Number Systems — Study Notes

1. Introduction

Numbers are the language of mathematics. Since ancient times, human beings have needed numbers to count objects, measure lengths, share things and compare quantities. Every time a new problem arose that the existing numbers could not solve, a new kind of number was invented. The natural numbers 1, 2, 3, ... helped us count. The whole numbers added zero. The integers added negative numbers so that we could represent debts, temperatures below zero and opposite directions. Fractions and rational numbers helped us express parts of a whole. Yet even the rational numbers were not enough, because the length of the diagonal of a unit square, which is the square root of 2, could not be written as p/q.

This chapter introduces the irrational numbers and combines them with rational numbers to form the system of real numbers. We will learn how to locate irrational numbers on the number line, how to perform operations on real numbers, how to rationalise denominators, and the laws of exponents for real numbers. Understanding the real number system is the foundation for algebra, geometry, trigonometry and calculus, so every concept in this chapter must be learnt very carefully.

2. Natural Numbers, Whole Numbers and Integers

The most basic collection of numbers is the set of natural numbers: N = {1, 2, 3, 4, ...}. Natural numbers are used for counting objects. When we include zero with the natural numbers, we get the set of whole numbers: W = {0, 1, 2, 3, ...}. When we include the negative of every natural number as well, we get the set of integers: Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}.

• Every natural number is a whole number, but every whole number is not a natural number (zero is the exception).
• Every whole number is an integer, and every integer is also a rational number because any integer p can be written as p/1.
• Integers have no fractional part; they are the numbers that can be represented by points at integer positions on the number line.

Rational Numbers

A number of the form p/q, where p and q are integers and q is not equal to zero, is called a rational number. Examples include 3/4, -7/9, 5 (= 5/1), and 0 (= 0/1). A rational number p/q is positive when p and q have the same sign and negative when they have opposite signs.

Irrational Numbers

A number that cannot be written in the form p/q, where p and q are integers and q is not zero, is called an irrational number. For example, the square root of 2 is irrational because it cannot be expressed as a ratio of two integers. Other famous irrational numbers are the square root of 3, pi, and 0.1011011101111... (a number with a non-terminating, non-repeating decimal expansion).

Real Numbers

The collection of all rational numbers together with all irrational numbers is called the set of real numbers, denoted by R. Every point on the number line corresponds to exactly one real number, and every real number can be represented by a point on the number line. For this reason, the number line is often called the real number line.

3. Decimal Expansions of Real Numbers

Terminating Decimals

A decimal expansion that terminates after a finite number of digits is called a terminating decimal. For example, 1/2 = 0.5 and 3/4 = 0.75. A rational number p/q, in its lowest form, has a terminating decimal expansion only when the denominator q has no prime factors other than 2 and 5.

Repeating (Non-Terminating Recurring) Decimals

A decimal expansion that goes on forever but repeats a block of digits is called a non-terminating recurring decimal. For example, 1/3 = 0.333... = 0.3 (bar on 3), and 1/7 = 0.142857142857... = 0.142857 (bar on the block). Every rational number has either a terminating or a non-terminating recurring decimal expansion.

Non-Terminating Non-Recurring Decimals

A decimal expansion that neither terminates nor repeats is called a non-terminating, non-recurring decimal. Such a decimal always represents an irrational number. For example, 0.101001000100001... and the decimal expansion of pi = 3.14159265... are non-terminating and non-recurring, so they represent irrational numbers.

Rule to remember: Every rational number has a terminating or recurring decimal expansion, while every irrational number has a non-terminating non-recurring decimal expansion.

4. Representing Real Numbers on the Number Line

Every real number can be located on the number line. To represent a number like the square root of 2, we can use a geometric construction. We take a unit square of side 1 unit. Its diagonal has length equal to the square root of (1^2 + 1^2) = square root of 2. With a compass, we transfer this length onto the number line, and the point obtained represents the square root of 2. In the same way we can locate the square root of 3, the square root of 5 and so on.

It is also possible to locate a decimal like 3.245 on the number line by successive magnification. We first locate 3 on the number line, then zoom into the portion between 3 and 4, and locate 3.2, then zoom again to locate 3.24, and so on. This process is called successive magnification and it shows that real numbers are "dense" on the number line.

5. Operations on Real Numbers

Real numbers satisfy the same properties of addition, subtraction, multiplication and division as rational numbers, provided we never divide by zero.

• The sum, difference, product and quotient of two rational numbers are rational numbers.
• The sum or difference of a rational number and an irrational number is always irrational.
• The product or quotient of a non-zero rational number and an irrational number is always irrational.
• The product or quotient of two irrational numbers may be rational or irrational.

Example: 2 + square root of 3 is irrational, and 2 x square root of 3 is irrational, but (square root of 2) x (square root of 2) = 2, which is rational.

Rationalising the Denominator

It is often convenient to write an irrational number like 1/(square root of 2) with a rational denominator. To rationalise the denominator of an expression like 1/(a + square root of b), we multiply the numerator and denominator by (a - square root of b), using the identity (a + b)(a - b) = a^2 - b^2.

Example: 1/(2 + square root of 3) = (2 - square root of 3)/((2 + square root of 3)(2 - square root of 3)) = (2 - square root of 3)/(4 - 3) = 2 - square root of 3.

6. Laws of Exponents for Real Numbers

If a is a positive real number and p and q are rational numbers, then the following laws of exponents hold:

• a^p x a^q = a^(p + q)
• a^p / a^q = a^(p - q)
• (a^p)^q = a^(p x q)
• a^p x b^p = (ab)^p
• a^0 = 1
• a^(-p) = 1/a^p
• a^(1/n) is the nth root of a; for example, a^(1/2) is the square root of a and a^(1/3) is the cube root of a.

Example: 2^(2/3) x 2^(1/5) = 2^((2/3) + (1/5)) = 2^((10 + 3)/15) = 2^(13/15).

Example: (64)^(1/6). Since 64 = 2^6, (2^6)^(1/6) = 2^(6/6) = 2^1 = 2.

Finding the Value of a^(m/n)

To evaluate a number like 9^(3/2), we first write 9^(3/2) = (9^(1/2))^3 = (square root of 9)^3 = 3^3 = 27. Alternatively, 9^(3/2) = (9^3)^(1/2) = (729)^(1/2) = 27. Both methods give the same answer.

Quick Revision Tables

Table 1: The Family of Numbers

Table 2: Decimal Expansions and Laws of Exponents

Mind Map

Important Diagrams (SVG)

Diagram 1: The Number System Hierarchy

Diagram 2: Representing Square Root of 2 on the Number Line

Common Mistakes

1. Students often write that pi is equal to 22/7. This is wrong. 22/7 is only an approximation of pi; pi is an irrational number while 22/7 is rational.
2. Many students believe that the square of every irrational number is irrational. In fact, (sqrt 2)^2 = 2, which is rational, so always check by actual computation.
3. A frequent error is to write 0.1010010001... as a rational number because it has a pattern. A number is rational only if its decimal is terminating or recurring; this expansion does not recur.
4. When rationalising, students forget to multiply both the numerator and the denominator by the conjugate, which changes the value of the expression.
5. Students confuse a^(1/2) with a^(-1/2). Remember that a^(1/2) is the square root of a, while a^(-1/2) = 1/(sqrt a).
6. While applying the quotient law a^p / a^q = a^(p - q), students sometimes subtract the wrong way when p is smaller than q, ending up with the wrong sign in the exponent.
7. Many students wrongly conclude that the sum of two irrationals is always irrational. For example, (2 + sqrt 3) + (2 - sqrt 3) = 4, which is rational.
8. Students forget that the denominator can never be zero in a rational number p/q; they sometimes write q = 0 by mistake.

Exam Tips

1. Always state whether a given decimal is terminating, recurring or non-recurring before classifying the number as rational or irrational, since the justification earns marks.
2. In rationalisation questions, first identify the conjugate of the denominator and then multiply numerator and denominator by it in one clean step.
3. Learn the laws of exponents thoroughly; most one-mark questions are direct applications of a^p x a^q = a^(p+q) or (a^p)^q = a^(pq).
4. For finding a rational number between two given numbers, use the formula (a + b)/2, and repeat the process if more numbers are required.
5. Always reduce final answers: write a number like 4/2 as 2, and simplify (sqrt 8) as 2 sqrt 2 before giving the final answer.
6. In the representation questions, draw a neat figure and mention Pythagoras theorem steps so that the examiner can award partial marks.
7. Practise writing the decimal expansions of common fractions like 1/3, 2/3, 1/7 so that you can quickly identify recurring patterns in the exam.

Conclusion

In this chapter we learnt that the number system grows step by step: natural numbers, whole numbers, integers, rational numbers and finally irrational numbers, which together form the real numbers. We studied that rational numbers have terminating or recurring decimal expansions while irrational numbers have non-terminating non-recurring expansions. We learnt how to represent real numbers, including irrationals like the square root of 2, on the number line, and how to rationalise denominators. Finally, we studied the laws of exponents for real numbers, which allow us to simplify expressions with rational powers. These ideas form the backbone of all further algebra and coordinate geometry, so a clear understanding of the real number system will serve the student throughout the rest of mathematics.