Lesson Notes By Weeks and Term v3 - Senior Secondary 1
Number Base System
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Number Base System
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Subject: General Mathematics
Class: Senior Secondary 1
Term: 1st Term
Week: 1
Theme: Number and Numeration
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.
Convert numbersfrom other bases to base 10 Convert decimal fraction from other bases to base 10 Convert from one base to another base Perform basic operations on number bases (with the exeception of base 2) Apply number base system to computer programming
the multiplicand by each digit of the multiplier. Convert intermediate products (if they exceed the base) and carry over. Sum the partial products, ensuring all operations are done in the specified base. Worked
Example:
1. Calculate $23_4 \times 13_4$. Multiply $23_4$ by $3_4$: $3_4 \times 3_4 = 9_{10}$. $9 \div 4 = 2$ remainder $1$. Write $1$, carry $2$. $3_4 \times 2_4 = 6_{10}$. Add carry $2$: $6+2 = 8_{10}$. $8 \div 4 = 2$ remainder $0$. Write $0$, carry $2$. Result of $23_4 \times 3_4 = 201_4$. Multiply $23_4$ by $1_4$ (effectively $10_4$): $1_4 \times 3_4 = 3_4$. Write $3$ (shifted one position left). $1_4 \times 2_4 = 2_4$. Write $2$. Result of $23_4 \times 10_4 = 230_4$.
Add the partial products: $201_4$ $+ 230_4$ ------- Units: $1+0 = 1_4$.
Fours: $0+3 = 3_4$.
Sixteens: $2+2 = 4_{10}$. $4 \div 4 = 1$ remainder $0$. Write $0$, carry $1$.
Result: $1031_4$. (Verification: $23_4 = 11_{10}$, $13_4 = 7_{10}$. $11 \times 7 = 77_{10}$. $1031_4 = 1 \times 64 + 0 \times 16 + 3 \times 4 + 1 \times 1 = 64+0+12+1 = 77_{10}$. Correct.) 2.7.
4. Division: For SS1, the simplest method for division is to convert both the dividend and divisor to base 10, perform the division, and then convert the result back to the desired base if required. Direct long division in other bases can be complex and is often introduced at a later stage or for more advanced learners. Worked
Example:
1. Calculate $221_3 \div 2_3$.
Step 1: Convert $221_3$ to base 10. $2 \times 3^2 + 2 \times 3^1 + 1 \times 3^0 = 2 \times 9 + 2 \times 3 + 1 \times 1 = 18 + 6 + 1 = 25_{10}$.
Step 2: Convert $2_3$ to base 10. $2 \times 3^0 = 2_{10}$.
Step 3: Perform division in base 10. $25_{10} \div 2_{10} = 12.5_{10}$. (If integer division is expected, note the remainder: $25 \div 2 = 12$ remainder $1$.)
Step 4: Convert the result ($12_{10}$ and the remainder $1_{10}$ if applicable) back to base 3. $12 \div 3 = 4$ remainder $0$ $4 \div 3 = 1$ remainder $1$ $1 \div 3 = 0$ remainder $1$ So, $12_{10} = 110_3$. The quotient is $110_3$ with a remainder of $1_3$. (
Note: If the question implies a complete division with a fractional part, converting $0.5_{10}$ to base 3 would be: $0.5 \times 3 = 1.5 \rightarrow 1$; $0.5 \times 3 = 1.5 \rightarrow 1$; so $0.5_{10} = 0.11..._3$). For SS1, focus on integer quotients and remainders for division problems in other bases. 2.
1. Definition of Number Base System A number base system is a way of representing numbers using a set of unique digits and a specific base (radix). The base indicates the number of unique digits (including zero) used in the system and determines the value of each digit based on its position (place value).
Base 10 (Decimal System): Uses 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). Each position represents a power of
1
0. This is the everyday system for counting.
Other Bases: Base 2 (Binary System): Uses 2 digits (0, 1). Each position represents a power of
2. Crucial for computers.
Base 5: Uses 5 digits (0, 1, 2, 3, 4). Each position represents a power of
5. Base 8 (Octal System): Uses 8 digits (0, 1, 2, 3, 4, 5, 6, 7). Each position represents a power of
8. Base 16 (Hexadecimal System): Uses 16 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F). A=10, B=11, ..., F=
1
5. Each position represents a power of 16. 2.
2. Place Value In any number base, the position of a digit determines its value. For a number $d_n d_{n-1} ... d_1 d_0 . d_{-1} d_{-2} ... d_{-m}$ in base $b$, its value is: $d_n \times b^n + d_{n-1} \times b^{n-1} + ... + d_1 \times b^1 + d_0 \times b^0 + d_{-1} \times b^{-1} + d_{-2} \times b^{-2} + ... + d_{-m} \times b^{-m}$ 2.
3. Conversion of Numbers from Other Bases to Base 10 To convert a number from any base $b$ to base 10, expand the number using its place values (powers of the base) and sum the products. Worked
Examples:
1. Convert $432_5$ to base
1
0. Identify the place values: $4 \times 5^2 + 3 \times 5^1 + 2 \times 5^0$ Calculate the powers: $4 \times 25 + 3 \times 5 + 2 \times 1$ Multiply and sum: $100 + 15 + 2 = 117_{10}$ Therefore, $432_5 = 117_{10}$.
2. Convert $1011_2$ to base 10. $1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0$ $1 \times 8 + 0 \times 4 + 1 \times 2 + 1 \times 1$ $8 + 0 + 2 + 1 = 11_{10}$ Therefore, $1011_2 = 11_{10}$. 2.
4. Conversion of Decimal Fractions from Other Bases to Base 10 For the fractional part, use negative powers of the base. Worked
Example:
1. Convert $17.53_8$ to base
1
0. Separate the whole number and fractional parts: Whole number part: $17_8 = 1 \times 8^1 + 7 \times 8^0 = 8 + 7 = 15_{10}$ Fractional part: $0.53_8 = 5 \times 8^{-1} + 3 \times 8^{-2}$ $= 5 \times \frac{1}{8} + 3 \times \frac{1}{64}$ $= \frac{5}{8} + \frac{3}{64}$ $= \frac{40}{64} + \frac{3}{64} = \frac{43}{64}$ To convert $\frac{43}{64}$ to decimal: $43 \div 64 \approx 0.671875_{10}$ Combine: $15_{10} + 0.671875_{10} = 15.671875_{10}$ Therefore, $17.53_8 = 15.671875_{10}$. 2.
5. Conversion of Numbers from Base 10 to Other Bases There are two main methods, one for whole numbers and one for fractions. 2.5.
1. For Whole Numbers (Division Method): Continuously divide the base 10 number by the desired new base, keeping track of the remainders. The new number is formed by reading the remainders from the last to the first (upwards). Worked
Example:
1. Convert $58_{10}$ to base 5. $58 \div 5 = 11$ remainder $3$ $11 \div 5 = 2$ remainder $1$ $2 \div 5 = 0$ remainder $2$ Reading the remainders upwards: $213_5$ Therefore, $58_{10} = 213_5$. 2.5.
2. For Decimal Fractions (Multiplication Method): Continuously multiply the fractional part by the desired new base. Take the whole number part of the product as the next digit in the new base. The new number is formed by reading the whole number parts downwards. Worked
Example:
1. Convert $0.375_{10}$ to base 2. $0.375 \times 2 = 0.750 \rightarrow 0$ $0.750 \times 2 = 1.500 \rightarrow 1$ $0.500 \times 2 = 1.000 \rightarrow 1$ Reading the whole numbers downwards: $0.011_2$ * Therefore, $0.375_{10} $213_5$ Therefore, $58_{10} = 213_5$. 2.5.
2. For Decimal Fractions (Multiplication Method): Continuously multiply the fractional part by the desired new base. Take the whole number part of the product as the next digit in the new base. The new number is formed by reading the whole number parts downwards. Worked
Example:
1. Convert $0.375_{10}$ to base 2. $0.375 \times 2 = 0.750 \rightarrow 0$ $0.750 \times 2 = 1.500 \rightarrow 1$ $0.500 \times 2 = 1.000 \rightarrow 1$ Reading the whole numbers downwards: $0.011_2$ Therefore, $0.375_{10} = 0.011_2$.
2. Convert $45.625_{10}$ to base
8. Whole number part ($45_{10}$ to base 8): $45 \div 8 = 5$ remainder $5$ $5 \div 8 = 0$ remainder $5$ Whole number part is $55_8$. Fractional part ($0.625_{10}$ to base 8): $0.625 \times 8 = 5.000 \rightarrow 5$ Fractional part is $0.5_8$.
Combine: $55.5_8$ Therefore, $45.625_{10} = 55.5_8$. 2.
6. Conversion from One Base to Another Base (via Base 10) To convert a number from base $A$ to base $B$:
1. Convert the number from base $A$ to base 10.
2. Convert the resulting base 10 number to base $B$. Worked
Example:
1. Convert $135_8$ to base
5. Step 1: Convert $135_8$ to base 10. $1 \times 8^2 + 3 \times 8^1 + 5 \times 8^0$ $= 1 \times 64 + 3 \times 8 + 5 \times 1$ $= 64 + 24 + 5 = 93_{10}$ Step 2: Convert $93_{10}$ to base 5. $93 \div 5 = 18$ remainder $3$ $18 \div 5 = 3$ remainder $3$ $3 \div 5 = 0$ remainder $3$ Reading remainders upwards: $333_5$ Therefore, $135_8 = 333_5$. 2.
7. Basic Operations on Number Bases (excluding base 2) The principles of addition, subtraction, multiplication, and division remain the same, but 'carrying over' or 'borrowing' occurs when the value reaches or exceeds the base number. 2.7.
1. Addition: Add digits column by column from right to left. If the sum in a column is equal to or greater than the base, divide the sum by the base. Write down the remainder and carry the quotient to the next column. Worked
Example:
1. Calculate $43_5 + 24_5$.
Units column: $3_5 + 4_5 = 7_{10}$. $7 \div 5 = 1$ remainder $2$. Write $2$, carry $1$.
Fives column: $4_5 + 2_5 + (\text{carry } 1) = 7_{10}$. $7 \div 5 = 1$ remainder $2$. Write $2$, carry $1$. The final carry forms the leftmost digit.
Result: $122_5$. (Verification: $43_5 = 23_{10}$, $24_5 = 14_{10}$, $23+14 = 37_{10}$. $122_5 = 1 \times 25 + 2 \times 5 + 2 \times 1 = 25+10+2 = 37_{10}$. Correct.) 2.7.
2. Subtraction: Subtract digits column by column from right to left. If a digit is too small to subtract from, borrow from the digit to its left. When borrowing 1 from the next column, it adds the value of the base to the current column's digit. Worked
Example:
1. Calculate $65_8 - 27_8$.
Units column: $5_8 - 7_8$. Cannot subtract. Borrow 1 from the 6 in the eights column. The borrowed 1 becomes $8_{10}$ in the units column. So, $ (5+8)_8 - 7_8 = 13_{10} - 7_{10} = 6_{10}$. Write $6$.
Eights column: The 6 becomes
5. So, $5_8 - 2_8 = 3_8$. Write $3$.
Result: $36_8$. (Verification: $65_8 = 53_{10}$, $27_8 = 23_{10}$, $53-23 = 30_{10}$. $36_8 = 3 \times 8 + 6 \times 1 = 24+6=30_{10}$. Correct.) 2.7.
3. Multiplication: Perform multiplication similar to base 10, multiplying each digit of the multiplicand by each digit of the multiplier. Convert intermediate products (if they exceed the base) and carry over. Sum the partial products, ensuring all operations are done in the specified base. Worked
Example:
1. Calculate $23_4 \times 13_4$. Multiply $23_4$ by $3_4$: $3_4 \times 3_4 = 9_{10}$. $9 \div 4 = 2$ remainder $1$. Write $1$, carry $2$. $3_4 \times 2_4 = 6_{10}$. Add carry $2$: $6+2 = 8_{10}$. $8 \div 4 = 2$ remainder $0$. Write $0$, carry $2$. * Result of $23_4 \times 3_4 = 201_4$. 3.
1. Introduction (5-10 minutes)
Teacher Activity: Begin by asking learners how they count objects (e.g., "How many fingers do you have? How do we count money in naira?"). Emphasize that we usually count in groups of ten. Introduce the idea that there are other ways to group and count. Show examples of tally marks (groups of 5) or using fingers on one hand (base 5 concept).
Student Activity: Learners engage in a quick counting exercise. Discuss how they count in their local language or for specific items (e.g., counting yams in heaps). 3.
2. Lesson Development (40-50 minutes)
Phase 1: Conversion to Base 10 Teacher Activity: Explain the concept of place value using base 10 numbers first (e.g., $245 = 2 \times 10^2 + 4 \times 10^1 + 5 \times 10^0$). Introduce other bases (e.g., base 5, base 8) and explain that digits must always be less than the base. Demonstrate step-by-step how to convert whole numbers from other bases to base 10 using the expansion method, providing clear examples like $432_5$ to base
1
0. Extend this to decimal fractions, explaining the use of negative powers of the base, using an example like $17.53_8$ to base
1
0. Student Activity: Learners follow the examples, taking notes. Solve simple conversion problems individually or in pairs on mini-whiteboards. Ask questions for clarification.
Phase 2: Conversion from Base 10 to Other Bases Teacher Activity: Demonstrate the division method for converting whole numbers from base 10 to other bases (e.g., $58_{10}$ to base 5). Emphasize reading remainders upwards. Demonstrate the multiplication method for converting decimal fractions from base 10 to other bases (e.g., $0.375_{10}$ to base 2). Emphasize reading whole number parts downwards. Combine both methods for mixed numbers (e.g., $45.625_{10}$ to base 8).
Student Activity: Learners practice conversion problems from base 10 to various bases. Engage in peer-to-peer checking of solutions.
Phase 3: Conversion Between Non-Decimal Bases Teacher Activity: Explain the two-step process: Convert to base 10 first, then from base 10 to the target base. Work through an example like $135_8$ to base
5. Student Activity: Learners attempt similar problems and present their solutions.
Phase 4: Basic Operations on Number Bases Teacher Activity: Demonstrate addition in a given base (e.g., base 5), highlighting the 'carrying over' rule when sums exceed the base. Demonstrate subtraction, emphasizing the 'borrowing' rule where a borrowed unit is equivalent to the base value. Demonstrate multiplication, explaining how to handle intermediate sums and carries within the target base. For division, explain the method of converting to base 10, performing the division, and converting the result back. Provide an example. Reiterate that base 2 operations are excluded for now.
Student Activity: Learners practice one or two examples for each operation. Discuss common errors or misunderstandings in their groups. 3.
3. Conclusion (5-10 minutes)
Teacher Activity: Summarize the key conversion methods and operational rules. Link the topic to computer systems and other real-life applications in Nigeria.
Student Activity: Learners ask final clarifying questions and provide a brief verbal summary of one key learning point from the lesson.
Convert $432_5$ to base
1
0. Identify the place values:
$4 \times 5^2 + 3 \times 5^1 + 2 \times 5^0$
Calculate the powers:
$4 \times 25 + 3 \times 5 + 2 \times 1$
Multiply and sum:
$100 + 15 + 2 = 117_{10}$
Therefore, $432_5 = 117_{10}$.
Convert $1011_2$ to base 10.
$1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0$
$1 \times 8 + 0 \times 4 + 1 \times 2 + 1 \times 1$
$8 + 0 + 2 + 1 = 11_{10}$
Therefore, $1011_2 = 11_{10}$.
2. 4. Conversion of Decimal Fractions from Other Bases to Base 10
For the fractional part, use negative powers of the base.
Worked
Example:
Convert $17.53_8$ to base
1
0.
Separate the whole number and fractional parts:
Whole number part: $17_8 = 1 \times 8^1 + 7 \times 8^0 = 8 + 7 = 15_{10}$
Fractional part: $0.53_8 = 5 \times 8^{-1} + 3 \times 8^{-2}$
$= 5 \times \frac{1}{8} + 3 \times \frac{1}{64}$
$= \frac{5}{8} + \frac{3}{64}$
$= \frac{40}{64} + \frac{3}{64} = \frac{43}{64}$
To convert $\frac{43}{64}$ to decimal: $43 \div 64 \approx 0.671875_{10}$
Combine: $15_{10} + 0.671875_{10} = 15.671875_{10}$
Therefore, $17.53_8 = 15.671875_{10}$.
2. 5. Conversion of Numbers from Base 10 to Other Bases
There are two main methods, one for whole numbers and one for fractions.
Computer Programming and Digital Devices: The most prominent application is in computing. Computers operate using the binary system (base 2). Understanding number bases helps learners grasp how mobile phones, ATMs, computers, and other digital gadgets process information. For instance, an ATM transaction involves numbers converted to binary for internal processing and then back to decimal for display. Hexadecimal (base 16) is also used in web design (color codes) and memory addresses. Traditional Counting Systems and Local Markets: In some Nigerian cultures, traditional counting methods may involve grouping objects differently than in base
1
0. For example, a market woman selling oranges might count them in heaps of 5 or
1
0. While not strict mathematical bases, this illustrates the concept of different grouping systems. Understanding number bases helps conceptualize such variations, improving numerical flexibility.
Inventory and Stock Management: Businesses and traders, from small kiosks to large warehouses, deal with inventory. Items are often packaged in specific quantities (e.g., cartons of 12 items, bags of 50 tubers). Understanding number bases can facilitate calculations when stock is managed in non-standard groups, helping prevent errors in stocktaking and sales. For example, a beverage distributor receives drinks in crates of
2
4. They might think in "base 24" units for easier stock management.