Real Numbers

The One Thing

Real numbers comprise both rational and irrational numbers. Their arithmetic properties are analysed using Euclid’s division lemma and algorithm, prime factorisation, the HCF–LCM relationship, and the denominator-based classification of rational decimal expansions.

Definitions and Results

• Real numbers: All rational and irrational numbers together form the set of real numbers.

• Rational number: A number that can be written in the form $p/q$, where $p$ and $q$ are integers and $q\ne 0$.

• Irrational number: A number that cannot be expressed as $p/q$, and whose decimal expansion is non-terminating and non-recurring.

• Euclid’s division lemma: For positive integers $a$ and $b$, there exist unique integers $q$ and $r$ such that

$$a=bq+r,$$ where $0\le r<b$. Here, $q$ and $r$ are whole numbers.

• Euclidean algorithm: A repeated application of Euclid’s division lemma to find the HCF of two positive integers.

• HCF or GCD: The greatest positive integer that divides each of two or more given numbers exactly.

• LCM: The least positive number that is exactly divisible by each of the given numbers.

• Prime factorisation: Writing a number as a product of its prime factors.

• Terminating decimal: A decimal expansion that ends after a finite number of digits, such as

$$\mathrm{0.125.}$$

• Non-terminating recurring decimal: A decimal expansion that continues indefinitely but repeats a pattern, such as

$$0.333\dots .$$

• Non-terminating non-recurring decimal: A decimal expansion that continues forever without repeating any fixed pattern. Such an expansion represents an irrational number.

• HCF transformation: If

$$a=bq+r,$$ then $$\mathrm{HCF}(a,b)=\mathrm{HCF}(b,r).$$

• HCF–LCM relationship: For two positive integers $a$ and $b$,

$$\mathrm{HCF}(a,b)\times \mathrm{LCM}(a,b)=a\times b.$$

• HCF from prime factorisation: When numbers are expressed in prime factorised form, their HCF is obtained by taking the lowest powers of all common prime factors.

• LCM from prime factorisation: Their LCM is obtained by taking the highest powers of all prime factors appearing in either number.

• Decimal criterion for rational numbers: If $p/q$ is in lowest form, its decimal expansion terminates if and only if the prime factorisation of $q$ contains only the primes $2$ and/or $5$.

• Non-terminating recurring criterion: If the denominator $q$, in lowest form, has any prime factor other than $2$ or $5$, then $p/q$ has a non-terminating recurring decimal expansion.

• Number of decimal places: If

$$q=2^m\times 5^n,$$ then the decimal expansion of $p/q$ terminates after at most $$\max (m,n)$$ decimal places.

• Classification of rational decimals: Every terminating decimal and every non-terminating recurring decimal represents a rational number.

• Irrational square roots: The square root of a prime number is irrational. More generally, the square root of a positive integer is irrational when the integer is not a perfect square.

• Decimal form of irrational numbers: The decimal expansion of an irrational number is non-terminating and non-recurring.

Worked Methods

1. Applying Euclid’s division lemma

• Identify positive integers $a$ and $b$, with $a$ as the dividend and $b$ as the divisor.
• Divide $a$ by $b$.
• Write the result in the form

$$a=bq+r.$$

• Check that $0\le r<b$.
• Use the uniqueness of the quotient and remainder guaranteed by Euclid’s division lemma.

Every division of a positive integer by another positive integer produces a unique quotient and a remainder smaller than the divisor.

2. Finding the HCF using Euclid’s algorithm

• Divide the larger number by the smaller number.
• Record the quotient and remainder.
• Divide the previous divisor by the remainder.
• Continue this process, replacing the divisor and remainder at each stage.
• Stop when the remainder becomes zero.
• The last non-zero remainder is the HCF.

The process must eventually stop because the positive remainders decrease. The last non-zero remainder divides both original numbers.

At each stage, use $$\mathrm{HCF}(a,b)=\mathrm{HCF}(b,r)$$ when $$a=bq+r.$$

3. Finding the HCF and LCM by prime factorisation

• Write each number as a product of prime factors.
• For the HCF, identify the prime factors common to all the numbers.
• Select the lowest power of each common prime factor.
• Multiply these selected powers to obtain the HCF.
• For the LCM, include every prime factor appearing in either number.
• Select the highest power of each prime factor.
• Multiply these powers to obtain the LCM.
• Verify the result, where appropriate, using

$$\mathrm{HCF}(a,b)\times \mathrm{LCM}(a,b)=a\times b.$$

This relationship follows from prime factorisation: for each prime, the product of the lowest and highest powers equals the product of the corresponding powers in the two original numbers.

4. Determining whether a rational decimal terminates

• Express the rational number as $p/q$.
• Reduce the fraction to lowest form.
• Prime-factorise the denominator $q$.
• If $q$ contains only the primes $2$ and/or $5$, the decimal terminates.
• If $q$ contains any prime factor other than $2$ or $5$, the decimal is non-terminating recurring.
• If

$$q=2^m\times 5^n,$$ the decimal terminates after at most $\max (m,n)$ decimal places.

Examples:

$$\frac{3}{8}=0.375$$ is terminating because $8=2^3$, so its denominator contains only the prime $2$.

$$\frac{7}{12}=0.58333\dots$$ is non-terminating recurring because $12=2^2\times 3$, and the denominator contains the prime $3$.

The denominator determines the decimal type only after the fraction has been reduced to lowest form.

5. Proving that $\sqrt{2}$ is irrational

• Assume, for contradiction, that

$$\sqrt{2}=\frac{p}{q},$$ where $p$ and $q$ are integers, $q\ne 0$, and $p/q$ is in lowest form.

• Square both sides:

$$2=\frac{p^2}{q^2}.$$

• Rearrange:

$$p^2=2q^2.$$

• Since $p^2$ is even, $p$ must be even. Write $p=2k$.
• Substitute:

$$(2k{)}^2=2q^2.$$

• Simplify:

$$4k^2=2q^2,q^2=2k^2.$$

• Therefore $q$ is also even.
• Both $p$ and $q$ are even, contradicting the assumption that $p/q$ is in lowest form.
• Hence $\sqrt{2}$ is irrational.

Where It Goes Wrong

• In Euclid’s division lemma, the remainder condition $0\le r<b$ is often omitted or incorrectly stated.
• In Euclid’s algorithm, the process must continue until the remainder becomes zero; the HCF is the last non-zero remainder, not the final zero.
• In prime-factor methods, the HCF requires the lowest powers of common prime factors, whereas the LCM requires the highest powers of every factor appearing in either number.
• The HCF–LCM formula applies to two positive integers and must be used as

$$\mathrm{HCF}(a,b)\times \mathrm{LCM}(a,b)=a\times b.$$

• The denominator rule for decimal expansions applies only after $p/q$ has been reduced to lowest form.
• In the proof that $\sqrt{2}$ is irrational, the assumption that $p/q$ is in lowest form is essential; deriving that both $p$ and $q$ are even creates the contradiction.

What Gets Asked

• State Euclid’s division lemma and identify the restrictions on $a$, $b$, $q$, and $r$.
• Use Euclid’s algorithm to find the HCF of two positive integers.
• Justify why the Euclidean algorithm terminates.
• Find the HCF and LCM from prime factorisations.
• Use

$$\mathrm{HCF}(a,b)\times \mathrm{LCM}(a,b)=a\times b$$ to determine an unknown HCF or LCM.

• Decide whether a rational number has a terminating or non-terminating recurring decimal expansion.
• Determine the maximum number of decimal places when

$$q=2^m\times 5^n.$$

• Classify $0.125$, $0.333\dots$, and other decimal expansions as terminating, non-terminating recurring, or non-terminating non-recurring.
• Explain why

$$\frac{3}{8}=0.375$$ is terminating and why $$\frac{7}{12}=0.58333\dots$$ is non-terminating recurring.

• Prove that $\sqrt{2}$ is irrational.
• State why the square root of a prime number, or of a positive integer that is not a perfect square, is irrational.
• Distinguish between rational and irrational numbers using their decimal expansions.