Computer Studies / Data Processing · Lesson 4

Binary, octal, hexadecimal, character codes and logic gates

JAMBWAECNECO

What the syllabus asks

JAMB: Section B 1(d) Logic circuits (AND, NOT, OR, NOR, NAND; logic equations for AND, NOT, OR; symbols; truth tables; comparator using XOR and NOR); 2(b) Data representation: types of data (integers, real numbers, strings, multimedia); number bases with special reference to binary, decimal, hexadecimal; 2(c) digitisation

WAEC: Logic circuits (definition, types, uses, symbols, logic equations, truth tables for AND, NOT, OR, NAND, NOR; comparator with XOR and NOR)

NECO: Logic circuits (definition, types, uses, symbols, logic equations, truth tables for AND, NOT, OR, NAND, NOR; comparator with XOR and NOR)

What you will learn

Explain why computers use binary and list data types.
Convert decimal numbers to binary, octal and hexadecimal.
Convert binary, octal and hexadecimal numbers to decimal.
Convert directly between binary, octal and hexadecimal by grouping.
Convert simple binary fractions.
Add, subtract and multiply binary numbers.
Describe character codes (ASCII, Unicode) and BCD.
Draw the symbols and write the equations for AND, OR, NOT, NAND, NOR and XOR.
Construct truth tables for single gates and simple circuits.
Construct a simple comparator using XOR and NOR/NOT gates.

Key points

Computers use binary because switches have two reliable states.
Data types: integer, real, string, Boolean, multimedia.
Bases: binary 2 (0–1), octal 8 (0–7), decimal 10, hexadecimal 16 (0–9, A–F).
A digit must be less than its base.
Decimal → base b: repeated division by b, remainders read bottom up.
Base b → decimal: multiply digits by place values (powers of b) and add.
Binary ↔ octal: groups of 3 bits; binary ↔ hex: groups of 4 bits.
Binary fractions: places ½, ¼, ⅛; decimal fraction → binary by repeated ×2, read top down.
Binary addition: 1+1 = 10, 1+1+1 = 11; subtraction uses borrowing.
Two’s complement: invert bits, add 1.
ASCII: A = 65, a = 97, 0 = 48; Unicode covers all scripts; BCD codes each decimal digit in 4 bits.
AND: all 1 → 1 (A·B). OR: any 1 → 1 (A + B). NOT: inverts (Ā).
NAND: 0 only when all 1. NOR: 1 only when all 0. XOR: 1 when inputs differ.
Truth table rows = 2ⁿ for n inputs; NAND and NOR are universal gates.
Comparator: XOR detects difference; NOT/NOR of XOR outputs gives 1 for equality.

Start here: Why binary? Data types and representation

Inside a computer, every piece of data, whether a number, a letter, a photo or a song, is stored as patterns of binary digits (bits): 0 and 1. The reason is physical: electronic components such as transistors work reliably as switches with two states, off (0) and on (1). Two states are easy to tell apart even when voltages vary slightly, so binary circuits are simpler, cheaper and more reliable than circuits that would have to tell ten voltage levels apart.

Types of data (JAMB syllabus)

Integers: whole numbers, positive or negative: 25, −7, 2026.
Real (floating-point) numbers: numbers with fractional parts: 3.75, −0.5, 98.6.
Strings (alphanumeric): sequences of characters: “Abuja”, “SS3B”, “NIN 1234”.
Boolean: true or false.
Multimedia: images (grids of pixels, each a number), audio (sound waves sampled into numbers), video (a series of images plus sound), and signals from sensors.

Number systems in computing

Programmers use hexadecimal and octal as shorthand for long binary strings, because 16 = 2⁴ and 8 = 2³; each hex digit stands for exactly four bits and each octal digit for three. Hex appears in colour codes for web pages (#FF0000 is pure red), in memory addresses and in MAC addresses of network cards.

Decimal	Binary	Octal	Hex
0	0000	0	0
1	0001	1	1
2	0010	2	2
3	0011	3	3
4	0100	4	4
5	0101	5	5
6	0110	6	6
7	0111	7	7
8	1000	10	8
9	1001	11	9
10	1010	12	A
11	1011	13	B
12	1100	14	C
13	1101	15	D
14	1110	16	E
15	1111	17	F

Learn this table: it makes grouping conversions quick. A subscript shows the base: 1010₂ = 10₁₀ = 12₈ = A₁₆.

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Practice questions

1. Convert 13₁₀ to binary.

1101
1011
1110
1001

Show answer

A: 1101
13 = 8 + 4 + 1.

2. Convert 10110₂ to decimal.

Show answer

D: 22
16 + 4 + 2 = 22.

3. Convert 255₁₀ to hexadecimal.

Show answer

C: FF
255 = 15×16 + 15.

4. Convert 1F₁₆ to decimal.

Show answer

B: 31
1×16 + 15 = 31.

5. Convert 37₈ to decimal.

Show answer

A: 31
3×8 + 7 = 31.

Key words

Number base (radix): The number of distinct digits used in a place-value number system.
Binary: Base 2, using digits 0 and 1.
Octal: Base 8, using digits 0–7.
Decimal: Base 10, using digits 0–9.
Hexadecimal: Base 16, using 0–9 and A–F.
Place value: The value of a position: a power of the base.
Most significant bit (MSB): The leftmost bit, with the highest place value.
Least significant bit (LSB): The rightmost bit, with place value 1.
Carry: A digit passed to the next column when a column sum reaches the base.
Two’s complement: A way of representing negative binary numbers: invert all bits and add 1.
ASCII: American Standard Code for Information Interchange: 7-bit codes for characters.
Unicode: A character code standard covering the scripts of the world’s languages.

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