Number and Numeration

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Number and Numeration — Quick Facts

Types of Numbers:

• Natural numbers (N): 1, 2, 3, 4, … (positive integers)
• Whole numbers: 0, 1, 2, 3, …
• Integers (Z): … -3, -2, -1, 0, 1, 2, 3, …
• Rational numbers (Q): Numbers of the form $\frac{p}{q}$ where $q \neq 0$ (includes fractions, integers, terminating/repeating decimals)
• Irrational numbers: Cannot be expressed as $\frac{p}{q}$; decimal expansion is non-terminating, non-repeating (e.g., $\pi$, $\sqrt{2}$, $e$)
• Real numbers: Rational + Irrational numbers combined

Index Notation:

• $a^m \times a^n = a^{m+n}$
• $a^m \div a^n = a^{m-n}$
• $(a^m)^n = a^{mn}$
• $a^0 = 1$ (for $a \neq 0$)
• $a^{-n} = \frac{1}{a^n}$

Standard Form: Any number expressed as $A \times 10^n$ where $1 \leq A < 10$ and $n$ is an integer.

Example: $4500 = 4.5 \times 10^3$

Number Base Systems: Base $b$ uses digits 0, 1, 2, …, $(b-1)$

Base	Name	Digits
2	Binary	0, 1
8	Octal	0-7
10	Decimal	0-9
16	Hexadecimal	0-9, A-F

⚡ JAMB Exam Tip: When converting from base $b$ to decimal, multiply each digit by its place value ($b^n$). When converting to base $b$, use repeated division by $b$.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Number and Numeration — Study Guide

Conversion: Base $b$ to Decimal:

Example: Convert $312_4$ to base 10.

$3 \times 4^2 + 1 \times 4^1 + 2 \times 4^0$ $= 3 \times 16 + 1 \times 4 + 2 \times 1$ $= 48 + 4 + 2 = 54_{10}$

Conversion: Decimal to Base $b$:

Example: Convert $167_{10}$ to base 5.

$167 \div 5 = 33$ remainder $2$ (LSD) $33 \div 5 = 6$ remainder $3$ $6 \div 5 = 1$ remainder $1$ $1 \div 5 = 0$ remainder $1$ (MSD)

Reading remainders upward: $167_{10} = 1132_5$

Binary ↔ Decimal Conversions:

$101101_2$ to decimal: $1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0$ $= 32 + 0 + 8 + 4 + 0 + 1 = 45_{10}$

Binary ↔ Octal (Quick Method):

Group binary digits in sets of 3 from the right: $101101_2 = 101\ 101_2 = 55_8$

Surds (Square Roots):

Simplify $\sqrt{72}$: $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$

Rationalising $\frac{1}{\sqrt{3} + \sqrt{2}}$: Multiply numerator and denominator by $(\sqrt{3} - \sqrt{2})$: $\frac{\sqrt{3} - \sqrt{2}}{3 - 2} = \sqrt{3} - \sqrt{2}$

⚡ Common Student Mistake: Forgetting to rationalise the denominator when simplifying fractions with surds. Always eliminate square roots from denominators.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Number and Numeration — Comprehensive Notes

Laws of Indices (Full):

1. $a^m \times a^n = a^{m+n}$
2. $a^m \div a^n = a^{m-n}$
3. $(a^m)^n = a^{mn}$
4. $a^0 = 1$
5. $a^{-n} = \frac{1}{a^n}$
6. $a^{m/n} = \sqrt[n]{a^m}$
7. $(ab)^n = a^n b^n$
8. $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

Compound Index:

Example: Simplify $8^{2/3}$

$8^{2/3} = (8^{1/3})^2 = (\sqrt[3]{8})^2 = 2^2 = 4$

Alternatively: $8^{2/3} = (2^3)^{2/3} = 2^{3 \times 2/3} = 2^2 = 4$

Logarithms:

If $a^x = y$, then $\log_a y = x$

Laws of Logarithms:

1. $\log_a (mn) = \log_a m + \log_a n$
2. $\log_a (m/n) = \log_a m - \log_a n$
3. $\log_a (m^n) = n \log_a m$
4. $\log_a 1 = 0$
5. $\log_a a = 1$
6. $\log_a b = \frac{\log_c b}{\log_c a}$ (change of base)

Worked Logarithm Example:

If $\log 2 = 0.3010$ and $\log 3 = 0.4771$, find $\log 48$.

$\log 48 = \log (16 \times 3) = \log 16 + \log 3$ $= \log (2^4) + 0.4771 = 4(0.3010) + 0.4771$ $= 1.2040 + 0.4771 = 1.6811$

Antilogarithm:

If $\log x = 1.6811$, find $x$.

Split: $1.6811 = 1 + 0.6811$

Antilog of $1 = 10$ Antilog of $0.6811$: First fixed decimal point, then use antilog tables.

$x = 10 \times 4.8 \approx 48$

Number Bases: Hexadecimal:

Hexadecimal (base 16) uses 0-9 and A-F where A=10, B=11, …, F=15.

Example: Convert $2F_{16}$ to decimal. $= 2 \times 16^1 + 15 \times 16^0 = 32 + 15 = 47_{10}$

Laws of Divisibility:

• A number is divisible by 2 if its last digit is even
• A number is divisible by 3 if the sum of its digits is divisible by 3
• A number is divisible by 4 if the last two digits form a number divisible by 4
• A number is divisible by 5 if it ends in 0 or 5
• A number is divisible by 9 if the sum of its digits is divisible by 9

Highest Common Factor (HCF) and Lowest Common Multiple (LCM):

Using prime factorisation:

• $HCF = \text{product of common prime factors (lowest power)}$
• $LCM = \text{product of all prime factors (highest power)}$

Example: Find HCF and LCM of 24 and 36. $24 = 2^3 \times 3^1$ $36 = 2^2 \times 3^2$

$HCF = 2^2 \times 3^1 = 4 \times 3 = 12$ $LCM = 2^3 \times 3^2 = 8 \times 9 = 72$

Note: $24 \times 36 = HCF \times LCM = 12 \times 72 = 864$ ✓

JAMB Pattern Analysis (2015-2024):

• 2015: Binary to decimal conversion
• 2017: Simplifying surds with rationalisation
• 2019: Logarithm laws and calculation
• 2021: HCF/LCM using prime factorisation
• 2023: Standard form calculations
• 2024: Compound indices ($8^{2/3}$ type)

⚡ Exam Strategy: For base conversion problems, always show your division/multiplication steps clearly. For logarithm problems, start by identifying which law to apply based on the structure of the expression.