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 24
means 3 8 60
56
40
40
0
To change a fraction to a decimal, take
the top divided by the bottom, or
numerator divided by the denominator. 10
12. Repeating Decimals
Fraction Decimal
1 0 3 33...
3 3 1.000 0.3
9
means 1 3 10 0.33
9
10
9
1
Every rational number (fraction) either terminates
or repeats when written as a decimal.
12
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 decimals
Out 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 the
Examples:
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 Operations
Under Addition
1. a b b a Commutative law of addition
2. a b c a b c Associative law of addition
3. a 0 0 a Identity law of addition
4. a a =0 Inverse Law of addition
Under Multiplication
1. ab ba Commutative law of multiplication
2. a bc ab c Associative law of multiplication
3. a 1a
1 Identity law of multiplication
1
4. a =1 Inverse Law of multiplication
a
Under Addition and Multiplication
1. 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 Zero
1. a0 0
2. If ab 0 then a 0, b 0 or both
25
26. Properties of Quotients
a c
1. if ad bc b, d 0
b d
ca a
2. b, c 0
cb b
a a a
3. b 0
b b b
a c ac
4. b, d 0
b d bd
a c a d ad
5. b, c , d 0
b d b c bc
a c ad bc
6. b, d 0
b d bd
a c ad bc
7. b, d 0
b d bd
26
27. Open & Closed Interval
Open Half Interval
Open Interval
xa x b or a, b
xa 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
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
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
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
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 Numbers
For 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
