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Unit A6 Section 4
Adding and Subtracting Binary Numbers

It is possible to add and subtract binary numbers in a similar way to base 10 numbers.
For example, 1 + 1 + 1 = 3 in base 10 becomes 1 + 1 + 1 = 11 in binary.
In the same way, 3 − 1 = 2 in base 10 becomes 11 − 1 = 10 in binary. When you add and subtract binary numbers you will need to be careful when 'carrying' or 'borrowing' as these will take place more often.

Key Addition Results for Binary Numbers

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

Key Subtraction Results for Binary Numbers

1−0=1
10−1=1
11−1=10

Worked Examples

1

Calculate, using binary numbers:

(a)
111 + 100
111
+100
1011
1

Note how important it is to 'carry' correctly.
(b)
101 + 110
101
+110
1011
1
(c)
1111 + 111
1111
+111
10110
111
2

Calculate the binary numbers:

(a)
111 − 101
111
−101
10
(b)
110 − 11
110
−11
11
(c)
1100 − 101
1100
−101
111

Exercises

Calculate the binary numbers:

(a)
11 + 1
(b)
11 + 11
(c)
111 + 11
(d)
111 + 10
(e)
1110 + 111
(f)
1100 + 110
(g)
1111 + 10101
(h)
1100 + 11001
(i)
1011 + 1101
(j)
1110 + 10111
(k)
1110 + 1111
(l)
11111 + 11101

Calculate the binary numbers:

(a)
11 − 10
(b)
110 − 10
(c)
1111 − 110
(d)
100 − 10
(e)
100 − 11
(f)
1000 − 11
(g)
1101 − 110
(h)
11011 − 110
(i)
1111 − 111
(j)
110101 − 1010
(k)
11011 − 111
(l)
11110 − 111

Calculate the binary numbers:

(a)
11 + 11
(b)
111 + 111
(c)
1111 + 1111
(d)
11111 + 11111

The answers always end in a single zero and all the other digits are ones. Also, the number of ones increases by 1 each time.

Alternatively, the answer is the binary number in the question with an extra zero on the right hand end. This is because adding a number to itself is the same as doubling, which in binary means multiplying by 10, so you just add a zero onto the right hand end of the number you are adding to itself.

Solve the following equations, where all numbers, including x, are binary:

(a)

x + 11 = 1101

x =
(b)

x − 10 = 101

x =
(c)

x − 1101 = 11011

x =
(d)

x + 1110 = 10001

x =
(e)

x + 111 = 11110

x =
(f)

x − 1001 = 11101

x =

Calculate the binary numbers:

(a)
10 − 1
(b)
100 − 1
(c)
1000 − 1
(d)
10000 − 1

The answers only involve the digit one. Also, the number of ones increases by 1 each time. The number of ones is equal to the number of zeros in the original calculation.

Here are 3 binary numbers:

111010110111101010011

Working in binary,

(a)

add together the two smaller numbers,

(b)

add together the two larger numbers,

(c)

take the smallest number away from the largest number,

(d)

add together all three numbers.