Lesson Notes By Weeks and Term v3 - Junior Secondary 1
Subtraction of numbers in base 2 numerals.
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Subtraction of numbers in base 2 numerals.
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Subject: General Mathematics
Class: Junior Secondary 1
Term: 3rd Term
Week: 6
Theme: Basic Operations
This page supports the lesson note with a companion video and a short classroom-ready summary.
For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.
This lesson focuses on the essential arithmetic operation of subtraction within the binary number system (base 2). Understanding binary subtraction is fundamental as it underpins the operations of digital computers and electronic devices, which are increasingly integral to modern Nigerian society, from mobile banking (USSD, POS) to communication technologies and computing. Mastery of this concept develops students' logical reasoning and prepares them for advanced topics in computer science and technology.
Specific Performance Objectives:
'1' from the leftmost column. The '1' in the leftmost column becomes '0'. The '0' in the middle column effectively becomes $0 + 2 = 2$.
Step 4: Now, subtract in the middle column: $2 - 1 = 1$. $$ \begin{array}{r} ^0\cancel{1}^201_2 \\ - 011_2 \\ \hline \quad 10_2 \end{array} $$ Step 5: Subtract the leftmost column: $0 - 0 = 0$. $$ \begin{array}{r} ^0\cancel{1}^201_2 \\ - 011_2 \\ \hline 010_2 \end{array} $$ Result: $101_2 - 011_2 = 010_2 = 10_2$ Example 4: Subtract $101_2$ from $110_2$ (3-digit binary numbers with borrowing) $$ \begin{array}{r} 110_2 \\ - 101_2 \\ \hline \end{array} $$ Step 1: Subtract the rightmost column: $0 - 1$. This requires borrowing.
Step 2: Borrow '1' from the middle column. The '1' in the middle column becomes '0'. The '0' in the rightmost column effectively becomes $0 + 2 = 2$.
Step 3: Now, subtract in the rightmost column: $2 - 1 = 1$. $$ \begin{array}{r} 1^0\cancel{1}^20_2 \\ - 101_2 \\ \hline \quad \quad 1 \end{array} $$ Step 4: Subtract the middle column: $0 - 0 = 0$. $$ \begin{array}{r} 1^0\cancel{1}^20_2 \\ - 101_2 \\ \hline \quad 01_2 \end{array} $$ Step 5: Subtract the leftmost column: $1 - 1 = 0$. $$ \begin{array}{r} 1^0\cancel{1}^20_2 \\ - 101_2 \\ \hline 001_2 \end{array} $$ Result: $110_2 - 101_2 = 001_2 = 1_2$ 2.1 Introduction to Binary Subtraction Subtraction in base 2 (binary subtraction) follows the same general principles as subtraction in base 10, but with a crucial difference in the digits used (0 and 1) and the value borrowed. 2.2 Basic Binary Subtraction Rules: The elementary subtraction facts in base 2 are: $0 - 0 = 0$ $1 - 0 = 1$ $1 - 1 = 0$ $0 - 1 = ?$ (This is where borrowing becomes necessary) 2.3 The Concept of Borrowing in Base 2: When a smaller digit needs to subtract a larger digit in a particular column (e.g., $0 - 1$), a '1' must be borrowed from the next higher place value column to the left. In base 10, borrowing '1' from the next column means adding 10 to the current column. In base 2, borrowing '1' from the next column means adding $2_{10}$ (which is $10_2$) to the current column.
Therefore, if a '1' is borrowed from the column to the left, the digit in the current column effectively becomes $0 + 2 = 2$. So, $0 - 1$ with borrowing becomes $(0+2) - 1 = 1$. The digit in the column from which '1' was borrowed is reduced by '1'. If it was '1', it becomes '0'. If it was '0', it would have borrowed from its left, and so on. 2.4 Step-by-Step Procedure for Binary Subtraction:
1. Align the binary numbers vertically according to their place values.
2. Start subtracting from the rightmost column (least significant digit).
3. Apply the basic subtraction rules.
4. If a digit is smaller than the digit being subtracted, borrow '1' from the digit in the column immediately to its left.
5. Remember that a borrowed '1' means adding $2_{10}$ (or $10_2$) to the current column.
6. Adjust the digit in the column from which the '1' was borrowed (reduce it by 1).
7. Continue this process column by column until all digits are subtracted. 2.5 Worked
Examples: Example 1: Subtract $10_2$ from $11_2$ (2-digit binary numbers) $$ \begin{array}{r} 11_2 \\ - 10_2 \\ \hline \end{array} $$ Step 1: Subtract the rightmost column: $1 - 0 = 1$. $$ \begin{array}{r} 11_2 \\ - 10_2 \\ \hline \quad 1 \end{array} $$ Step 2: Subtract the next column to the left: $1 - 1 = 0$. $$ \begin{array}{r} 11_2 \\ - 10_2 \\ \hline 01_2 \end{array} $$ Result: $11_2 - 10_2 = 01_2 = 1_2$ Example 2: Subtract $01_2$ from $10_2$ (2-digit binary numbers with borrowing) $$ \begin{array}{r} 10_2 \\ - 01_2 \\ \hline \end{array} $$ Step 1: Subtract the rightmost column: $0 - 1$. This cannot be done directly.
Step 2: Borrow '1' from the next column to the left. The '1' in the left column becomes '0'. The '0' in the right column effectively becomes $0 + 2 = 2$.
Step 3: Now, subtract in the rightmost column: $2 - 1 = 1$. $$ \begin{array}{r} ^0\cancel{1}^20_2 \\ - \quad 01_2 \\ \hline \quad \quad 1 \end{array} $$ Step 4: Subtract the next column to the left: $0 - 0 = 0$. $$ \begin{array}{r} ^0\cancel{1}^20_2 \\ - \quad 01_2 \\ \hline 01_2 \end{array} $$ Result: $10_2 - 01_2 = 01_2 = 1_2$ Example 3: Subtract $011_2$ from $101_2$ (3-digit binary numbers with borrowing) $$ \begin{array}{r} 101_2 \\ - 011_2 \\ \hline \end{array} $$ Step 1: Subtract the rightmost column: $1 - 1 = 0$. $$ \begin{array}{r} 101_2 \\ - 011_2 \\ \hline \quad \quad 0 \end{array} $$ Step 2: Subtract the middle column: $0 - 1$. This requires borrowing.
Step 3: Borrow '1' from the leftmost column. The '1' in the leftmost column becomes '0'. The '0' in the middle column effectively becomes $0 + 2 = 2$.
Step 4: Now, subtract in the middle column: $2 - 1 = 1$. $$ \begin{array}{r} ^0\cancel{1}^201_2 \\ - 011_2 \\ \hline \quad 10_2 \end{array} $$ Step 5: Subtract the leftmost column: $0 - 0 = 0$. $$ \begin{array}{r} ^0\cancel{1}^201_2 \\ - 011_2 \\ \hline 010_2 \end{array} $$ Result: $101_2 - 011_2 = 010_2 = 10_2$ Example 4: Subtract $101_2$ from $110_2$ (3-digit binary numbers 3.1 Introduction (5 minutes)
Teacher Activity: Begins by reviewing the concept of binary numbers (base 2) and their digits (0 and 1). Briefly revisits binary addition to refresh memory on basic operations and carrying.
States the lesson objective: to learn how to subtract binary numbers.
Student Activity: Students respond to questions on binary numbers and addition, recalling previous knowledge. 3.2 Explanation and Demonstration (20 minutes)
Teacher Activity: Explains the basic rules of binary subtraction ($0-0, 1-0, 1-1$). Introduces the challenge of $0-1$ and thoroughly explains the concept of 'borrowing' in base
2. Emphasizes that borrowing '1' means adding $2_{10}$ ($10_2$) to the current column. Demonstrates worked examples (e.g., Examples 1 and 2 above) on the board, explaining each step clearly and slowly, drawing lines to show borrowing. Asks guiding questions to check for understanding (e.g., "What happens when we borrow '1' from the next column?").
Student Activity: Students listen attentively, take notes, and ask clarifying questions about the rules and the borrowing process. Students observe the step-by-step demonstrations and attempt to follow the logic. 3.3 Guided Practice (15 minutes)
Teacher Activity: Presents a few more 2-digit and 3-digit subtraction problems involving borrowing (e.g., Example 3 and 4). Guides students through these problems collectively, prompting them for each step (e.g., "What do we do in the rightmost column?", "Do we need to borrow here?"). Divides the class into small groups (e.g., 3-4 students per group) and assigns one problem for group discussion and solution. Circulates among groups, providing support, clarification, and correcting misconceptions.
Student Activity: Students actively participate in solving the problems under teacher guidance. Students work in groups to solve assigned problems, discussing strategies and arriving at a consensus solution. One student from each group presents their solution on the board, explaining their steps. 3.4 Application and Consolidation (10 minutes)
Teacher Activity: Reviews common errors observed during group work. Summarizes the key rules for binary subtraction, especially borrowing. Assigns independent practice questions for students to work on individually.
Student Activity: Students note corrections and reinforce their understanding. Students attempt the independent practice questions.
Instruction: Perform the following binary subtractions.
Question: Subtract $01_2$ from $11_2$.
Solution: $$ \begin{array}{r} 11_2 \\ 01_2 \\ \hline \end{array} $$ Rightmost column: $1 - 1 = 0$.
Leftmost column: $1 - 0 = 1$.
Answer: $10_2$
Commentary: A straightforward subtraction without borrowing.
Question: Subtract $10_2$ from $100_2$.
Solution: $$ \begin{array}{r} 100_2 \\ 010_2 \\ \hline \end{array} $$ Rightmost column: $0 - 0 = 0$.
Middle column: $0 - 1$. Borrow '1' from the leftmost column. The '1' in the leftmost column becomes '0'. The '0' in the middle column becomes $0 + 2 = 2$. Now, $2 - 1 = 1$.
Leftmost column: $0 - 0 = 0$.
Answer: $010_2 = 10_2$
Commentary: This involves borrowing across one column. The '0' in the hundreds place also needs to be adjusted after borrowing.
Question: Subtract $011_2$ from $110_2$.
Solution: $$ \begin{array}{r} 110_2 \\ 011_2 \\ \hline \end{array} $$ Rightmost column: $0 - 1$. Borrow '1' from the middle column. The '1' in the middle column becomes '0'. The '0' in the rightmost column becomes $0 + 2 = 2$. Now, $2 - 1 = 1$.
Middle column: Now $0 - 1$. Borrow '1' from the leftmost column. The '1' in the leftmost column becomes '0'. The '0' in the middle column becomes $0 + 2 = 2$. Now, $2 - 1 = 1$.
Leftmost column: $0 - 0 = 0$.
Answer: $011_2 = 11_2$
Commentary: This example demonstrates sequential borrowing across multiple columns.
Question: Subtract $11_2$ from $101_2$.
Solution: $$ \begin{array}{r} 101_2 \\ 011_2 \\ \hline \end{array} $$ Rightmost column: $1 - 1 = 0$.
Middle column: $0 - 1$. Borrow '1' from the leftmost column. The '1' in the leftmost column becomes '0'. The '0' in the middle column becomes $0 + 2 = 2$. Now, $2 - 1 = 1$.
Leftmost column: $0 - 0 = 0$.
Answer: $010_2 = 10_2$
Commentary: A 3-digit subtraction where the subtrahend is a 2-digit number (implicitly padded with a leading zero).
Digital Electronics and Computing: Binary subtraction is a fundamental operation performed by the Arithmetic Logic Unit (ALU) of computers and microprocessors. For example, when a smartphone user in Nigeria makes a calculation, sends a message, or processes a payment via a POS terminal, these high-level operations are broken down into basic binary arithmetic, including subtraction, at the hardware level. This knowledge is crucial for anyone considering a career in ICT in Nigeria.
Data Processing and Storage: In digital systems, data (like images, text, or financial records) is represented as sequences of binary digits (bits). Operations like deleting a file or updating a database entry involve complex binary manipulations where subtraction might be implicitly used to manage memory addresses or data blocks. Students can relate this to how their photos are stored on their phones or how their academic records are kept in a school database. Error Detection and Correction in Telecommunication: While advanced, the underlying principles of error detection and correction codes used in Nigerian telecommunication networks (GSM, internet) rely heavily on binary arithmetic. Simple parity checks, which involve counting bits (which can be seen as binary arithmetic), help ensure data integrity during transmission, for instance, when making calls or browsing the internet from a rural village.