Addition of Binary Numbers for JSS 1

Addition of Numbers in Base 2

Performance Objectives

By the end of the lesson, students should be able to:

1. Add two- or three-digit binary numbers.

Introduction to Binary Numbers

A binary number is a number expressed in the base-2 numeral system, which uses only two symbols: 0 and 1.

In everyday life, counting is done in base 10 (denary), which uses ten digits from $0$ to $9$. In base 2 (binary), counting is based on groups of twos. The place values in base 2 increase by multiplying by $2$ at each step from right to left.

The place values for a three-digit binary number are:

1. Fours ($2^2$ or $4$) in the third column from the right.
2. Twos ($2^1$ or $2$) in the second column from the right.
3. Units ($2^0$ or $1$) in the first column on the right.

Observe the diagram below showing how base 2 place values are represented using blocks.

Rules of Binary Addition

Adding binary numbers follows a simple set of rules. Because $1$ is the largest digit in base 2, any sum that reaches two ($2_{10}$) must be written as $10_2$. This means writing down $0$ and carrying $1$ to the next column on the left.

The rules for adding binary digits are:

1. $0_2 + 0_2 = 0_2$
2. $0_2 + 1_2 = 1_2$
3. $1_2 + 0_2 = 1_2$
4. $1_2 + 1_2 = 10_2$ (write down $0$ and carry $1$ to the next column)
5. $1_2 + 1_2 + 1_2 = 11_2$ (write down $1$ and carry $1$ to the next column)

Adding Two-Digit Binary Numbers

When adding two-digit binary numbers, the digits are aligned vertically according to their place values (Twos and Units). Addition begins from the rightmost column (Units) and moves to the left.

Worked Example 1

Calculate $10_2 + 11_2$.

Solution:

Align the numbers vertically:

$$
\begin{array}{ccc}
& 1 & 0_2 \\
+ & 1 & 1_2 \\
\hline
1 & 0 & 1_2 \\
\hline
\end{array}
$$

The steps for this addition are:

1. Add the Units column: $0 + 1 = 1$. Write down $1$ in the Units column.
2. Add the Twos column: $1 + 1 = 10$. Write down $0$ in the Twos column and carry $1$ to the Fours column.
3. Bring down the carried $1$ into the Fours column.

Therefore, $10_2 + 11_2 = 101_2$.

Worked Example 2

Calculate $11_2 + 11_2$.

Solution:

Align the numbers vertically:

$$
\begin{array}{ccc}
& 1 & 1_2 \\
+ & 1 & 1_2 \\
\hline
1 & 1 & 0_2 \\
\hline
\end{array}
$$

The steps for this addition are:

1. Add the Units column: $1 + 1 = 10$. Write down $0$ and carry $1$ to the Twos column.
2. Add the Twos column: $1 + 1 = 10$. Add the carried $1$ to get $11$. Write down $1$ and carry $1$ to the Fours column.
3. Bring down the carried $1$ into the Fours column.

Therefore, $11_2 + 11_2 = 110_2$.

Worked Example 3

Calculate $11_2 + 01_2$.

Solution:

Align the numbers vertically:

$$
\begin{array}{ccc}
& 1 & 1_2 \\
+ & 0 & 1_2 \\
\hline
1 & 0 & 0_2 \\
\hline
\end{array}
$$

The steps for this addition are:

1. Add the Units column: $1 + 1 = 10$. Write down $0$ and carry $1$ to the Twos column.
2. Add the Twos column: $1 + 0 = 1$. Add the carried $1$ to get $10$. Write down $0$ and carry $1$ to the Fours column.
3. Bring down the carried $1$ into the Fours column.

Therefore, $11_2 + 01_2 = 100_2$.

Adding Three-Digit Binary Numbers

Adding three-digit binary numbers follows the same method. The columns are aligned under Fours, Twos, and Units.

Worked Example 1

Calculate $101_2 + 110_2$.

Solution:

Align the numbers vertically:

$$
\begin{array}{cccc}
& 1 & 0 & 1_2 \\
+ & 1 & 1 & 0_2 \\
\hline
1 & 0 & 1 & 1_2 \\
\hline
\end{array}
$$

The steps for this addition are:

1. Add the Units column: $1 + 0 = 1$. Write down $1$.
2. Add the Twos column: $0 + 1 = 1$. Write down $1$.
3. Add the Fours column: $1 + 1 = 10$. Write down $0$ and carry $1$ to the next column (Eights column).
4. Bring down the carried $1$.

Therefore, $101_2 + 110_2 = 1011_2$.

Worked Example 2

Calculate $111_2 + 101_2$.

Solution:

Align the numbers vertically:

$$
\begin{array}{cccc}
& 1 & 1 & 1_2 \\
+ & 1 & 0 & 1_2 \\
\hline
1 & 1 & 0 & 0_2 \\
\hline
\end{array}
$$

The steps for this addition are:

1. Add the Units column: $1 + 1 = 10$. Write down $0$ and carry $1$.
2. Add the Twos column: $1 + 0 = 1$. Add the carried $1$ to get $10$. Write down $0$ and carry $1$.
3. Add the Fours column: $1 + 1 = 10$. Add the carried $1$ to get $11$. Write down $1$ and carry $1$.
4. Bring down the carried $1$.

Therefore, $111_2 + 101_2 = 1100_2$.

Worked Example 3

Calculate $110_2 + 011_2$.

Solution:

Align the numbers vertically:

$$
\begin{array}{cccc}
& 1 & 1 & 0_2 \\
+ & 0 & 1 & 1_2 \\
\hline
1 & 0 & 0 & 1_2 \\
\hline
\end{array}
$$

The steps for this addition are:

1. Add the Units column: $0 + 1 = 1$. Write down $1$.
2. Add the Twos column: $1 + 1 = 10$. Write down $0$ and carry $1$.
3. Add the Fours column: $1 + 0 = 1$. Add the carried $1$ to get $10$. Write down $0$ and carry $1$.
4. Bring down the carried $1$.

Therefore, $110_2 + 011_2 = 1001_2$.

Using Binary Charts and Blocks for Addition

Binary charts and base 2 blocks help to show how regrouping works during addition. In base 2, whenever two blocks of the same size are gathered, they must be exchanged for one larger block of the next place value.

The block values are:

1. A single unit block represents $1$.
2. A block of two units joined together represents $2$ (a “two-block”).
3. A block of four units joined together represents $4$ (a “four-block”).

Look at the diagram below showing how blocks are combined and exchanged during addition.

How to Add Binary Numbers Using Blocks

This practical activity demonstrates how to add $11_2$ and $01_2$ using physical base 2 blocks.

The required materials are:

1. Single unit blocks.
2. Two-blocks (two units joined together).
3. Four-blocks (four units joined together).

The procedure involves the following steps:

1. Represent the first number ($11_2$) by placing one two-block in the Twos column and one unit block in the Units column.
2. Represent the second number ($01_2$) below the first by placing one unit block in the Units column.
3. Gather all the blocks in the Units column. This gives a total of two unit blocks.
4. Exchange the two unit blocks for one two-block. Place this new two-block at the top of the Twos column as a carry. There are now zero unit blocks left in the Units column.
5. Gather all the blocks in the Twos column. This includes the original two-block and the carried two-block, giving a total of two two-blocks.
6. Exchange the two two-blocks for one four-block. Place this four-block in the Fours column. There are now zero two-blocks left in the Twos column.
7. Count the final blocks in each column: one four-block, zero two-blocks, and zero unit blocks. This represents the binary number $100_2$.

Binary Number Chart

A binary number chart can also be used to keep digits aligned during addition. The table below shows how the addition of $111_2 + 101_2$ is structured on a place value chart.

The final answer read from the chart is $1100_2$.

Questions for Evaluation

Solve the following binary addition problems:

1. $10_2 + 11_2$
2. $11_2 + 01_2$
3. $11_2 + 11_2$
4. $101_2 + 100_2$
5. $110_2 + 111_2$
6. $111_2 + 011_2$