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

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P. 2 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. Case Study The point of intersection of the arc and the number line is the position of (i.e., point C). 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? I think I can do it by drawing a square of side 1 first. In junior forms, we learnt from Pythagoras theorem that the diagonal of a square of side 1 is.

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

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P. 4 A. Integers 1.1 Real Number System 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. Integers 7, 4, 0, 1, 2 Negative IntegersPositive Integers (Natural Numbers) Zero is neither positive nor negative.

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P. 5 B. Rational Numbers All of them are rational numbers. Recurring decimals are also called repeating decimals. is a fraction, 2.5 is a terminating decimal and 0.16 is a recurring decimal.. A rational number is a number that can be written in the form, where p and q are integers and q 0. Note that and. Any integer n can be written as. Therefore, integers are also rational numbers. Recurring decimals can be converted into fractions, as shown in the next page. 1.1 Real Number System

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P. 6 B. Rational Numbers Express 0.16 as a fraction: … (1) Let n n … (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 and 1000b , then we obtain 99a 83 and 999b Real Number System

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P. 7 C. Irrational Numbers Irrational numbers can only be written as non-terminating and non-recurring decimals: Numbers that cannot be written in the form are irrational numbers. Examples:,, and sin 45 is just an approximation of. 1.1 Real Number System

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P. 8 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,,. Rational numbers 1, 2, 4, 7, 0,, 3.5, Fractions Terminating decimals Recurring decimals Integers Irrational numbers, Negative integersZeroPositive integers 1.1 Real Number System

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P. 9 D. Real Numbers We can represent any real number on a straight line called the real number line Real numbers have the following property: For example: is a real number since. is not a real number since. a 2 0 for all real numbers a. 1.1 Real Number System

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

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

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

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P. 13 B. Operations of Surds Example 1.1T Solution: Simplify. 1.2 Surds

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P. 14 Example 1.2T Solution: Simplify. B. Operations of Surds 1.2 Surds

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P. 15 Example 1.3T Solution: Simplify. B. Operations of Surds 1.2 Surds

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

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P. 17 C. Rationalization of the Denominator Example 1.4T Solution: Simplify. 1.2 Surds

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

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P The complex number system contains an imaginary unit, denoted by i, such that i 2 1. A. Introduction to Complex Numbers Properties of complex numbers: 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

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P. 20 Notes: 1.For a complex number a bi, a is called the real part and b is called the imaginary part. 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. Complex numbers do not have order. So we cannot compare which of the complex numbers 2 3i and 4 2i is greater. 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. A. Introduction to Complex Numbers 1.3 Complex Number System

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P 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. (1)Addition z 1 z 2 (a bi) (c di) a bi c di (a c) (b d)i For complex numbers z 1 a bi and z 2 c di, where a, b, c and d are real numbers, we have: e.g.(3 6i) (5 8i) (3 5) [6 ( 8)]i 8 2i

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P. 22 This term belongs to the real part because i 2 1. B. Operations of Complex Numbers (2)Subtraction z 1 z 2 (a bi) (c di) a bi c di (a c) (b d)i (3)Multiplication z 1 z 2 (a bi)(c di) ac adi bci bdi 2 (ac bd) (ad bc)i e.g.(9 7i) (2 3i) (9 2) [ 7 ( 3)]i 7 4i 1.3 Complex Number System

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P. 23 Example 1.5T Solution: Simplify (7 2i)(5 3i) 4i(3 i). (7 2i)(5 3i) 4i(3 i) (35 21i 10i 6i 2 ) (12i 4i 2 ) 35 21i 10i 6 12i i B. Operations of Complex Numbers 1.3 Complex Number System

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P. 24 (4)Division The process of division is similar to the rationalization of the denominator in surd. (p q)(p q) p 2 q 2 The denominator contains. B. Operations of Complex Numbers 1.3 Complex Number System

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P. 25 Example 1.6T Solution: Simplify and express the answer in standard form. B. Operations of Complex Numbers 1.3 Complex Number System

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P Real Number System Chapter Summary Real numbers Rational numbers Fractions Terminating decimals Recurring decimals Integers Irrational numbers Negative integersZeroPositive integers

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P Surds Chapter Summary 1.For any positive real numbers a and b: 2.For any positive real numbers a and b: (a) (b)

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P Complex Number System Chapter Summary 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.

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

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

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Follow-up 1.3 Solution: Simplify the following expressions. (a)(b) (a) (b) a 2 b 2 (a b)(a b) B. Operations of Surds 1.2 Surds

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

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

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

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