Types of Numbers (Cambridge (CIE) IGCSE International Maths)

Numbers form the cornerstone of all mathematical study, and understanding their classifications is fundamental for success in Cambridge (CIE) IGCSE International Maths. Whether you are dealing with natural numbers, integers, fractions, or irrational numbers, each type of number has specific properties and plays a unique role in various mathematical contexts. In these notes, we will explore the types of numbers you are expected to know at the IGCSE level, including detailed explanations, examples, and progressive exercises that start easy and become more challenging.

Our goal is to provide a thorough resource so you can understand the definitions, identify the sets of numbers, and confidently solve problems that require classifying numbers or operating within particular sets. We will also discuss some typical pitfalls or misconceptions students may have, especially when working between different sets of numbers.

By the end of these notes, you should have a clear picture of the number system hierarchy, from the very first numbers we learn as children (the counting numbers) to the more abstract realm of irrational numbers.

1. Introduction to Number Classifications

The classification of numbers typically follows a hierarchical structure. One way to visualize it is to imagine concentric sets (like nested circles), where each new set includes the previous set but extends it to include new elements. Below is a common classification you’ll find in textbooks:

Natural Numbers (also called counting numbers)
Whole Numbers
Integers
Rational Numbers
Irrational Numbers
Real Numbers
(Some courses also introduce Complex Numbers, but they are usually beyond the core IGCSE syllabus.)

Each of these sets expands upon the previous ones, except for some variations with the definitions of "natural numbers" (whether zero is included or not). In the Cambridge (CIE) IGCSE International Maths syllabus, the most essential sets to master are the real-number sets, which include the rationals and the irrationals.

Let us start by clarifying each set in turn, along with definitions and examples.

2. Natural Numbers

The natural numbers, often denoted by $$\mathbb{N}$$ or sometimes $$\mathbf{N}$$, are typically what we use for counting discrete objects. They are the simplest set of numbers children first learn:

$$\mathbb{N} = \{1, 2, 3, 4, 5, \dots\}.$$

Some mathematicians include zero in the set of natural numbers, writing $$\{0, 1, 2, 3, \dots\}$$, but for many IGCSE contexts, we start counting at 1. Check your syllabus or teacher’s instructions for whether zero belongs to the natural numbers or the whole numbers.

Key properties:

They are all positive.
They are discrete (no fractions or decimals included).
They have no upper bound in theory (the set goes to infinity).

2.1 Example (Easy)

Question: List the first 5 natural numbers.

Solution: The first 5 natural numbers (excluding zero) are:

$$1, 2, 3, 4, 5.$$

2.2 Example (Slightly More Challenging)

Question: If you are told you have n sweets in a jar and you remove 1 sweet, how many sweets remain, assuming n is a natural number and n ≥ 2?

Solution: If n is a natural number and you remove 1 sweet, you are left with n − 1 sweets, which is still a natural number as long as n ≥ 2.

3. Whole Numbers

The set of whole numbers is very similar to the natural numbers, but it includes zero. Many references define the whole numbers as:

$$\{0, 1, 2, 3, 4, \dots\}.$$

Conceptually, if your curriculum states that the natural numbers start at 1, then the set of whole numbers is simply the natural numbers plus 0. Some educators and textbooks call them the “non-negative integers.”

3.1 Example (Easy)

Question: Is 0 a whole number?

Answer: Yes, 0 is a whole number by definition in most standard references.

3.2 Example (Moderate)

Question: You start with 0 marbles and add 3 marbles one at a time. Represent this using whole numbers at each step.

Solution:

Initially: 0 marbles (a whole number).
After adding 1 marble: 1 marble (natural number & whole number).
After adding a second marble: 2 marbles.
After adding a third marble: 3 marbles.

This simple scenario highlights that 0 belongs to the whole numbers, whereas 1, 2, and 3 belong to both the whole numbers and the natural numbers.

4. Integers

The set of integers, typically denoted by $$\mathbb{Z}$$, includes all whole numbers and also their negatives. In other words:

$$\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}.$$

This set is important for many algebraic operations, especially when dealing with expansions, factorizations, and more advanced manipulations.

4.1 Example (Easy)

Question: Which of the following numbers are integers? $$-4, 0.5, 0, 2.718, -5.25, 10$$

Answer and Reasoning:

$$-4$$ is an integer, because negative whole numbers are in the set of integers.
$$0.5$$ is not an integer, because 0.5 is a fraction (half).
$$0$$ is an integer, as zero belongs to $$\mathbb{Z}$$.
$$2.718$$ (approx. e) is not an integer, it’s an irrational number in fact (though you might only be asked whether it’s an integer or not at this level).
$$-5.25$$ is not an integer, because it has a decimal part.
$$10$$ is an integer because it is a whole number.

4.2 Example (Moderate)

Question: Evaluate the following expressions and determine if the result is an integer:

$$5 + (-7)$$
$$-8 - 4$$
$$6 \times 3$$
$$10 \div 2$$
$$9 \div 4$$

Solution with Reasoning:

$$5 + (-7) = -2$$, which is an integer (negative integer).
$$-8 - 4 = -12$$, also an integer.
$$6 \times 3 = 18$$, an integer.
$$10 \div 2 = 5$$, still an integer (division results in a whole number).
$$9 \div 4 = 2.25$$, not an integer; it becomes a decimal.

5. Rational Numbers

A rational number is any number that can be expressed as $$\frac{p}{q}$$, where p and q are integers and q ≠ 0. The set of rational numbers includes:

$$\mathbb{Q} = \left\{\frac{p}{q} : p \in \mathbb{Z},\ q \in \mathbb{Z},\ q \neq 0\right\}.$$

Rational numbers encompass all integers (since any integer $$n$$ can be written as $$\frac{n}{1}$$), fractions (proper and improper), and decimal numbers that either terminate or have repeating decimal expansions.

Examples of rational numbers: $$\frac{1}{2}, \frac{3}{4}, -\frac{7}{3}, 0.5, -2.3333\ldots (\text{a repeating decimal}), 7.$$

Non-examples: Numbers like $$\sqrt{2}$$ or $$\pi$$ are not rational, because they cannot be expressed as a ratio of two integers.

5.1 Example (Easy)

Question: Identify which numbers are rational from the set: $$\{0.75, -2, \sqrt{3}, 3.5, \pi\}.$$

Solution:

0.75 is rational because it can be written as $$\frac{3}{4}.$$
-2 is rational because it can be written as $$\frac{-2}{1}.$$
$$\sqrt{3}$$ is not rational (it is an irrational number).
3.5 is rational because $$3.5 = \frac{7}{2}.$$
$$\pi$$ is not rational (irrational transcendental number).

5.2 Example (Moderate)

Question: Convert the following decimal numbers to fraction form (in simplest terms) if they are rational:

0.2
-0.6
4.375
1.333... (repeating 3)

Solution:

$$0.2 = \frac{2}{10} = \frac{1}{5}.$$
$$-0.6 = \frac{-6}{10} = -\frac{3}{5}.$$
$$4.375 = 4 \frac{3}{8} = \frac{35}{8}.$$ (Detailed steps: 4.375 = 4 + 0.375, 0.375 = $\frac{3}{8}$. So total is $\frac{32}{8} + \frac{3}{8} = \frac{35}{8}$.)
$$1.333\ldots = 1.\overline{3} = \frac{4}{3}.$$ (If you are unfamiliar with converting repeating decimals to fractions, let x = 1.\overline{3}, then 10x = 13.\overline{3}, subtract x from 10x: 9x = 12, x = $\frac{12}{9} = \frac{4}{3}$.)

5.3 Example (Hard)

Question: Prove that 0.999... (repeating 9) is actually 1, and hence rational.

Solution Outline:

Let $$x = 0.999\ldots$$ (repeating 9 infinitely).
Then $$10x = 9.999\ldots$$ (shifting the decimal point one place to the right).
Subtract the original number: $$10x - x = 9.999\ldots - 0.999\ldots = 9.$$
Hence $$9x = 9$$, so $$x = 1.$$

Therefore, $$0.999\ldots$$ is exactly 1, which is a rational integer. This sometimes surprises students, but it’s a standard result in mathematics.

6. Irrational Numbers

Irrational numbers are real numbers that cannot be expressed as a ratio of two integers. Their decimal expansions neither terminate nor become purely repetitive. Examples include:

$$\sqrt{2}, \sqrt{3}, \sqrt{5}$$
$$\pi$$ (pi), $$e$$ (Euler’s number), $$\phi$$ (the golden ratio)

A critical concept is understanding that while rational numbers might have non-terminating decimals, those decimals repeat in a pattern. Irrational numbers have infinite, non-repeating decimal expansions.

At the IGCSE level, you typically work with simpler irrational numbers (square roots of non-perfect squares), and knowledge of $$\pi$$ is also used extensively in trigonometry and circle geometry. The distinction from rational numbers is crucial for certain proofs and for approximate calculations.

6.1 Example (Easy)

Question: Which of the following are irrational?

$$\sqrt{2}, \quad \sqrt{9}, \quad \frac{1}{\sqrt{2}}, \quad 3.14159\ldots (\pi)$$

Answer:

$$\sqrt{2}$$ is irrational.
$$\sqrt{9} = 3$$, which is rational (an integer).
$$\frac{1}{\sqrt{2}}$$ is also irrational (it’s a rational multiple of an irrational, which remains irrational).
$$\pi$$ is a well-known irrational number.

6.2 Example (Moderate)

Question: Explain why $$\sqrt{2}$$ is irrational in simple terms (no heavy proof required at IGCSE, but a brief intuition can be helpful).

Intuitive Sketch: If $$\sqrt{2}$$ were rational, it could be written as $$\frac{p}{q}$$ with p and q integers having no common factors. Squaring both sides would give $$2 = \frac{p^2}{q^2} \implies 2 q^2 = p^2.$$ This would imply that p has a factor of 2, and by further reasoning, q must also have a factor of 2, contradicting the assumption that p and q share no common factors. Hence no such fraction exists.

6.3 Example (Hard)

Question: Sometimes we approximate $$\pi$$ as 3.14 or 22/7. Explain why none of these approximations can be exact.

Solution Idea: Since $$\pi$$ is irrational, any rational fraction like 22/7 is only an approximation. The decimal expansion of $$\pi$$ goes on forever without repeating. Hence, while 3.14 or 22/7 might be good approximations for practical purposes, they can never represent $$\pi$$ exactly.

7. Real Numbers

The real numbers, denoted by $$\mathbb{R}$$, include all the rational and all the irrational numbers. Essentially, any number that can occupy a position on the number line is considered a real number. In IGCSE mathematics, unless specified otherwise, we typically assume we are working within the set of real numbers.

Key points:

All integers are real.
All fractions (rationals) are real.
All surds (simplest forms of square roots of non-perfect squares) are real and usually irrational.
Numbers like $$\pi$$ and $$e$$ are real.

7.1 Example (Moderate)

Question: Is $$\sqrt{2} \approx 1.414213562...$$ a real number, and where does it lie on the number line?

Solution: Yes, $$\sqrt{2}$$ is real (though irrational), and it lies between 1.4142 and 1.4143 on the number line. You can refine your approximation to as many decimal places as you like, but the exact value is impossible to express as a fraction or a finite decimal.

8. Prime and Composite Numbers

Within the set of integers, specifically the positive integers (natural numbers if you include 1), we also classify numbers as prime or composite.

A prime number is a positive integer greater than 1 that has exactly two distinct factors: 1 and itself.
A composite number is a positive integer greater than 1 that has more than two factors. In other words, it can be formed by multiplying two smaller positive integers.

Typical prime numbers encountered in lower-level maths: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, etc. Note that 1 is neither prime nor composite, and 2 is the only even prime number.

8.1 Example (Easy)

Question: Identify the prime numbers in the set $$\{1, 2, 3, 4, 9, 11, 12\}.$$

Solution:

1: neither prime nor composite.
2: prime (only factors: 1 and 2).
3: prime (only factors: 1 and 3).
4: composite (factors: 1, 2, 4).
9: composite (factors: 1, 3, 9).
11: prime (only factors: 1 and 11).
12: composite (factors: 1, 2, 3, 4, 6, 12).

8.2 Example (Moderate)

Question: List all prime numbers between 10 and 30.

Solution: 11, 13, 17, 19, 23, 29.

8.3 Example (Hard)

Question: Prove that for any integer $$n > 1$$, it must be either prime or composite (except for 1 which is neither). Then, show how to use prime factorization for 60.

Solution Sketch:

By the fundamental theorem of arithmetic (beyond the direct IGCSE scope for a “proof,” but recognized as a fact), every integer greater than 1 can be expressed uniquely as a product of prime factors (ignoring the order).
1 is a special case, not prime or composite.
Prime factorization of 60: $$60 = 2 \times 30 = 2 \times 2 \times 15 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5.$$

9. Surds (Irrational Roots)

Surds are often encountered in IGCSE Maths as expressions involving square roots (or roots of higher order) that cannot be simplified to remove the root sign. For instance, $$\sqrt{2}, \sqrt{3}, \sqrt{5}, \dots$$ are common surds.

Operations with surds typically include simplifying, rationalizing denominators, or combining like terms.

9.1 Simplifying Surds (Easy Example)

Question: Simplify $$\sqrt{50}.$$

Solution: $$\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}.$$

9.2 Rationalizing the Denominator (Moderate Example)

Question: Simplify $$\frac{3}{\sqrt{5}}.$$

Solution: Multiply the numerator and denominator by $$\sqrt{5}$$ to get:

$$\frac{3}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{3\sqrt{5}}{5}.$$

9.3 Combining Surds (Hard Example)

Question: Simplify $$\sqrt{8} + 3\sqrt{2} - 2\sqrt{18}.$$

Solution Steps:

Simplify each surd:
- $$\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}.$$
- $$3\sqrt{2}$$ remains as it is.
- $$2\sqrt{18} = 2\sqrt{9 \times 2} = 2 \times 3 \sqrt{2} = 6\sqrt{2}.$$
So the expression becomes: $$2\sqrt{2} + 3\sqrt{2} - 6\sqrt{2}.$$
Combine like terms: $$2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2},$$ and $$5\sqrt{2} - 6\sqrt{2} = -\sqrt{2}.$$

Final Answer: $$-\sqrt{2}.$$

10. Index Notation and Exponents

While not strictly a separate “type of number,” understanding integer exponents and fractional exponents is vital in dealing with real numbers in the IGCSE syllabus.

Rules of Exponents (briefly):

$$a^m \times a^n = a^{m+n}.$$
$$\frac{a^m}{a^n} = a^{m-n}.$$ (for $$a \neq 0$$)
$$(a^m)^n = a^{mn}.$$
$$a^{-n} = \frac{1}{a^n}.$$ (for $$a \neq 0$$)
$$a^{\frac{p}{q}} = \sqrt[q]{a^p}.$$ (fractional exponents)

These exponents can be applied to both rational and irrational bases, and the result typically remains within the set of real numbers (except for cases involving negative bases and fractional exponents leading to complex numbers, but that is generally beyond IGCSE).

11. Rounding and Approximation

Since we often deal with irrational numbers (like $$\sqrt{2}$$, $$\pi$$, etc.), we rely on approximations and rounding in real-life and test calculations. Remember that rational numbers can also be expressed in decimal form, where you might choose to round to a certain number of decimal places or significant figures.

Examples:

$$\pi \approx 3.142$$ to 4 significant figures.
$$\sqrt{2} \approx 1.4142$$ to 5 significant figures.
$$\frac{1}{3} \approx 0.3333$$ to 4 decimal places.

11.1 Rounding Example (Moderate)

Question: Round $$\pi$$ to 3 significant figures.

Solution: $$\pi \approx 3.14.$$ Since $$\pi = 3.1415926535...,$$ to 3 significant figures we stop at the hundredths place (because the digit after 4 is 1, which does not round the 4 up).

11.2 Rounding Example (Hard)

Question: Suppose you have $$\sqrt{5} \approx 2.2360679775...$$. Round to 4 decimal places and also express the same approximation in 2 significant figures.

Solution Steps:

To 4 decimal places: $$2.2361$$ (the next digit is 0, so 2.2360 remains 2.2361).
To 2 significant figures: The first 2 digits in 2.2360... are 2 and 2, so we look at the third digit (3) to decide if we round the second digit up. The result is $$2.2.$$

Notice that rounding can significantly change the outcome depending on the required decimal places or significant figures, especially for more complex calculations.

12. Hierarchy Recap

The following sequence shows how the types of numbers are nested from “smallest” (in the sense of the most exclusive set) to “largest” (in the sense of the most inclusive set):

Natural Numbers ⊂ Whole Numbers ⊂ Integers ⊂ Rational Numbers ⊂ Real Numbers.

Here, “⊂” indicates “is a subset of.” In practice, you may also see “⊆,” but the idea is the same: each set is contained within the next. The only addition that typically goes beyond the real numbers is the set of complex numbers, which are beyond the usual scope of IGCSE.

13. Mixed Examples and Exercises

Let us practice with a range of examples that integrate what we have learned. We will go from relatively straightforward classification problems to more involved manipulations involving surds or exponents.

13.1 Exercise Set (Easy)

Classify the following numbers as either rational or irrational: $$\frac{4}{5}, \quad 0.3333\ldots, \quad \sqrt{7}, \quad 0.125, \quad \pi, \quad 6.023.$$
Write 0.4 as a fraction in simplest form.
Identify whether 1 is prime, composite, or neither.
True or false: $$-3$$ is a natural number.

Solutions:

$$\frac{4}{5}$$ (rational), $$0.3333\ldots = \frac{1}{3}$$ (rational), $$\sqrt{7}$$ (irrational), $$0.125 = \frac{1}{8}$$ (rational), $$\pi$$ (irrational), 6.023 (rational if it terminates, but if it’s a measured value, we usually treat it as rational up to that decimal).
$$0.4 = \frac{2}{5}.$$
1 is neither prime nor composite.
False. $$-3$$ is not a natural number. It’s an integer, but natural numbers are positive (and possibly zero).

13.2 Exercise Set (Moderate)

Convert the recurring decimal 0.585858... to a fraction in simplest form.
Simplify the surd $$\sqrt{72}.$$
Evaluate and determine if the result is an integer: $$(-2)^4, \; (-3)^2, \; 2^5, \; 5 \div 2.$$
State the next prime number after 19.

Solutions (Outline):

Let $$x = 0.\overline{58}.$$ Then $$100x = 58.\overline{58}.$$ Subtract $$x$$ from $$100x$$ to get $$99x = 58,$$ so $$x = \frac{58}{99}.$$ Simplify if needed (58 and 99 share no common factors, so that’s simplest form).
$$\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}.$$
$$(-2)^4 = 16$$ (integer), $$(-3)^2 = 9$$ (integer), $$2^5 = 32$$ (integer), $$5 \div 2 = 2.5$$ (not an integer).
After 19, the next prime is 23.

13.3 Exercise Set (Hard)

Write a brief explanation of why $$\sqrt{3}$$ must be irrational without using the formal proof by contradiction.
Simplify: $$\frac{2\sqrt{18}}{3\sqrt{2}}.$$
Rationalize the denominator of $$\frac{5}{\sqrt{2} - \sqrt{3}}.$$
Convert 1.272727... (repeating 27) to a fraction in simplest form.

Solution Sketch:

Intuition for $$\sqrt{3}$$ being irrational: A repeating or terminating decimal pattern does not exist for $$\sqrt{3}$$ when you attempt long division or calculator approximation. Every attempt to express it as a fraction leads to contradictions.
Simplify $$\frac{2\sqrt{18}}{3\sqrt{2}} = \frac{2 \times 3\sqrt{2}}{3\sqrt{2}} = \frac{6\sqrt{2}}{3\sqrt{2}}.$$ Then $$\frac{6\sqrt{2}}{3\sqrt{2}} = \frac{6}{3} = 2.$$ (Since the $$\sqrt{2}$$ cancels out top and bottom.)
$$\frac{5}{\sqrt{2} - \sqrt{3}} \times \frac{\sqrt{2} + \sqrt{3}}{\sqrt{2} + \sqrt{3}} = \frac{5(\sqrt{2} + \sqrt{3})}{(\sqrt{2})^2 - (\sqrt{3})^2} = \frac{5(\sqrt{2} + \sqrt{3})}{2 - 3} = \frac{5(\sqrt{2} + \sqrt{3})}{-1} = -5(\sqrt{2} + \sqrt{3}).$$
Let $$x = 1.2\overline{72}.$$ Then $$x - 1 = 0.2\overline{72}.$$ Often, it’s easier to consider $$y = 0.\overline{72}.$$ Then convert that fraction and add 1.2 to the final result. Alternatively, you can consider the repeating block carefully:
If you interpret 1.272727... as 1 + 0.272727..., let $$z = 0.\overline{27} = \frac{27}{99} = \frac{3}{11}.$$ Then $$1 + 0.272727... = 1 + \frac{3}{11} = \frac{11}{11} + \frac{3}{11} = \frac{14}{11}.$$ So $$1.272727... = \frac{14}{11}.$$

14. Common Pitfalls and Misconceptions

1. Confusing Natural Numbers and Whole Numbers: Some textbooks include 0 in the natural numbers; some do not. The IGCSE syllabus generally focuses on the distinction that whole numbers include 0, while natural numbers start at 1.

2. Mixing Up Rational and Irrational: Remember that any decimal that repeats or terminates is rational. Non-repeating, non-terminating decimals are irrational.

3. Believing 1 Is Prime or Composite: 1 is neither prime nor composite. This is an important fact to avoid factorization errors.

4. Overlooking Negative Integers as Part of the Integers: $$\{-1, -2, -3, \dots\}$$ are all integers. Many students forget to check negative values in integer classification questions.

5. Failing to Simplify Surds Properly: Not pulling out perfect squares (like 9, 16, 25, etc.) can lead to losing marks in simplifying expressions involving roots.

6. Unclear on Rounding Rules: In problems where you are asked to give your answer to a certain number of decimal places or significant figures, carefully observe the next digit to decide on rounding up or not.

15. Summary and Final Notes

In the Cambridge (CIE) IGCSE International Maths course, a solid understanding of the types of numbers underpins much of the subsequent work in algebra, geometry, statistics, and even basic trigonometry. The number system can be visualized as a layered framework:

Natural Numbers ⊂ Whole Numbers ⊂ Integers ⊂ Rational Numbers ⊂ Real Numbers

We also explored how prime and composite numbers fit into the mix, and the significance of irrational numbers (such as surds, $$\sqrt{2}, \pi,$$ etc.) that do not fit neatly into fraction form. Moreover, dealing with exponents and surds requires a firm grip on the properties of real numbers, especially how irrational values behave under multiplication, division, and simplification.

A key takeaway is the ability to classify any given number quickly into its respective set and then use the properties of that set (e.g., whether a fraction is simpler to handle than a decimal, or whether an irrational expression can be simplified by factoring out perfect squares).

As you continue to study, practice classifying numbers, simplifying expressions involving surds, and converting between decimal and fraction forms. This practice will help solidify your number sense, which is vital not just for exams, but for deeper mathematical understanding at higher levels.

We hope these comprehensive notes have given you the clarity and confidence needed to tackle Types of Numbers questions in your IGCSE International Maths course. Remember: mastery of these foundational concepts is a gateway to more advanced topics. Keep practicing and exploring!