Introduction To Number Theory | Cambridge IGCSE Exam Prep

Number theory

[Please watch the video attached at the end of this blog for a visual explanation of number theory]

What is number theory in mathematics?

Number theory is a branch of mathematics that studies the types of numbers, especially whole numbers (integers). It looks at prime numbers, how numbers can be divided into smaller parts, the patterns that emerge when working with numbers and so much more. It’s a way of exploring the wonders of numbers, and finding out how they behave! ?

The number theory sub-topic in the IGCSE curriculum aims to aid students to understand the different types of numbers and develop their analytical skills.

In IGCSE exams, number theory questions may appear as single questions or as part of a bigger mathematical problem that requires an understanding of number theory concepts. In this blog post, I will take you through the different types of numbers we should know about. 

Types of numbers in mathematics

There are several types of numbers in mathematics and each type has its own unique uses, and they are all interconnected in various ways. Let’s have a look at some different types of numbers we use in mathematics.

  • Natural numbers

Natural numbers are also known as counting numbers, usually written as {1, 2, 3, 4, 5, …}. They are used for counting and measuring and include whole numbers from 1 up to infinity.

In mathematics, the set of natural numbers is represented as N = {1, 2, 3, 4, 5, …}.

Natural numbers are used widely in real-world situations, such as calculating the number of items in a grocery list, the number of students in a classroom, or the number of ducks in the lake.

Natural numbers are one of the earliest mathematical concepts known to mankind and have been used for many years to quantify things. It is basically the foundation of mathematics that help us understand more advanced mathematical concepts.

  • Integers

Integers are a set of numbers that include all the natural numbers and the negative whole numbers including zero. In mathematics, the set of integers is represented as Z = {…,-3, -2, -1, 0, 1, 2, 3, …}.

Positive numbers are called positive integers and negative numbers are called negative integers.

Integers are used in many areas of mathematics, as well as in real-world situations. For example, they are used to represent quantities such as the distance below sea level, movie or song ratings, or the temperature in degrees Celsius.


  • Prime numbers 

Prime numbers are a set of positive integers that are only divided by 1 and themselves. For example, the number 5 is a prime number because the only numbers it can be divided by evenly are 1 and 5. The number 10, on the other hand, is not a prime number because it can be divided by 1, 2, and 5 [5*2=10]. 

Think of prime numbers as building blocks for all other numbers. Every number can be broken down into a combination of prime numbers, just like how we build bigger things out of smaller parts. 

  • Square numbers

A number that is multiplied by itself is called a square number.

For example, to find the square of 6,  you should multiply 6 x 6, which is 36. This indicates that 36 is the square number of 6.

Square numbers

  • Cube numbers

Cube numbers are similar to square numbers. The only difference is that these numbers are represented as the result of multiplying a number by itself twice. For example, to find the cube of 2, you should multiply 2 by 2 and again by 2 which is 8. So the cube of 2 is 8.

cube numbers

  • Triangular numbers

Triangular numbers are numbers that can be arranged into a triangle pattern. To find the triangular number, you can use the following formula: n * (n + 1) / 2. For example, to find the 6th triangular number, you would use the formula 6 * (6 + 1) / 2, which gives you 21. So the 6th triangular number is 21.

Triangular numbers

  • Rational numbers

Rational numbers are numbers that can be expressed as a fraction of two integers. They can be either positive, negative, or zero.

Rational numbers are important in mathematics because they can be used to represent many real-world quantities, such as distances, temperatures, and money. 

  • Irrational numbers

Irrational numbers are numbers that can not be expressed as a fraction of two integers. In simple terms, they cannot be written as a/b where a and b are integers and b is not equal to zero. 

Some examples of irrational numbers are pi (3.14159…), the square root of 2 (1.414213…), and the square root of 3 (1.732050…).

Tips for practicing number theory for your upcoming Cambridge IGCSE math exams

Here are a few tips that can help you prepare for your upcoming Cambridge IGCSE math exams.

Study the basic concepts of number theory including the formulas.

Practice problems – Try solving as many problems as you can, focusing on both easy and difficult problems. Head over to our question bank to practice number theory problems. You can filter out the difficulty level as per your preference.

Review your notes and textbooks

Use study guides and study groups. You can register for the Tutopiya 2023 revision group class for IGCSE exam prep for live online group learning.

Know the format of the exam.

Seek help if needed – It is never too late to get the help you need. Reach out to our team to find the right tutor that fits your needs to help you ace your upcoming exams!

Watch the video below for a visual explanation of number theory and the different types of numbers!

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