BINARY NUMBERS

CONTENT

1. Addition of Numbers in Base \(2\) Numerals: Addition of Two- or Three-digit Binary Numbers
2. Subtraction of Numbers in Base \(2\) Numerals: Subtraction of Two- or Three-digit Binary Numbers

Addition of Numbers in Base 2 Numerals

Addition of Two-digit Binary Numbers

When adding binary numbers, it is important to note that:

\(0 + 0 = 0\)

\(1 + 0 = 1\)

\(0 + 1 = 1\)

\(1 + 1 = 10\)

Examples:

Find the sum of the following binary numbers:

(a) \(10_2 , 11_2\)  (b) \(11, 11\)  (c) \(10_2 , 01_2\)  (d) \(10_2 , 10_2\)  (e) \(101_2 , 111_2\)  (f) \(111_2, 101_2\) and \(100_2\)

Solution:

(a) \(\begin{array}{@{}rrr} 1 0_2  \\  + 1 1_2 \\ \hline = 1 0 1_2 \\ \hline \end{array}\)    (b) \(\begin{array}{@{}rrr} 1 1_2  \\  + 1 1_2 \\ \hline = 1 1 0_2 \\ \hline \end{array}\)

(c) \(\begin{array}{@{}rrr} 1 0_2  \\  + 0 1_2 \\ \hline = 1 1_2 \\ \hline \end{array}\)    (d) \(\begin{array}{@{}rrr} 1 0_2  \\  + 1 0_2 \\ \hline = 1 0 0_2 \\ \hline \end{array}\)

(e) \(\begin{array}{@{}rrr} 1 0 1_2  \\  + 1 1 1_2 \\ \hline = 1 1 1 0_2 \\ \hline \end{array}\)    (f) \(\begin{array}{@{}rrr} 1 1 1_2  \\  1 0 1_2 \\ + 1 0 0_2 \\ \hline = 1 0 1 0 0 0_2 \\ \hline \end{array}\)

CLASS ACTIVITY  

1. Find the sum of the following binary numbers: \(110_2, 101_2\)
2. Obtain the sum of \(111_2, 101_2\)

Subtraction of Two- and Three-digit Binary Numbers

When subtracting binary numbers:

\(0 -0 = 0\)

\(1 -0 = 1\)

\(1 -1 = 0\)

\(10 -1 = 1\)

Examples:

Find the difference of the following binary numbers:  (a) \(11_2, 10_2\)  (b) \(111_2, 110_2\)  (c) \(110_2, 100_2\)  (d) \(11011_2, 10011_2\)

Solutions:

(a) \(\begin{array}{@{}rrr} 1 1_2  \\  – 1 0_2 \\ \hline = 0 1_2 \\ \hline \end{array}\)    (b) \(\begin{array}{@{}rrr} 1 1 1_2  \\  – 1 1 0_2 \\ \hline = 0 0 1_2 \\ \hline \end{array}\)

(c) \(\begin{array}{@{}rrr} 1 1 0_2  \\  – 1 0 0_2 \\ \hline = 1 0 1 0_2 \\ \hline \end{array}\)    (d) \(\begin{array}{@{}rrr} 1 1 0 1 1_2  \\  – 1 0 0 1 1_2 \\ \hline = 0 1 0 0 0_2 \\ \hline \end{array}\)

CLASS ACTIVITY   

1. Obtain the difference between \(110110_2, 101101_2\)
2. Calculate the value of \(10011_2, -1011_2\)

ASSIGNMENT

Calculate the sum or difference of the following set of binary numbers:

(a) \(\begin{array}{@{}rrr} 1 1 011010  \\  – 1 1010111 \\ \hline   \\ \hline \end{array}\)    (b) \(\begin{array}{@{}rrr} 1 1 0 1 0111  \\  – 011 0 1 100 \\ \hline \\ \hline \end{array}\)

(c) \(\begin{array}{@{}rrr} 1 1 011010  \\  – 1 1010111 \\ \hline  \\ \hline \end{array}\)    (d) \(\begin{array}{@{}rrr} 1 0001100 \\  + 10010010 \\ \hline \\ \hline \end{array}\)

(e) \(\begin{array}{@{}rrr} 1 0010010  \\  + 1 0010111 \\ \hline  \\ \hline \end{array}\)    (f) \(\begin{array}{@{}rrr} 1 00 1 0111  \\  + 1100 0 1 01 \\ \hline \\ \hline \end{array}\)

(g) \(\begin{array}{@{}rrr} 1 11 00011 \\  – 1 1000101 \\ \hline  \\ \hline \end{array}\)    (h) \(\begin{array}{@{}rrr} 1 11 00011  \\  – 0111100 0 \\ \hline \\ \hline \end{array}\)

PRACTICE QUESTIONS

1.