Binary Numbers (Base 2) – JSS3 Mathematics Lesson Note (Third Term)

Binary numbers are numbers written in base 2, using only the digits 0 and 1.

Behavioural objectives

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

1. state the place values of a binary number;
2. add binary numbers using the base 2 identities;
3. add binary numbers that have a binary point;
4. subtract binary numbers using the base 2 identities;
5. subtract binary numbers that have a binary point.

What is base 2?

Base 2 is a number system that counts in groups of two, so every place value is a power of 2.

• Digits used: only 0 and 1.
• Another name: the binary system.
• Binary point: the dot that separates whole-number places from fraction places, like the decimal point in base 10.
• Writing the base: a small 2 below the number, as in 101₂.

What are the place values in base 2?

Each place is worth twice the place on its right.

• Places after the binary point: worth 1/2, 1/4, 1/8 and so on.
• Example: 10111₂ = 16 + 0 + 4 + 2 + 1 = 23 in base 10.

How do we add binary numbers?

Binary numbers are added column by column, from right to left, in the same way as ordinary base 10 numbers.

• Rule: if a column total is more than 1, write the last digit and carry the rest to the next column.
• Where to start: the right-hand column.

What are the identities for addition?

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 10: write 0, carry 1.
• 1 + 1 + 1 = 11: write 1, carry 1.
• 1 + 1 + 1 + 1 = 100: write 0, carry 10.

Worked examples

Simplify the following:

1. 1110 + 1001
2. 1111 + 1101 + 101

Solution 1

1. Right column: 0 + 1 = 1. Write 1.
2. Second column: 1 + 0 = 1. Write 1.
3. Third column: 1 + 0 = 1. Write 1.
4. Fourth column: 1 + 1 = 10. Write 0 and carry 1 into a new column.
5. Answer: 10111₂.

Solution 2

1. First column: 1 + 1 + 1 = 11. Write 1, carry 1.
2. Second column: 1 + 0 + 0 + 1 (carried) = 10. Write 0, carry 1.
3. Third column: 1 + 1 + 1 + 1 (carried) = 100. Write 0, carry 10.
4. Fourth column: 1 + 1 + 10 (carried) = 100. Write 0, carry 10.
5. Write the last carry, 10, in front.
6. Answer: 100001₂.

How do we add binary numbers with a binary point?

To add binary numbers with a binary point, place the binary points directly under each other, then add as before.

• Fill gaps with zeros: write 0 at the end of a shorter fraction, such as 10.01 as 10.0100.
• Start at the far right: carry across the binary point as in decimals.
• Keep the point: write it in the answer directly below the points above.

Examples

1. 1.1011₂ + 10.1001₂ + 10.01₂
2. 10.001₂ + 101.111₂

Solution 1

• Answer: 110.1000₂, which is the same as 110.1₂.

Solution 2

• Answer: 1000.000₂, which is the same as 1000₂.

How do we subtract binary numbers?

Binary numbers are subtracted column by column, from right to left, borrowing from the next column when the top digit is smaller.

• Borrowing: take 1 from the next column on the left, which adds 10 to the column you are working in.
• Lender column: it loses 1, so a 1 becomes 0.

What are the identities for subtraction?

• 0 – 0 = 0
• 1 – 0 = 1
• 1 – 1 = 0
• 10 – 1 = 1
• 11 – 1 = 10
• 100 – 1 = 11

Worked examples

Simplify the following:

1. 1110 – 1001
2. 101010 – 111

Solution (a)

1. First column: 0 – 1 is not possible. Borrow from the second column to make 10. Then 10 – 1 = 1.
2. Second column: it is now 0. So 0 – 0 = 0.
3. Third column: 1 – 0 = 1.
4. Fourth column: 1 – 1 = 0.
5. Answer: 101₂.

Solution (b)

• Answer: 100011₂.

How do we subtract binary numbers with a binary point?

To subtract binary numbers with a binary point, place the binary points directly under each other, then subtract as before.

Example

101.101₂ – 11.011₂

• Answer: 10.010₂, which is the same as 10.01₂.

What mistakes should students avoid?

• Using the digit 2: it does not exist in base 2. Write 1 + 1 as 10.
• Forgetting to carry: carry whenever a column total is 10 or more.
• Misplacing the binary point: line up the points before you start.
• Forgetting the borrowed 1: the lender column drops by 1.
• Dropping the base: write the small 2 after the answer.

Evaluation

1. Simplify 101 + 101 + 111.
2. Simplify 10101 + 111.
3. Simplify 11011 – 101.
4. Simplify 1.101₂ + 10.011₂.
5. Simplify 110.11₂ – 10.01₂.

Assignment

Reading assignment

• New Gen Maths Book 3, chapter 1, Exercise 1e, page 18, Nos 1 to 12
• Essential Mathematics for J.S.S.3, pages 8 to 10

Weekend assignment

1. Express 342₆ as a number in base 10. (a) 342 (b) 3420 (c) 134
2. Change the number 10010₂ to base 10. (a) 18 (b) 34 (c) 40
3. Express in base two the square of 11. (a) 1001 (b) 1010 (c) 1011
4. Find the value of (101)2 in base two. (a) 1010 (b) 1111 (c) 1001
5. Multiply 100001₂ by 11. (a) 1001 (b) 1100011 (c) 10111

Theory

1. Calculate 110₂ × (1011₂ + 1001₂ – 101₂).
2. Convert 110111 to base five.