Real Numbers and Complex Numbers 1 Real Numbers and Complex Numbers Case Study 1.1 Real Number System 1.2 Surds 1.3 Complex Number System Chapter Summary

Case Study I think I can do it by drawing a square of side 1 first. If you are given a pair of compasses and a ruler only, do you know how to represent the irrational number on a number line? In junior forms, we learnt from Pythagoras’ theorem that the diagonal of a square of side 1 is . As shown in the figure, after drawing the diagonal of the square, we can use a pair of compasses to draw an arc with radius OB and O as the centre. The point of intersection of the arc and the number line is the position of (i.e., point C).

1.1 Real Number System We often encounter different numbers in our calculations, For example, 0.16 means 0.166 666… . 1, 2, 4, 7, 0, , 2.5, 0.16, p, , … . These numbers can be classified into different groups.

7, 4, 0, 1, 2 1.1 Real Number System A. Integers 1, 2, 4, 7 and 0 are all integers. Positive integers (natural numbers) are integers that are greater than zero. Negative integers are integers that are less than zero. Zero is neither positive nor negative. Integers 7, 4, 0, 1, 2 Negative Integers Positive Integers (Natural Numbers)

1.1 Real Number System B. Rational Numbers is a fraction, 2.5 is a terminating decimal and 0.16 is a recurring decimal. . Recurring decimals are also called repeating decimals. All of them are rational numbers. A rational number is a number that can be written in the form , where p and q are integers and q  0. Recurring decimals can be converted into fractions, as shown in the next page. Note that and . Any integer n can be written as . Therefore, integers are also rational numbers.

1.1 Real Number System B. Rational Numbers Express 0.16 as a fraction: Let n  0.16 .  0.166 666… ............ (1) 10n  1.666 66… ............ (2) (2)  (1): 10n  n  1.5 9n  1.5 n   In other recurring decimals, such as a  0.83 and b  0.803, . . . . consider 100a  83.83 and 1000b  803.803, then we obtain 99a  83 and 999b  803. . . . .

1.1 Real Number System C. Irrational Numbers Numbers that cannot be written in the form are irrational numbers. Examples: p, , and sin 45 Irrational numbers can only be written as non-terminating and non-recurring decimals: is just an approximation of p.

1.1 Real Number System D. Real Numbers If we group all the rational numbers and irrational numbers together, we have the real number system. That is, a real number is either a rational number or an irrational number. Real numbers 1, 2, 4, 7, 0, , 3.5, 0.16, p, . Rational numbers 1, 2, 4, 7, 0, , 3.5, 0.16 Fractions Terminating decimals Recurring Integers Irrational numbers p, Negative integers Zero Positive integers

a2  0 for all real numbers a. 1.1 Real Number System D. Real Numbers We can represent any real number on a straight line called the real number line. . . . 1 . . 2.5 . p Real numbers have the following property: a2  0 for all real numbers a. For example:  is a real number since .  is not a real number since .

1.2 Surds In junior forms, we learnt the following properties for surds: For any real numbers a and b, we have In general, 1. 2.

1.2 Surds A. Simplification of Surds For any surds, when we reduce the integer inside the square root sign to the smallest possible integer, such as: then the surd is said to be in its simplest form.

1.2 Surds B. Operations of Surds Like surds are surds with the same integer inside the square root sign, such as and . When two surds are like surds, we can add them or subtract them:

1.2 Surds B. Operations of Surds Example 1.1T Simplify . Solution:

1.2 Surds B. Operations of Surds Example 1.2T Simplify . Solution:

1.2 Surds B. Operations of Surds Example 1.3T Simplify . Solution:

1.2 Surds C. Rationalization of the Denominator Rationalization of the denominator is the process of changing an irrational number in the denominator into a rational number, such as:

Example 1.4T 1.2 Surds Solution: C. Rationalization of the Denominator Simplify . Solution:

a2  0 for all real numbers a. 1.3 Complex Number System A. Introduction to Complex Numbers In Section 1.1, we learnt that a2  0 for all real numbers a. For example:  is a real number since .  is not a real number since . Therefore, in a real number system, equations such as x2  1 and (x  1)2  4 have no real solution: Define .  i Then Complex numbers  1  2i

1.3 Complex Number System A. Introduction to Complex Numbers Properties of complex numbers: 1. The complex number system contains an imaginary unit, denoted by i, such that i2  1. 2. The standard form of a complex number is a  bi, where a and b are real numbers. 3. All real numbers belong to the complex number system.

1.3 Complex Number System A. Introduction to Complex Numbers Notes: 1. For a complex number a  bi, a is called the real part and b is called the imaginary part. Complex numbers do not have order. So we cannot compare which of the complex numbers 2  3i and 4  2i is greater. 2. When a  0, a  bi  0  bi  bi, which is a purely imaginary number. 3. When b  0, a  bi  a  0i  a, so any real number can be considered as a complex number. 4. When a  b  0, a  bi  0  0i  0. Two complex numbers are said to be equal if and only if both of their real parts and imaginary parts are equal. If a, b, c and d are real numbers, then a  bi  c  di if and only if a  c and b  d.

1.3 Complex Number System B. Operations of Complex Numbers The addition, subtraction, multiplication and division of complex numbers are similar to the operations of algebraic expressions. In the operation of algebraic expressions, only like terms can be added or subtracted. We classify the real part and the imaginary part of the complex number as unlike terms in algebraic expressions. For complex numbers z1  a  bi and z2  c  di, where a, b, c and d are real numbers, we have: (1) Addition z1  z2  (a  bi)  (c  di) e.g. (3  6i)  (5  8i)  a  bi  c  di  (3  5)  [6  (8)]i  (a  c)  (b  d)i  8  2i

1.3 Complex Number System B. Operations of Complex Numbers (2) Subtraction z1  z2  (a  bi)  (c  di) e.g. (9  7i)  (2  3i)  a  bi  c  di  (9  2)  [7  (3)]i  (a  c)  (b  d)i  7  4i (3) Multiplication This term belongs to the real part because i2  1. z1z2  (a  bi)(c  di)  ac  adi  bci  bdi2  (ac  bd)  (ad  bc)i

Example 1.5T 1.3 Complex Number System Solution: B. Operations of Complex Numbers Example 1.5T Simplify (7  2i)(5  3i)  4i(3  i). Solution: (7  2i)(5  3i)  4i(3  i)  (35  21i  10i  6i2)  (12i  4i2)  35  21i  10i  6  12i  4  33  19i

1.3 Complex Number System B. Operations of Complex Numbers (4) Division The denominator contains . The process of division is similar to the rationalization of the denominator in surd. (p  q)(p  q)  p2  q2

Example 1.6T 1.3 Complex Number System Solution: B. Operations of Complex Numbers Example 1.6T Simplify and express the answer in standard form. Solution:

Chapter Summary 1.1 Real Number System Real numbers Rational numbers Fractions Terminating decimals Recurring Integers Irrational numbers Negative integers Zero Positive integers

Chapter Summary 1.2 Surds 1. For any positive real numbers a and b:

Chapter Summary 1.3 Complex Number System 1. Every complex number can be written in the form a  bi, where a and b are real numbers. 2. The operations of complex numbers obey the same rules as those of real numbers.

Follow-up 1.1 1.2 Surds Solution: B. Operations of Surds Simplify the following expressions. (a) (b) Solution: (a) (b)

Follow-up 1.2 1.2 Surds Solution: B. Operations of Surds Simplify the following expressions. (a) (b) Solution: (a) (b)

Follow-up 1.3 1.2 Surds Solution: B. Operations of Surds Simplify the following expressions. (a) (b) Solution: (a) a2  b2  (a  b)(a  b) (b)

Follow-up 1.4 1.2 Surds Solution: C. Rationalization of the Denominator Follow-up 1.4 Simplify the following expressions. (a) (b) Solution: (a) (b)

Follow-up 1.5 1.3 Complex Number System Solution: B. Operations of Complex Numbers Follow-up 1.5 Simplify (5  2i)(7  3i)  3(2  5i). Solution: (5  2i)(7  3i)  3(2  5i)  (35  15i  14i  6i2)  (6  15i)  35  15i  14i  6  6  15i  23  44i

Follow-up 1.6 1.3 Complex Number System Solution: B. Operations of Complex Numbers Follow-up 1.6 Simplify and express the answer in standard form. Solution: