Mathematics REVISION OF BINARY NUMBERS FOR NDA

Binary Numbers

ln NDA exam, generally 1-2 questions are asked from this chapter which are based on conversion of binary to decimal and decimal to binary. In the binary system, only two symbols `0` and `1` are used as digits, called binary digits or bits. Since, in this system only two numbers are used. so its base or radix is 2.

DECIMAL SYSTEM

In the decimal system, we use 10 digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Since, 10 basic symbols are used in this system, so its base or radix is 10.

Decimal to Binary Conversion

1. Conversion of Integral Decimal Numbers

`"Step 1:"` Take the LCM of the given decimal number taking only 2 as divisor (as the base or radix of
binary number is 2).

`"Step 2 : "` write the remainder at each step in bracket as shown in the given example.

`"Step 3 : "` Repeat this process until we obtain quotient less than the divisor.

`"Step 4 : "` When we obtain 1 as quotient, start writing the number from there to upward direction,

`:. (21)_10 = (10101)_2`

Therefore, `"21"` of decimal system is equal to `(10101)` of binary system.


`"2. Conversion of Fractional Decimal Numbers-"` Fractional numbers can be converted to binary form by successive multiplication by `2.` In each step, the digit before the decimal point is being transfered to the binary record and the process is repeated with the remaining fractional number. The last step is reached, if the fractional part is zero or it is terminated, when the desired accuracy is attained. The first bit obtained is the most significant and the last is the least significant.

Example, Write `(21) _(10)` into binary number system.

When we obtain 1 as quotient, start writing the number from there to upward direction, as shown

`:. (21)_10 = (10101)_2`

BINARY TO DECIMAL CONVERSION

`"1. Conversion of Integral Binary Numbers-"` 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 `2^0, 2^1, 2^2` 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 number.

`"2. Conversion of Fractional Binary Numbers-"` In order to convert the binary fractions to decimal numbers, we use negative powers of `2` to the right of the binary point.

Arithmetic Operations of Binary Numbers

The arithmetic operations of binary numbers namely addition, subtraction, multiplication and division are almost similar to those of decimal.

Binary Addition

The rules of binary addition are :

(i) `0 + 0 = 0`
(ii) `0 + 1=1`
(iii) `1 + 0=1`
(iv) `1 + 1 = 10` (one-zero, not ten)

`"Note:"` The last rule is often written as `1 + 1 = 0` with a carry of `1`.

Binary Subtraction

The rules of binary subtraction are

(i) `0 - 0 = 0`
(ii) `1 - 0 = 1`
(iii) `1- 1 =0`
(iv) `0-1=1`

Binary Multiplication

The rules of binary multiplication are

(i) `0 xx 0 = 0`
(ii) `0 xx 1 = 0`
(iii) `1 xx 0 = 0`
(iv) `1 xx 1 = 1`

Binary Division

The rules of binary division are

(i) `1 div 1 = 1`
(ii) `0 div 1 = 0`
(iii) ` 0 div 0=` Not defined
(iv) `1 div 0 =` Not defined

 
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