Grade 10 Core Mathematics Study Notes

Numbers and Algebra

At Grade 10, Numbers and Algebra builds on the number work from Junior School in three directions:

1. Real Numbers — classifying numbers into families (odd, even, prime, composite, rational, irrational) and working with reciprocals.
2. Indices and Logarithms — shorthand for repeated multiplication, and the tool that reverses it.
3. Quadratic Expressions and Equations — expressions and equations where the highest power of the unknown is 2.

This lesson covers the first sub-strand, Real Numbers.

Application of Odd and Even Numbers

The odd/even rule answers a practical question: can a group of items be shared equally into two?

• If the count is even, the items share equally into two groups with nothing left over.
• If the count is odd, one item is always left over.

The rule also makes it easy to pick out all the odd or even numbers in a given range — just scan the ones digits. Between 9 and 29, for instance, the even numbers are 10, 12, 14, 16, 18, 20, 22, 24, 26, and 28 — ten numbers in all.

Examples from real life

• A coach has 28 players — even — so two equal teams of 14 are possible.
• A farmer has 149 mangoes — odd — one mango is always left over when split into two groups.
• A parking lot holds 88 cars — even — two equal sections of 44.

Key Points

• Even counts share equally into two groups; odd counts leave one item over.
• To count odd/even numbers in a range, scan the ones digits.

Odd and Even Numbers

An odd number is a whole number that cannot be divided by 2 without a remainder. Its ones place value is 1, 3, 5, 7, or 9.

An even number is a whole number that can be divided by 2 without a remainder. Its ones place value is 0, 2, 4, 6, or 8.

To decide whether a whole number is odd or even, you only need to look at the digit in the ones place. The classification matters whenever items need to be shared equally into two — pairing shoes, matching players, splitting a packet of biscuits between two children.

Examples

• 141 is odd — ones digit is 1.
• 208 is even — ones digit is 8.
• 6 124 is even — ones digit is 4.
• 20 687 is odd — ones digit is 7.
• 36 120 is even — ones digit is 0.

Key Points

• Check only the ones digit.
• Ones digit 1, 3, 5, 7, 9 → odd.
• Ones digit 0, 2, 4, 6, 8 → even.

Prime and Composite Numbers

A prime number is a number with exactly two factors: 1 and itself. A factor of a number is any whole number that divides into it without a remainder — for example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

A composite number is a number with more than two factors.

The numbers 0 and 1 are neither prime nor composite. Every composite number can be written as a product of prime numbers.

Prime numbers are the building blocks of cryptography — the system that keeps online banking, mobile-money transactions, and private messages secure. The difficulty of splitting a very large composite number back into its prime factors is what makes digital communication hard to crack.

Examples

• 47 is prime. Factors: 1 and 47.
• 99 is composite. Factors: 1, 3, 9, 11, 33, 99.
• 173 is prime.
• 377 is composite. Factors: 1, 13, 29, 377.
• 991 is prime.

The prime numbers between 1 and 40 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37.

Key Points

• Prime = exactly two factors (1 and itself).
• Composite = more than two factors.
• 0 and 1 are neither prime nor composite.
• Every composite can be written as a product of primes.

Application of Prime and Composite Numbers

The prime/composite rule answers a different practical question: in how many ways can items be arranged in equal rows?

• A prime number of items fits in one row only — the only factor pair is 1 and the number itself.
• A composite number of items fits in several different rectangles, because it has several factor pairs.

Examples from real life

• A farmer plants 17 trees — prime — only one row of 17 is possible.
• A teacher has 24 learners — composite — can be grouped into 2 × 12, 3 × 8, 4 × 6, and more.
• A shopkeeper has 29 oranges — prime — only one row of 29 on the display shelf.

Key Points

• Prime count → one row only.
• Composite count → multiple rectangular arrangements.
• The number of arrangements equals the number of factor pairs.

Rational and Irrational Numbers

Real numbers are the combination of rational numbers and irrational numbers.

A rational number is any number that can be written as a fraction p/q, where q is not zero. This includes positive numbers, negative numbers, and zero.

An irrational number is a number that cannot be written as p/q. Irrational numbers are non-terminating (they never end) and non-repeating (they never settle into a repeating pattern) when written as decimals.

Both types appear in construction and design. Rationals show up as whole metres, standard fractions, and neat decimals on a tape measure. Irrationals appear whenever a square root is involved — the diagonal of a 1 m × 1 m floor tile is exactly √2 metres, which no fraction can capture perfectly. Builders round to a rational approximation for cutting, but the exact value is irrational.

Examples of rational numbers

• 1/2, 5/4, 2 1/6 — already in fraction form.
• 9 — a whole number, written as 9/1.
• −12 — written as −12/1.
• 0.777… — a recurring decimal, which can be written as a fraction.
• √36 — simplifies to 6.

Examples of irrational numbers

• √5, √7 — do not simplify, decimals go on forever without repeating.
• 8.861528… — decimal does not terminate and has no repeating pattern.

Key Points

• Real = Rational + Irrational (every real number is one or the other, never both).
• Rational: writable as p/q; includes all fractions, terminating decimals, and recurring decimals.
• Irrational: non-terminating, non-repeating decimals; many square roots.

Reciprocal by Division

The reciprocal of a number is 1 divided by that number. For any non-zero number x, the reciprocal is 1/x.

Two key facts:

• The product of a number and its reciprocal is 1.
• Zero has no reciprocal, because dividing by zero is undefined.

Reciprocals are the tool behind everyday rates and unit costs. If a bus covers 60 km in one hour, its reciprocal tells you one kilometre takes 1/60 of an hour. If 5 kg of sugar costs Ksh 650, the reciprocal of 5 gives the cost per kilogram.

Examples

• Reciprocal of 4: 1 ÷ 4 = 1/4 = 0.25.
• Reciprocal of −8: 1 ÷ −8 = −1/8 = −0.125.
• Reciprocal of 4/5: flip to get 5/4 = 1.25. Check: 4/5 × 5/4 = 1. ✓
• Reciprocal of 5 1/3: convert to 16/3, then flip to 3/16 = 0.1875.

The procedure

1. If the number is a mixed number, convert to an improper fraction first.
2. Write 1 divided by the number, or flip the fraction.
3. Simplify the result.
4. Check: original × reciprocal = 1.

Key Points

• Reciprocal = 1/x.
• For fractions, flip.
• Zero has no reciprocal.
• Product of a number and its reciprocal is always 1.

Reciprocal from Mathematical Tables (1 to 10)

Reciprocal tables are printed tables that let you read off the reciprocal of a number without dividing. They are part of the KNEC Four-Figure Mathematical Tables booklet used throughout Senior School. Before calculators became common, these tables were the main tool scientists and engineers used to handle long calculations quickly and consistently — and they remain a reliable backup when a calculator is unavailable.

The table is arranged for numbers between 1 and 10 only. For other numbers, express them in standard form first (covered in the next block).

Reading the layout

An extract from a reciprocal table:

• The x column holds the number to one decimal place (2.4, 2.5, …).
• The main columns 0–9 give the reciprocal when the second decimal is 0, 1, 2, … 9.
• The subtract columns 1–9 give the mean differences — small adjustments for the third decimal place.

The procedure — for 2.457

1. Locate 2.4 in the x column.
2. Move to the column headed 5 (second decimal) → 0.4082.
3. Move to mean difference column 7 (third decimal) → 12.
4. Subtract 12 from 0.4082 on the right-hand side: 0.4082 − 0.0012 = 0.4070.
5. Reciprocal of 2.457 is 0.4070.

For a number with only one or two decimal places, steps 3 and 4 are not needed.

Examples

• Reciprocal of 2.4: row 2.4, column 0 → 0.4167.
• Reciprocal of 2.45: row 2.4, column 5 → 0.4082.
• Reciprocal of 2.457: 0.4082 − 12 → 0.4070.

Key Points

• Tables work only for numbers between 1 and 10.
• Three lookups: row (first decimal) → main column (second decimal) → mean difference (third decimal).
• Always subtract the mean difference, never add.

Reciprocals Outside 1 to 10

Reciprocal tables only cover numbers between 1 and 10. For any other number, rewrite it in standard form first.

Standard form reminder: A number in standard form is written as A × 10ⁿ, where A is between 1 and 10 and n is an integer. For example, 274.6 = 2.746 × 10² and 0.0007431 = 7.431 × 10⁻⁴.

The procedure

1. Write the number in standard form A × 10ⁿ.
2. Use the reciprocal table to find 1/A.
3. The reciprocal of 10ⁿ is 10⁻ⁿ — flip the sign of the power.
4. Multiply the two parts together.

Example 1 — reciprocal of 274.6

1. 274.6 = 2.746 × 10².
2. Reciprocal of 2.746: row 2.7, column 4 → 0.3650. Mean difference column 6 → 8. Subtract: 0.3650 − 0.0008 = 0.3642.
3. Reciprocal of 10² is 10⁻².
4. 0.3642 × 10⁻² = 0.003642.

Reciprocal of 274.6 is 0.003642.

Example 2 — reciprocal of 0.0007431

1. 0.0007431 = 7.431 × 10⁻⁴.
2. Reciprocal of 7.431 from the table is 0.1346.
3. Reciprocal of 10⁻⁴ is 10⁴.
4. 0.1346 × 10⁴ = 1 346.

Reciprocal of 0.0007431 is 1 346.

Key Points

• Always convert to standard form first.
• Reciprocal of 10ⁿ is 10⁻ⁿ — just flip the sign of the power.
• Multiply the two reciprocals together for the final answer.