.321 to binary form is:
(0.321)10=(0.0101001000)2

Binary to Decimal conversion : → For converting binary number to decimal number, we start from the least significant bit, i.e. from right, by multiplying them with the powers of 2 in increasing order, i.e. with 2⁰, 2¹, 2² and so on. This process is repeated until the most significant bit, i.e. left bit has been processed. Adding all of them, we get the required decimal no.
E.g. conversion of (10101)₂ to its equivalent decimal no. is
(10101)₂ = 1×2⁴ + 0×2³ + 1×2² + 0×2¹ + 1×2⁰
= 1×16 + 0×8 + 1×4 + 0×2 + 1×1
= 16 + 4 + 1
= 21
(10101)2 = (21)10

Conversion of fractional Binary no. into decimal

In order to convert the binary fractions to decimal numbers, we use negative powers of 2 to the right of the binary point.
e.g. find the decimal equivalent to (0.10110)₂
2(–1)× 1 + 2(–2)× 0 + 2(–3)× 1 + 2(–4)× 1 + 2(–5)× 0
2(–1) + 2(–3)+ 2(–4)

Operations on Binary no.

(a) 0 + 0 = 0
(b) 0 + 1 = 1
(c) 1 + 0 = 1
(d) 1 + 1 = 0 with a carry of 1.

Subtraction:
(a) 0 – 0 = 0
(b) 1 – 0 = 1
(c) 1 – 1 = 0
(d) 0 – 1 = 1

Multiplication:

(a) 0 × 0 = 0
(b) 0 × 1 = 0
(c) 1 × 0 = 0
(d) 1 × 1 = 1

Division:
(a) 1 ÷ 1 = 1
(b) 0 ÷ 1 = 0
(c) 0 ÷ 0 = not defined
(d) 1 ÷ 0 = not defined