Presentation on theme: "Real Numbers and Complex Numbers"— Presentation transcript:
1 Real Numbers and Complex Numbers 1Real Numbers and Complex NumbersCase Study1.1 Real Number System1.2 Surds1.3 Complex Number SystemChapter Summary
2 Case StudyI think I can do it by drawing a squareof side 1 first.If you are given a pair of compasses and a ruler only, do you know how to represent the irrational numberon a number line?In junior forms, we learnt from Pythagoras’ theorem that the diagonal of a square of side 1 isAs 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).
3 1.1 Real Number SystemWe often encounter different numbers in our calculations,For example,0.16 means ….1, 2, 4, 7, 0, , 2.5, 0.16, p, , ….These numbers can be classified into different groups.
4 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.Integers7, 4, , , 2Negative IntegersPositive Integers(Natural Numbers)
5 1.1 Real Number System B. Rational Numbers is a fraction, 2.5 is a terminating decimal and0.16 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 inthe form , where p and q are integers and q 0.Recurring decimals can be converted into fractions, as shown in the next page.Note thatandAny integer n can be written as Therefore, integers are alsorational numbers.
6 1.1 Real Number System B. Rational Numbers Express 0.16 as a fraction: Let n 0.16. … (1)10n … (2)(2) (1): 10n n 1.59n 1.5n In other recurring decimals, such as a 0.83 and b 0.803,. .. .consider 100a and 1000b , then we obtain99a 83 and 999b 803.. .. .
7 1.1 Real Number System C. Irrational Numbers Numbers that cannot be written in the form are irrationalnumbers.Examples: p, , and sin 45Irrational numbers can only be written as non-terminating and non-recurring decimals:is just an approximationof p.
8 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 numbers1, 2, 4, 7, 0, , 3.5, 0.16, p,.Rational numbers1, 2, 4, 7, 0, , 3.5, 0.16FractionsTerminatingdecimalsRecurringIntegersIrrational numbersp,Negative integersZeroPositive integers
9 a2 0 for all real numbers a. 1.1 Real Number SystemD. Real NumbersWe can represent any real number on a straight line called the real number line....1..2.5.pReal 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
10 1.2 SurdsIn junior forms, we learnt the following properties for surds:For any real numbers a and b, we haveIn general,1.2.
11 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.
12 1.2 Surds B. Operations of Surds Like surds are surds with the same integer inside the square root sign, such as andWhen two surds are like surds, we can add them or subtract them:
13 1.2 SurdsB. Operations of SurdsExample 1.1TSimplifySolution:
14 1.2 SurdsB. Operations of SurdsExample 1.2TSimplifySolution:
15 1.2 SurdsB. Operations of SurdsExample 1.3TSimplifySolution:
16 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:
17 Example 1.4T 1.2 Surds Solution: C. Rationalization of the Denominator SimplifySolution:
18 a2 0 for all real numbers a. 1.3 Complex Number SystemA. Introduction to Complex NumbersIn Section 1.1, we learnt thata2 0 for all real numbers a.For example: is a real number since is not a real number sinceTherefore, in a real number system, equations such as x2 1 and (x 1)2 4 have no real solution:Define iThenComplex numbers 1 2i
19 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 thati2 1.2. The standard form of a complex number isa bi,where a and b are real numbers.3. All real numbers belong to the complex number system.
20 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, thena bi c diif and only if a c and b d.
21 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) Additionz1 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
22 1.3 Complex Number System B. Operations of Complex Numbers (2) Subtractionz1 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) MultiplicationThis term belongs to the real part because i2 1.z1z2 (a bi)(c di) ac adi bci bdi2 (ac bd) (ad bc)i
24 1.3 Complex Number System B. Operations of Complex Numbers (4) DivisionThe denominator containsThe process of division is similar to the rationalization of the denominator in surd.(p q)(p q) p2 q2
25 Example 1.6T 1.3 Complex Number System Solution: B. Operations of Complex NumbersExample 1.6TSimplify and express the answer in standard form.Solution:
26 Chapter Summary 1.1 Real Number System Real numbers Rational numbers FractionsTerminatingdecimalsRecurringIntegersIrrational numbersNegative integersZeroPositive integers
27 Chapter Summary 1.2 Surds 1. For any positive real numbers a and b:
28 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.
29 Follow-up 1.1 1.2 Surds Solution: B. Operations of Surds Simplify the following expressions.(a) (b)Solution:(a)(b)
30 Follow-up 1.2 1.2 Surds Solution: B. Operations of Surds Simplify the following expressions.(a) (b)Solution:(a)(b)
31 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)
32 Follow-up 1.4 1.2 Surds Solution: C. Rationalization of the DenominatorFollow-up 1.4Simplify the following expressions.(a) (b)Solution:(a)(b)