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MATHJHS 1 • Term 1Topic 4Free Trial Lesson

Number Bases and Binary System (Base 2)

Place values in base two, conversion between base ten and base two, and binary arithmetic.

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Curated Video Lesson

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Comprehensive Study Notes

Aligned with Ghana NaCCA & WAEC BECE syllabus standards

Topic Introduction & Real-World Context:

Human beings naturally count in Base 10 (the denary or decimal system) because we have ten fingers. However, modern digital computers, smartphones, and the internet operate on the binary system (Base 2), which uses only two digits: 0 and 1. An electronic transistor is either off (0) or on (1). Understanding number bases allows students to comprehend how data is stored, processed, and calculated in computing while mastering place values and numeral conversions.

What You Will Master in This Lesson (NaCCA Objectives):

Understand the concept of place values in Base 10, Base 2, and other numeral bases.

State the allowable digits for any given base (e.g. Base n uses digits 0 to n - 1).

Convert numbers from decimal (Base 10) to binary (Base 2) using repeated division.

Convert numbers from binary (Base 2) to decimal (Base 10) using expanded notation.

Perform addition and subtraction of binary numbers.

Solve simple number base equations for unknown bases.

1. What is a Number Base and Place Value?

A number base (or radix) indicates how many distinct digits are used to count before moving to the next place value. In Base 10: Digits used are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. In Base 2 (Binary): Digits used are strictly 0 and 1. Rule: A number in base n can NEVER contain the digit n or any digit greater than n! For example, 102_two is invalid because the digit 2 cannot exist in base 2.

Base 2 place values from right to left: 2⁰ = 1, 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32, 2⁶ = 64.

Subscript notation: The base of a number is written as a subscript or word (e.g., 1101_two or 1101₂).

If no base is written, the number is assumed to be in Base 10.

2. Converting Base 10 (Decimal) to Base 2 (Binary)

To convert an integer from Base 10 to Base 2: 1. Divide the decimal number continuously by 2. 2. Record the quotient and write the remainder (which will always be either 0 or 1) on the side. 3. Continue dividing the quotients by 2 until the quotient becomes 0. 4. Read the remainders from BOTTOM TO TOP.

Example: Convert 19_ten to Base 2:

19 ÷ 2 = 9 remainder 1

9 ÷ 2 = 4 remainder 1

4 ÷ 2 = 2 remainder 0

2 ÷ 2 = 1 remainder 0

1 ÷ 2 = 0 remainder 1

Reading from bottom to top gives: 10011_two.

3. Converting Base 2 (Binary) to Base 10 (Decimal)

To convert a binary number to Base 10: 1. Assign powers of 2 to each digit, starting with 0 at the extreme right-hand digit. 2. Multiply each binary digit by 2 raised to its corresponding position power. 3. Sum the resulting products together to obtain the decimal value. Formula: (d_n × 2^n) + ... + (d_1 × 2¹) + (d_0 × 2⁰)

Example: Convert 11011_two to Base 10:

= (1 × 2⁴) + (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰)

= 16 + 8 + 0 + 2 + 1 = 27_ten.

4. Binary Addition (Base 2 Arithmetic)

Adding numbers in binary follows column addition principles, with regrouping when the sum reaches 2: • 0 + 0 = 0 • 0 + 1 = 1 • 1 + 0 = 1 • 1 + 1 = 10_two (write 0, carry over 1) • 1 + 1 + 1 = 11_two (write 1, carry over 1)

Common Mistakes Students Make in BECE Examinations:

⚠️Reading remainders from top to bottom instead of bottom to top during repeated division.

⚠️Writing the digit 2 or higher in a base 2 answer. In Base 2, 1 + 1 = 10_two.

⚠️Thinking that 2⁰ = 0. Remember: 2⁰ = 1.

Teacher's BECE Exam Pro-Tips:

⭐Always verify your conversion by converting your answer back to the original base.

⭐Write column place values (16, 8, 4, 2, 1) directly above the binary digits to avoid misalignment.

Quick Revision Summary Checklist:

Base 2 uses only 0 and 1.

Base 10 to Base 2: Repeated division by 2, read remainders from bottom to top.

Base 2 to Base 10: Expanded form using powers of 2 (2⁰=1, 2¹=2, 2²=4, 2³=8, 2⁴=16).

Step-by-Step Worked Examples (1)

Real BECE exam-standard problems with complete solution steps

Example 1: Converting Decimal to Binary

Problem StatementConvert 25_ten to Base 2.

Step-by-Step Solution:

Step 1: 25 ÷ 2 = 12 remainder 1.

Step 2: 12 ÷ 2 = 6 remainder 0.

Step 3: 6 ÷ 2 = 3 remainder 0.

Step 4: 3 ÷ 2 = 1 remainder 1.

Step 5: 1 ÷ 2 = 0 remainder 1.

Step 6: Reading remainders from bottom to top: 11001_two.

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Key Takeaway / Exam Rule: Always read remainders from the last division upwards to the first.

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