Question

1. Add the following binary number in 8-bit. Rewrite each problem in decimal notation to check you work. 10110101112 + 011001002

2. Find the decimal equivalent for the following binary numbers. 10010102/100000002

3. Find the decimal equivalent for the following binary numbers. 0.10112

4. Find the binary equivalent the following decimal numbers. 8.75

Answer 1

Binary Number System:

It has base '2' i.e it has two base numbers 0 and 1 these base numbers are called "Bits".

In this number system, group of 4 bits is known as "Nibble" and group of eight bits is known as "Byte".

Computers use the binary system for  implementation any digital circuit because It is very simple to design any hardware which needs only to detect two states, on and off .

Decimal Number System

It has 'base 10' i.e it has 10 distinct symbols(0,1,2,3,4,5,6,7,8,9)

Converting from the binary to the decimal system

Procedure: 1. Determine all of the place values where 1 occurs

2. find the sum of the those values.

EX: 10101 = (1 × 2⁴) + (0 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰) = 21

24	23	22	21	20
1	0	1	0	1
16	0	4	0	1

Hence: 16 + 4 + 1 = 21.

Binary Addition

Binary addition follows the same rules as addition in the decimal system only difference is that carrying a 1 over when the values added equal 10, carry over occurs when the result of addition equals 2.

Note that in the binary system:

• 0 + 0 = 0
  0 + 1 = 1
  1 + 0 = 1
  1 + 1 = 0, carry over the 1, i.e. 10

1 + 1 +1 = 1, carry over the 1, i.e. 11

EX:

		10	11	11	0	1
+		1	0	1	1	0
=	1	0	0	1	1	1

1. Add the following binary in 8 bit

Given Numbers are: 1011010111₂ + 01100100₂

carry bit		1	1				1
	1	0	1	1	0	1	0	1	1	1
			0	1	1	0	0	1	0	0
+
Result :	1	1	0	0	1	1	1	0	1	1

Sum : 1100111011

Decimal value of (1100111011₂₎ is

= (1 × 2⁹)+(1 × 2⁸)+(0 × 2⁷)+(0 × 2⁶)+(1 × 2⁵)+(1 × 2⁴) + (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰)

= 512+256+0+0+32+16+8+0+2+1

=827

Rewrite each problem in decimal to check your work.

1011010111₂ + 01100100₂

convert the given binary value into decimal

1011010111₂ = (1 × 2⁹)+(0 × 2⁸)+(1 × 2⁷)+(1 × 2⁶)+(0 × 2⁵)+(1 × 2⁴) + (0 × 2³) + (1 × 2²) + (1 × 2¹) + (1 × 2⁰)

= 512+0+128+64+0+16+0+4+2+1

=727

01100100₂ =(0 × 2⁷)+(1 × 2⁶)+(1 × 2⁵)+(0 × 2⁴) + (0 × 2³) + (1 × 2²) + (0× 2¹) + (0 × 2⁰)

= 0+64+32+0+0+4+0+0

= 100

Addition of two decimal value 727+100 = 827

Both decimal and binary results are same

2. Find the decimal equivalent for the following binary numbers 1001010₂/10000000₂

First here we divided that given binary numbers 1001010₂/10000000₂ using binary division method and then i converted that result into a decimal equivalent.

Binary Division

Binary division is similar to decimal division method

Each step of binary division having these four sub steps:

Divide: Divide the working portion of the dividend by the divisor.

Multiply: Multiply the quotient (a single digit) by the divisor.

Subtract: Subtract the product from the working portion of the dividend.

Bring down: Copy down the next digit of the dividend to form the new working portion.

In that given problem Dividend : 1001010₂

Divisor : 10000000₂

In a given value the Dividend have only 7 digits binary number, so make it to 8 digit number we can add 0 in the Most significant position, then it becomes 01001010₂.

10000000) 01001010 ( 0 (quotient value) 00000000 ______________ 01001010 (Remainder value)

Decimal equivalent of 01001010₂ is = 74

10000000₂ is = 128

74/128 = 0 (Remainder is 74)

3. Find the decimal equivalent for the following binary numbers. 0.1011₂

0.1011 =  (0 × 2⁰) . (1 × 2^-1) + (0 × 2^-2) + (1 × 2^-3) + (1× 2^-4)

= 0 . (1 * 0.5) + 0 + (1 * 0.125) + (1 * 0.0625)

= 0.6875

4. Find the binary Equivalent the following decimal numbers 8.75

Steps to convert the decimal into binary number

1. Divide the given number by 2.
2. Get the integer quotient for the next division.
3. take the remainder for the binary result.
4. Repeat the steps until the quotient is equal to 0.

In the given problem we have two part one is integral part and fractional part.

Now consider the integral part 8 and do the following steps

Divide by 2	Quotient	Remainder
8/2	4	0
4/2	2	0
2/2	1	0
1/2	0	1

Write the remainder in reverse order to get the Result (8)₁₀: 1000₂

Now the fractional part 0.75 and do the following steps

   Multiply the fractional part repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:

multiplying = integer + fractional part

0.75 × 2 = 1 + 0.5

0.5 × 2 =  1 + 0

(0.75)₁₀ = (11)₂

(8.75)₁₀ = (1000.11)₂