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Chapter 1: Real Numbers

    Introduction

    The Fundamental Theorem of Arithmetic

    Revisiting Irrational Numbers

    Exercise 1.1

    Exercise 1.2

    Summary

    Enhanced Curriculum Support

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Chapter 1: Real Numbers > Introduction

Introduction

In the previous grades, we have encountered and irrational numbers. Here, we continue our discussion on real numbers. We begin with two very important properties of positive integers: namely the Euclid’s division algorithm and the Fundamental Theorem of Arithmetic.

Have you ever divided one number by another and got a remainder? Imagine you have 23 chocolates and want to share them equally among 5 friends. You’ll see that each friend gets chocolates, and you’re left with .

This leftover part—the remainder—is what Euclid’s Division Algorithm is all about!Euclid’s division algorithm, deals with divisibility of integers. Stated simply, it says:

Any positive integer 'a' can be divided by another positive integer 'b' in such a way that it leaves a remainder 'r' that is smaller than b.

Many of us recognise this as the usual long division process. Although this result is quite easy to state and understand, it has many applications related to the divisibility properties of integers. We touch upon a few of them and use it mainly to compute the of two positive integers-something that’s helpful in solving real-world problems like sharing things equally or simplifying fractions.

The Fundamental Theorem of Arithmetic, on the other hand, has to do something with multiplication of positive integers. We already know that:

Every number can be expressed as a product of primes in a unique way.

This important fact is the Fundamental Theorem of Arithmetic.It’s like discovering the DNA of numbers—every number has its unique "prime signature"! Again, while it is a result that is easy to state and understand, it has some very deep and significant applications in the field of mathematics.

We use the Fundamental Theorem of Arithmetic for two main applications:

  • 1. We use it to prove the irrationality of many of the numbers we have studied, such as: 2, 3 and 5-numbers that cannot be written as fractions.

  • 2.Cracking the code of decimal numbers: Ever wondered why some fractions, like 1/4, give a neat decimal (0.25), while others, like 1/3, go on forever (0.333…)? The prime factorization of the denominator tells us whether the decimal will end or repeat forever! We apply this theorem to explore when exactly the decimal expansion of a rational number, say pq (where q ≠ 0) is terminating and when it is nonterminating- it is repeating.

We do so by looking at the prime factorisation of the denominator q of pq. We will see that the of q will completely reveal the nature of the decimal expansion of pq.

Why Does This Matter?

These concepts aren’t just math for exams—they’re tools that help us understand patterns, solve puzzles, and uncover the secrets hidden in numbers. From coding to construction, and even designing games, these ideas pop up everywhere!

So let's get started.