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- 1. Types of Numbers 1
- 2. ContentNatural numbersWhole numbersIntegersRational numbersIrrational numbersReal numbersRationalize the denominatorProperties of real numbersComplex numbers 2
- 3. Natural numbers (N)Set of natural numbers(N)={1,2,3,….} 1 2 3… 3
- 4. Whole Numbers (W)Set of Whole numbers(W)={0,1,2,3…} W 0 N 3 2 1 4
- 5. IntegersSet of Integers (Z) ={…-3,-2,-1,0,1,2,3,…} Z W -1 -2 N 0 3 2 -3 1 5
- 6. Rational Numbers (Q)Set of rational numbers(Q) ={x:x=p/q;p,q ЄZ and q≠0} Z W -1 -2 N 0 -1/2 3 2 5/25 -3 1 6
- 7. Irrational numbers REAL NUMBERS: a number that can be written as a decimal.RATIONAL NUMBERS: a number that can be written as a fraction.IRRATIONAL NUMBERS: a number that is not rational. It can not be written as a fraction 7
- 8. What this means…….• The number line goes on forever.• Every point on the line is a REAL number.• There are no gaps on the number line.• Between the whole numbers and the fractions there are numbers that are decimals but they don’t terminate and are not recurring decimals. They go on forever. 8
- 9. Examples of IRRATIONAL numbers , 2 , 3, 5 , 6 , 7 , 8 4 is not irrational because 4 2 a rational number 9
- 10. Converting Fractions and Decimals Fraction Decimal 0 37 5 3 8 3.000 0.375 8 24means 3 8 60 56 40 40 0To change a fraction to a decimal, takethe top divided by the bottom, ornumerator divided by the denominator. 10
- 11. Complete the table. Fraction Decimal 4 5 0.8 3 0.03 100 7 20 0.35 6 7 6.7 10 9 1 9.125 8 11
- 12. Repeating Decimals Fraction Decimal 1 0 3 33... 3 3 1.000 0.3 9means 1 3 10 0.33 9 10 9 1Every rational number (fraction) either terminatesor repeats when written as a decimal. 12
- 13. Repeating Decimals Fraction Decimal 5 0 45454 ... 11 5.00000 0.454 11 44means 5 11 60 0.454 55 50 0.45 44 60 55 50 44 6 13
- 14. Repeating Decimals Fraction Decimal 5 0 8 33... 6 5.000 0.83 6 48means 5 6 20 0.833 18 20 18 0.83 2 14
- 15. PLACE YOUR ANSWERS IN THESE + Rational Irrational - Rational IrrationalRational Rationalirrational Irrational 15
- 16. + Rational Irrational - Rational IrrationalRational Rational Irrational Rational Rational Irrationalirrational Irrational Either Irrational Irrational Either 16
- 17. PLACE YOUR ANSWERS IN THESE x Rational Irrational Rational IrrationalRational Rationalirrational Irrational 17
- 18. x Rational Irrational Rational IrrationalRational Rational Irrational Rational Rational Irrationalirrational Irrational Rational Irrational Irrational Rational 18
- 19. Relationship among various sets of number Real Numbers R Rational Numbers Q Irrational Irrational Integers Z numbers Numbers Whole numbers W H Natural numbers N N W Z Q R 19
- 20. • Two ways of representing real numbers. As is evident, all rational numbers can be written as fractions. Decimals which are presented on the place value system, are two types. – 1. Finite decimals – 2. infinite decimals Out of these finite decimals are rational. For example 3.467 is a finite decimal which is equal to 3467/1000 and is of the form p/q. 20
- 21. • Infinite decimals are also of two types. 1. recurring decimals 2. Non-recurring decimalsOut of these recurring decimals are rational and non recurring decimals are irrational. 21
- 22. Rationalizing the Denominator This process transfers the surd from the denominator to the numerator. Follow theExamples:1. Rationalize the Denominator(a) 2/√3(b) 10/√5 22
- 23. Examples:2. Rationalize the Denominator 2/(√7+2).3. Rationalize the Denominator of (3√2+ 2√3)/(3√2- √2) and simplify. 23
- 24. Properties of the Real Number System Rules of OperationsUnder Addition1. a b b a Commutative law of addition2. a b c a b c Associative law of addition3. a 0 0 a Identity law of addition4. a a =0 Inverse Law of additionUnder Multiplication1. ab ba Commutative law of multiplication2. a bc ab c Associative law of multiplication3. a 1a 1 Identity law of multiplication 14. a =1 Inverse Law of multiplication aUnder Addition and Multiplication1. a b c ab bc Distributive law for multiplication w.r.t addition 24
- 25. Properties of Negatives 1. a a 2. a b ab a b 3. a b ab 4. 1 a a Properties Involving Zero1. a0 02. If ab 0 then a 0, b 0 or both 25
- 26. Properties of Quotients a c1. if ad bc b, d 0 b d ca a2. b, c 0 cb b a a a3. b 0 b b b a c ac4. b, d 0 b d bd a c a d ad5. b, c , d 0 b d b c bc a c ad bc6. b, d 0 b d bd a c ad bc7. b, d 0 b d bd 26
- 27. Open & Closed Interval Open Half Interval Open Interval xa x b or a, bxa x b or a, b xa x b or a, b Closed Interval xx a or ,a xx a or ,a xa x b or a, b xx a or a, xx a or a, 27
- 28. Imaginary and Complex Numbers 28
- 29. What is a Complex Number• A number that can be expressed in the form a + bi where a and b are real numbers and i is the imaginary unit.• Imaginary unit is the number represented by i, where i 1 and i 2 1• Imaginary number is a number that can be expressed in the form bi, where b is a real number and i is the imaginary unit.• When written in the form a + bi , a complex number is said to be in Standard Form. 29
- 30. The Set of Complex Numbers Complex Numbers C Real Numbers R Rational Numbers Q Integers Z Imaginary R Numbers i Whole numbers W Irrational Natural Numbers Numbers N H C z:z a ib a, b R In Cartesian Form;a Re z the real parts of C while b Im z the imaginary parts of C 30
- 31. Imaginary Numbers• Consider if we use the product rule to rewrite as 16 1 16 – This step is called “poking out the i” – We know how to evaluate• Imaginary unit: 16 – Thus, 16 4i i 1 – Any number with an i is called an imaginary number – Also by definition: 2 i 1 31
- 32. Complex Numbers• Complex Number: a number written in the format a + bi where: – a and b are real numbers – a is the real part – bi is the imaginary part 32
- 33. Complex Numbers (Example)Ex 1: Simplify and write in a + bi format: a) 81 b) 12 48 33
- 34. Adding & Subtracting Complex Numbers 34
- 35. Adding & Subtracting Complex Numbers• To add complex numbers – Add the real parts – Add the imaginary parts – The real and imaginary parts cannot be combined any further• To subtract two complex numbers – Distribute the negative to the second complex number – Treat as adding complex numbers 35
- 36. Adding & Subtracting Complex Numbers (Example)Ex 2: Simplify and write in a + bi format: a) (8 – 3i) + (2 + i) b) (5 + 9i) – (4 – 8i) c) 5i – (-5 + 2i) 36
- 37. Multiplying Complex Numbers 37
- 38. Multiplying Complex Numbers• To multiply 3i · 2i – Multiply the real numbers first: 6 – Multiply the i s: i · i = i2 3i · 2i = 6i2 = -6 • Remember that it is only acceptable to leave i in the final answer• To multiply complex numbers in general – Use the distributive property or FOIL 38
- 39. Multiplying Complex Numbers (Example)Ex 3: Multiply and write in a + bi format: a) -3i · 5i b) 7i(9 – 4i) c) (3 – 2i)(7 + 6i) 39
- 40. Dividing Complex Numbers 40
- 41. Complex Conjugates• Consider (3 + i), (3 – i) – What do you notice?• Complex conjugate: the same complex number with real parts a and imaginary part bi except with the opposite sign – Very similar to conjugates when we discussed rationalizing – Ex: The complex conjugate of (2 – i) is (2 + i) 41
- 42. Dividing Complex Numbers• Goal is to write the quotient of complex numbers in the format a + bi – Multiply the numerator and denominator by the complex conjugate of the denominator (dealing with an expression) – The numerator simplifies to a complex number – The denominator simplifies to a single real number – Divide the denominator into each part of the numerator and write the result in a + bi format 42
- 43. Dividing Complex Numbers (Example)Ex 4: Divide – write in a + bi format: 6 i a) 3i 2 i b) 2 5i 43
- 44. SUMMERY Operations on Complex NumbersFor z a bi and z c di then ; 1 ,2• Adding complex numbers z1 z2 a bi c di a c b d i• Subtracting complex numbers z1 z2 a bi c di a c b d i• Multiplying complex numbers z1 z2 a bi c di ac bd ad bc i• Dividing complex numbers a bi c di ac bd bc ad i z1 z2 a bi c di c di c di c2 d 2 44
- 45. Same Complex Numbers 2 complex numbers z = a + bi and z = c + di 1 2 are same if a = c and b = d. Example: Given z = 2 + (3y+1)i and z = 2x + 7i 1 2 with z = z . Find the value of x and y. 1 2 45
- 46. Conjugate Complex A complex conjugate of a complex number z = a + bi is z* = a – bi If z and z are complex numbers, then 1 2 * 1. z1 z2 z1* z 2* * 2. z1 z2 z1* z2* * * 3. z 1 z1 * 1 1 4. z1* z1 46

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