9.1. Binary arithmetic operations

Direct Subtraction

There are four main subtraction expressions namely:

1. 0₂ -0₂ =0₂

2. 1₂ -0₂ =1₂

3. 1₂ -1₂ =0₂

4. 10₂ -1₂ =1₂ (borrow 1 from the next most significant digit to make 0 which

becomes 10₂, hence 10₂ -1₂ =1₂

Examples:

Here are some examples of binary subtraction. These are computed without regard to the word size, hence there can be no sense of "overflow" or "underflow". Work the columns right to left subtracting in each column. If you must subtract a one from a zero, you need to “borrow” from the left, just as in decimal subtraction.

Subtraction using ones complements

The main purpose of using the ones complement in computers is to perform binary subtractions. For example to get the difference in 5-3, using the ones compliment, we proceed as follows:

1. Rewrite the problem as 5 + (-3) to show that the computer performs binary subtraction by adding the binary equivalent of 5 to the ones compliment of 3.

2. Convert the absolute value of 3 into 8-bit or 4-bit for smaller numbers equivalent i.e. 00000011₂.

3. Take the ones compliment of 00000011₂ i.e. 11111100₂ which is the binary representation of -3₁₀.

4. Add the binary equivalent of 5 to the one’s complement of 3 i.e.

00000101

+ 11111100

(1)00000001

Looking at the difference of the two binary numbers, you will observe that:

1. It has a ninth bit. The ninth bit is known as an overflow bit.
2. The result shows that the difference between the two numbers is 00000001. This is not true; we know that it should be 00000010. To address this problem in a system that uses ones compliment the overflow digit is added back to the magnitude of the 8-bit difference. Therefore the difference becomes 00000001+1 = 00000010 which is the correct answer.

Example:

Let's consider how we would solve our problem of subtracting 1₁₀ from 7₁₀ using 1's complement.

Easier method

Using ones compliment evaluate 11₁₀ - 13₁₀

Step 1: Rewrite the equation

thus: 11₁₀ + (- 13₁₀)

Step 2: Convert decimals to binary

11₁₀ = 1011₂

13₁₀ = 1101₂

Step 3: Convert 13₁₀ = 1101₂ into ones compliment (-13₁₀)

in this case 1101₂ is converted to 0010₂ this means that 0s turn to ones and ones turn to zeros

Step 4: insert overflows to show sign bit i.e. positive or negative

11₁₀ = (0)1011₂

13₁₀ = (1)0010₂

zero (0) stands in for a positive sign bit and one (1) stand in for a negative sign bit

Step 5: add (0)1011₂ to (1)0010₂

(1)0010₂

(0)1011₂

Algebraic expression

Let's review the steps for subtracting x from y with an n-bit 1's complement representation.

1. Negate x using 1's complement. Reverse all the bits in x to form -x. Add -x and y.
2. If the sum exceeds n bits, add the extra bit to the result.
3. If the sum does not exceed n bits, leave the result as it is

Subtraction using twos complements

Like in one’s compliment, the two complement of a number is obtained by negating a positive number to its negative counterpart. For example, to get the difference in 5 – 3, using the two’s complement, we proceed as follows:

1. Rewrite the problem as 5 + (-3)

2. Convert the absolute value of 3 into 8-bit binary equivalent i.e. 00000011

3. Take the ones complement of 00000011 i.e. 11111100

4. Add a 1 to the ones compliment i.e. 11111100 to get 11111101

5. Add the binary equivalent of 5 to the twos complement of 3 i.e.

6. Ignoring the overflow bit, the resulting number is 0000010 which is directly read as a binary equivalent of +2

Example:

Now let's consider how we would solve our problem of subtracting 1₁₀ from 7₁₀ using 2's complement.

Trial Questions

1. Work out the following binary subtractions

i. 11001₂ – 1010₂ =

ii. 101₂ – 100₂ =

iii. 11011₂ – 111₂ =

iv. 1100₂ – 011₂ =

v. 111011₂ – 110₂ =

vi. 1000₂ – 11₂ =

vii. 01101₂ – 1011₂ =

viii. 11111111₂ – 10101101₂ =

ix. 11101101₂ – 100111₂ =

x. 100000₂ – 1111₂ =

2. Convert the decimal number -7 to 8-bit binary using ones and twos complement

3. Using 8-bit length, find the twos compliment of the following decimal numbers

A) -31₁₀

B) -28₁₀

C) -5₁₀

4. Using ones and twos complement perform the following arithmetic operations

a) 14₁₀ – 7₁₀ =

b) 28₁₀ – 12₁₀ =

c) 34₁₀ -33₁₀ =

d) 100₁₀ – 50₁₀ =

e) 102₁₀ -64₁₀ =

5. Perform the following subtractions

a) 1000₂ -10₂ =

b) 11111₂ – 1111₂ =

c) 11011₂ – 111₂ =

6. Perform the following in ones complement, showing that a-b is the same as a+(-b) where ‘a’ is the positive number and ‘b’ is the negative number

a) 1000₂ – 10₂ =

b) 11110₂ – 111₂ =

c) 100000₂ – 100₂ =

7. Repeat number 3 above using twos complement

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