Binary Calculation Course

Lesson 1 • Comparing Number Systems Side by Side

Equivalent values across decimal, binary, octal, and hexadecimal are mapped together. Students develop fluency in recognising the same quantity in multiple bases.

Lesson 2 • The Decimal System Reviewed

Decimal place values and digit roles are revisited as a familiar reference point. Reinforces positional logic before introducing unfamiliar bases.

Lesson 3 • Positional Notation Explained

Positional value and place-weight principles across any base are introduced. This anchors all subsequent number-system comparisons in the chapter.

Lesson 4 • Introduction to Binary

Binary as a base-2 system using only 0 and 1 is defined and contextualised. Students recognise why binary is fundamental to digital computing.

Lesson 1 • Converting Binary Fractions

Fractional binary digits use negative powers of two to the right of the point. Students convert binary fractions and combine them with integer parts.

Lesson 2 • Handling Large Binary Numbers

Systematic column-by-column tracking prevents errors in long binary strings. Students use structured worksheets to manage 16-bit and 32-bit conversions.

Lesson 3 • Positional Weight Method

Each binary digit is multiplied by its corresponding power of two and summed. This foundational method applies to all subsequent conversion techniques.

Lesson 4 • Conversion Accuracy and Verification

Cross-checking converted values using reverse conversion confirms accuracy. Students apply multiple verification strategies to build reliable conversion habits.

Lesson 1 • Converting Mixed Decimal Numbers

Integer and fractional parts are converted separately and then joined at the radix point. Students combine both algorithms into a unified workflow.

Lesson 2 • Subtraction Method for Integers

Subtracting the largest fitting power of two identifies each binary bit. Students use this method as an intuitive alternative to repeated division.

Lesson 3 • Repeated Division by Two

Dividing by two and recording remainders produces binary digits from LSB to MSB. Students practise the algorithm on multi-digit decimal integers.

Lesson 4 • Converting Decimal Fractions

Repeated multiplication by two extracts binary fractional digits one at a time. Students recognise terminating versus non-terminating binary fractions.

Lesson 1 • Unsigned Integer Encoding

Unsigned binary integers and their range limits per bit width are defined. Students calculate maximum values for 4-, 8-, and 16-bit fields.

Lesson 2 • Sign-Magnitude Representation

Sign-magnitude encoding uses one bit to indicate positive or negative. Students identify its limitations, including dual zero, before moving to better schemes.

Lesson 3 • Binary-Coded Decimal

Binary-coded decimal encodes each decimal digit in a four-bit group. Students apply BCD in contexts where decimal precision is required.

Lesson 4 • Two's Complement Encoding

Two's complement is the dominant signed-integer scheme in modern hardware. Students compute two's complement representations and verify correctness.

Lesson 1 • Carry Propagation and Ripple

Carries ripple left through columns, potentially affecting all higher bits. Students trace carry chains and identify where ripple causes delays.

Lesson 2 • Subtraction via Two's Complement

Subtracting by adding the two's complement eliminates borrow logic entirely. Students verify that addition and complement-based subtraction yield identical results.

Lesson 3 • Binary Subtraction Rules

Single-bit subtraction cases introduce the borrow concept analogous to carry. Students apply borrow propagation across multi-bit subtraction problems.

Lesson 4 • Binary Addition Rules

The four single-bit addition cases define all binary addition behaviour. Students apply these rules column by column in multi-bit problems.

Lesson 1 • Signed Binary Multiplication

Sign extension and two's complement rules govern signed binary multiplication. Students apply sign correction to products of mixed-sign operands.

Lesson 2 • Binary Long Division Algorithm

Binary long division mirrors decimal long division using shift and subtract steps. Students compute quotients and remainders for multi-bit operands systematically.

Lesson 3 • Binary Multiplication Basics

Binary multiplication reduces to AND operations and left shifts per multiplier bit. Students construct partial products and sum them to get the final product.

Lesson 4 • Binary Division by Repeated Subtraction

Repeated subtraction of the divisor from the dividend counts quotient bits. Students apply this method to small operands before advancing to long division.

Lesson 1 • XOR for Toggling and Comparison

XOR flips targeted bits and returns zero when two values are identical. Students apply XOR for toggling, difference detection, and simple encryption.

Lesson 2 • Bit Shifting Operations

Left and right shifts move bits by specified positions, multiplying or dividing by powers of two. Students distinguish logical from arithmetic right shifts.

Lesson 3 • Fundamental Logic Gates and Truth Tables

AND, OR, NOT, and XOR truth tables define all bitwise operation outcomes. Students complete truth tables and map gate symbols to their logical behaviour.

Lesson 4 • Applying AND for Bit Masking

AND with a mask clears selected bits while preserving others unchanged. Students design masks to isolate specific bit fields in a byte.

Lesson 1 • Special Values and Precision Limits

Reserved bit patterns encode infinity, NaN, and zero as special cases. Students identify rounding errors and representational gaps near precision boundaries.

Lesson 2 • Scientific Notation in Binary

Binary scientific notation expresses values as a mantissa times a power of two. Students normalise binary numbers into standard scientific form.

Lesson 3 • Standard Floating-Point Format Structure

The sign, biased exponent, and mantissa fields compose the standard format. Students map each field's bit range and understand the biasing convention.

Lesson 4 • Decoding Floating-Point to Decimal

Extracting sign, exponent, and mantissa fields reverses the encoding process. Students reconstruct decimal values from given 32-bit floating-point patterns.