Exact Arithmetic and Number Structure

Prerequisites: signed integers, multiplication tables and the meaning of a fraction. All practice questions in this lesson are original, not official TMUA questions.

What This Lesson Is About

This lesson develops exact arithmetic and number structure: how to read a calculation correctly, keep its value unchanged, and use the factors of a number to explain its properties. These skills help you decide not only what an answer is, but why your method is valid.

We begin with the meaning of an expression and the order in which its operations should be carried out. We then use that understanding to simplify and compare fractions. Next, we look inside whole numbers through prime factorisation, convert recurring decimals into fractions, and finish by testing claims about numbers. Each stage builds on the same habit: identify the structure before calculating.

1. Reading and Evaluating Numerical Expressions

A numerical expression is a combination of numbers and operations, such as 2 + 3 × 4 or (3/4 − 1/6)/(5/8). To evaluate it means to find the value it represents. Before calculating, identify which parts belong together and which operations must happen first; otherwise, you may calculate a different expression from the one given.

1.1 Exact values: keeping the original quantity

An exact value represents the whole quantity without rounding. For example, 2/3 is exact, but 0.667 is a rounded approximation to it. Replacing 2/3 by 0.667 changes the quantity slightly, which can affect a later calculation or a comparison between close values.

A decimal is not automatically approximate: 0.5 is exactly 1/2. The important distinction is whether digits have been rounded or discarded. In this lesson, keep fractions exact unless a question explicitly requests an approximation.

1.2 Order of operations: deciding what to calculate first

When an expression contains several operations, you need a shared rule for reading it. For instance, 2 + 3 × 4 means 2 plus the product of 3 and 4, so its value is 14. Adding 2 and 3 first would instead give 20, the value of a different expression, (2 + 3) × 4.

This shared rule is called the order of operations. Evaluate grouped expressions in brackets first, then powers. Next carry out multiplication and division from left to right, and finally addition and subtraction from left to right. Operations at the same level have equal priority: division does not have to wait until every multiplication is finished.

For example, 18 ÷ 3 × 2 is read from left to right at the multiplication-and-division stage: 18 ÷ 3 = 6, then 6 × 2 = 12. It is not 18 ÷ (3 × 2), which would be 3. Brackets are how you explicitly request that different grouping.

1.3 Fraction bars: grouping the numerator and denominator

The same grouping rule applies to a fraction bar. Its numerator is one complete expression, and its denominator is another. In (a + b)/(c + d), first form the sum a + b and the sum c + d, then divide the first by the second, provided the second is nonzero.

For example, (6 + 2)/(3 + 1) = 8/4 = 2. The parentheses in this one-line notation make clear that the whole sum 6 + 2 is being divided by the whole sum 3 + 1. When a numerator contains fractions of its own, simplify that grouped numerator before carrying out the outer division.

1.4 Negative signs and powers: identifying the base

A power acts on its base, so brackets also matter when negative numbers are involved. In (−3)², the brackets make −3 the base: (−3) × (−3) = 9. In −3², the base of the square is 3 and the negative sign remains outside, giving −(3²) = −9.

This is another application of reading the expression before calculating. The position of the brackets, rather than a change in the rule for multiplication, explains why the two answers differ.

1.5 Division by zero: checking whether a value exists

Before evaluating any quotient, check that its denominator is not zero. Division by zero is undefined. For example, 5/(3 − 3) has no value because its denominator is zero; it is not equal to zero.

This restriction will matter again when we simplify fractions by cancellation. A simplification must preserve a valid original value, so it cannot make an undefined division acceptable.

2. Simplifying and Comparing Fractions

Once you have read a fractional expression correctly, you can look for a simpler way to calculate it without changing its value. Two useful methods are cancellation, which removes shared factors, and a common denominator, which puts fractions on the same scale for comparison.

2.1 Terms and factors: knowing what can be cancelled

A term is a part of a sum or difference. In 6 + 9, the terms are 6 and 9. A factor is a quantity multiplying another quantity: writing 6 + 9 as 3(2 + 3) shows that 3 multiplies the whole sum in the brackets.

Cancellation removes the same nonzero factor from the whole numerator and the whole denominator. It works because that common factor divided by itself equals 1. You cannot remove a factor from just one term of a sum while leaving the other term unchanged.

2.2 Cancelling before multiplication: keeping the numbers small

When two fractions are multiplied, the numerator is a product and the denominator is a product. This makes shared factors easy to remove before large numbers are formed. The result is the same as multiplying first and simplifying afterwards.

For example, (14/15) × (5/7) = (14 × 5)/(15 × 7). Divide 14 and 7 by 7, and divide 5 and 15 by 5, to obtain (2 × 1)/(3 × 1) = 2/3. Multiplying first gives 70/105, which simplifies to the same answer.

2.3 Comparing fractions: expressing them in equal-sized parts

To compare two fractions exactly, rewrite them with a common positive denominator. They then count parts of the same size, so the larger numerator gives the larger value. This avoids rounding decimals that may be very close together.

3. Using Prime Factors to Understand Whole Numbers

The previous section used common factors to simplify fractions. We now develop a systematic way to find those factors: break each whole number into a product of primes. The resulting prime factorisation helps answer questions about divisibility, common multiples and perfect squares.

3.1 Prime factorisation: the building blocks of an integer

A prime is an integer greater than 1 with exactly two positive divisors, 1 and itself. For example, 2, 3 and 5 are prime. The number 1 is not prime because it has only one positive divisor.

Every positive integer greater than 1 can be written uniquely as a product of primes, apart from their order. For example, 72 = 8 × 9 = 2³ × 3². The exponent 3 records three copies of 2, and the exponent 2 records two copies of 3.

Divisibility can now be read from the factors. A positive integer divides 72 only if it uses no more copies of any prime than 72 contains. Thus 2² × 3 divides 72, but 2⁴ does not, because 72 contains only three copies of 2.

3.2 Highest common factor: the largest divisor shared by two numbers

The highest common factor, or HCF, is the largest positive integer that divides both numbers exactly. It can contain only primes present in both numbers, and it cannot use more copies of a prime than either number supplies. For each shared prime, therefore, choose the smaller exponent.

3.3 Lowest common multiple: the smallest number divisible by both

The lowest common multiple, or LCM, is the smallest positive integer that both original numbers divide. It must contain enough copies of every prime to accommodate either number. This time choose the larger exponent for each prime present in at least one of the numbers.

3.4 Perfect squares: pairing the prime factors

A positive perfect square is the product of a positive integer with itself. Every prime factor in the integer is duplicated when it is squared, so every prime exponent in the square is even. For example, 12² = (2² × 3)² = 2⁴ × 3².

Conversely, if every prime exponent is even, pair up the prime factors and take one from each pair to form the square root. This gives a way to find the smallest positive integer multiplier that turns a number into a square: supply one extra copy of each prime whose exponent is odd.

4. Converting Recurring Decimals into Fractions

So far, we have kept fractions exact rather than rounding them. Sometimes a number is supplied as a recurring decimal instead. Converting it to a fraction puts it back into a form that can be simplified, compared and used in exact calculations.

4.1 The repeating block: identifying which digits recur

A recurring decimal has a digit or block of digits that repeats indefinitely. In 0.272727…, the block 27 repeats from the first decimal place. In 0.166666…, the first digit 1 does not repeat; the recurring block is the single digit 6 that follows it. The dots mean the indicated repetition continues indefinitely.

The conversion method works by making two copies of the decimal with matching repeating tails. Multiplying by 10, 100 or another power of ten shifts the digits. Subtracting the aligned copies then removes the identical decimal tails and leaves an equation involving whole numbers.

4.2 Aligning the tails: choosing the power of ten

When the decimal begins with a nonrepeating digit, first shift that initial part to the left of the decimal point. Then shift once more by the length of the repeating block. The two repeating tails will align, even though the original decimal and a single shifted copy might not.

5. Testing Statements about Numbers

The earlier sections found particular values. A different kind of question asks whether a statement is true for every number in a specified group. Here, the task is to justify a claim rather than just perform a calculation. We can use the number properties already studied, but we need to be clear about what an example does and does not establish.

5.1 Examples and counterexamples: what one case can show

A successful example shows only that a claim works in that case. It does not prove a statement about every integer. A counterexample is an allowed case in which the claim fails; just one is enough to disprove a claim that says it always holds.

5.2 A general argument: explaining why a rule always holds

To prove a statement about every allowed integer, use a representation covering all of them. An even integer can be written as 2m for an integer m. An odd integer can be written as 2m + 1, because it is one more than a multiple of 2.

For example, let two odd integers be 2a + 1 and 2b + 1, where a and b are integers. Their sum is 2a + 1 + 2b + 1 = 2(a + b + 1). The quantity in brackets is an integer, so the sum is a multiple of 2 and therefore even. Unlike checking 3 + 5, this argument covers every pair of odd integers.

This explains the prime-number counterexample more broadly. Every prime other than 2 is odd: any even integer greater than 2 has 2 as an extra divisor and cannot be prime. Adding two odd primes therefore gives an even sum, while adding 2 to an odd prime gives an odd sum. The outcome depends on the kind of primes chosen.

6. Common Mistakes and How to Correct Them

7. Practice

Work through these questions in order. They revisit evaluating an expression, comparing fractions, using prime factors, converting a recurring decimal and judging a number claim. Keep answers exact, and include the reason for your method as well as the calculation.

1. Evaluate (5/6 − 1/4) ÷ (7/9).

2. Order 5/8, 7/12 and 2/3 from least to greatest.

3. Find the HCF and LCM of 84 and 126.

4. Find the least positive integer k such that 180k is a perfect square.

5. Convert 0.166666… to an exact fraction.

6. Which statement is always true for integers a and b? A: if ab is even, a and b are both even. B: if ab is odd, a and b are both odd. C: if a + b is even, a and b are both even. D: if a² is even, a is odd. Explain your choice and reject the others.

Worked Solutions

1. Evaluate the grouped numerator before dividing

The numerator is the difference 5/6 − 1/4. Using denominator 12 gives 10/12 − 3/12 = 7/12. Now divide by 7/9 by multiplying by its reciprocal: (7/12) × (9/7) = 9/12 = 3/4.

Check by reversing the operations: (3/4) × (7/9) = 7/12, and adding 1/4 gives 10/12 = 5/6. The answer is 3/4.

2. Compare fractions with a common denominator

Use denominator 24 so the parts have equal size: 5/8 = 15/24, 7/12 = 14/24 and 2/3 = 16/24. Ordering the numerators gives 14 < 15 < 16, so the requested order is 7/12 < 5/8 < 2/3.

3. Choose prime exponents for the HCF and LCM

Factorise the numbers: 84 = 2² × 3 × 7 and 126 = 2 × 3² × 7. For the HCF, take the smaller exponents of the shared primes: 2 × 3 × 7 = 42. For the LCM, take the larger exponents: 2² × 3² × 7 = 252.

Check the meanings directly: 84/42 = 2 and 126/42 = 3, so 42 divides both. Also 252/84 = 3 and 252/126 = 2, so both divide 252. The exponent choices ensure that these are the highest common factor and lowest common multiple.

4. Supply the missing partner in the prime factorisation

180 = 2² × 3² × 5. Only the exponent of 5 is odd, so multiply by one more 5 to make every prime exponent even. Thus k = 5 and 180k = 900 = 30².

Every valid multiplier must contain a factor of 5 to pair the existing unpaired 5. No positive integer smaller than 5 does, so this multiplier is the least.

5. Align the recurring tails after the initial digit

Let x = 0.166666…. The initial 1 does not repeat. Multiplying by 10 and 100 gives 10x = 1.666666… and 100x = 16.666666…; these have identical decimal tails.

Subtract: 100x − 10x = 15, so 90x = 15 and x = 1/6. Long division of 1 by 6 gives 0.166666…, checking the conversion.

6. Prove the true statement and reject the others

B is true: if ab is odd, then a and b must both be odd. To justify this, suppose either factor were even, say a = 2m for an integer m. Then ab = 2(mb) would be even, contrary to the given odd product. The same argument applies if b were even, so neither factor can be even.

For each other option, an allowed counterexample is enough. A fails when a = 2 and b = 3: the product 6 is even but b is odd. C fails when a = b = 1: their sum 2 is even but both integers are odd. D fails when a = 2: its square 4 is even but a is not odd.

8. Summary

• Read grouping and the order of operations before calculating; brackets also determine the base of a power.
• Keep exact values, cancel shared nonzero factors and use common denominators to compare fractions.
• Prime factorisation explains HCFs, LCMs and the paired factors of perfect squares.
• Convert recurring decimals by aligning identical tails with powers of ten and subtracting.
• Use a general argument to prove an “always” claim and an allowed counterexample to disprove one.