Real & Complex Numbers

Question 1

what are natural numbers

Answer 1

N={1,2,3,4,5,6,7,…..}

Question 2

what are whole numbers?

Answer 2

W={0,1,2,3,4,5,6,7………}

Question 3

Natural numbers are also called ____________?

Answer 3

Positive integers

Question 4

the set of integers is denoted by ______?

Answer 4

Z

Question 5

what are integers?

Answer 5

consist of +ve integers, 0, -ve integers

Question 6

the set of irrational numbers is denoted by ______?

Answer 6

Q’

Question 7

R=

Answer 7

Q U Q’

Question 8

W
Z
Q
N
arrange in subset form.

Answer 8

N⊂ W⊂ Z⊂ Q

Question 9

Q ∩ Q’ =

Answer 9

null

∩=common

Q=rational numbers
Q’= irrational numbers

Question 10

if a prime number has a root, would it be rational or irrational?

Answer 10

irrational number

Question 11

composite numbers?

Answer 11

numbers that have more than two factors.

Question 12

how many types of decimal representations of rational numbers are there?

Answer 12

2 types
1. terminating decimal fractions
2. non-terminating and recurring decimal fractions

Question 13

what are mixed recurring decimals?

Answer 13

the one in which at least one of the digits following the decimal point is not repeated, and subsequently, some digits are repeated.

Question 14

-4.893728… is it an integer?

Answer 14

No. An integer is a number that is whole and positive or negative. It does not have a decimal or fractional part.

Question 15

how can I convert recurring decimal into a rational number?

Answer 15

let it be x
multiply both sides with no. of repeating digits.
form 2 equations
subtract it

Question 16

closure property? example? with explanation?

Answer 16

a + b equals R
ab equals R

Question 17

commutative property? example? with explanation?

Answer 17

a+b=b+a
ab =ba

Question 18

associative property? example? with explanation?

Answer 18

(a+b)+c= a+(b+c)
(ab)c = a(bc)

Question 19

what is additive identity?

Answer 19

0
a + 0 = a

Question 20

what is additive inverse?

what is it in real and complex numbers?

Answer 20

pair up to give 0
3 + (-3)= 0

property of real numbers with respect to addition.

pg 40

Question 21

what is multiplicative identity?

Answer 21

1
a x 1= a

Question 22

what is multiplicative inverse?

Answer 22

a 1/a
a x 1/a = 1

Question 23

distributive property?

Answer 23

a * (b + c) = (a * b) + (a * c)
Multiplication Distributing Over Addition
Multiplication Distributing Over Subtraction

Question 24

how many properties of equality of real numbers are there?
Explain?

Answer 24

7
Reflexive Property
Symmetric Property
Transitive Property
Additive Property
Multiplicative Property
cancellation property for addition
cancellation property for multiplication

Question 25

symmetric property?

Question 26

reflexive property?

Question 27

how many properties of inequalities are there?
Explain?

Answer 27

5
Trichotomy
Transitive
Additive
Multiplicative
Multiplicative Inverse

Question 28

n√a
label it.
explain it.
write it in exponential form.

Answer 28

n is +ve integer GREATER than 1
a is real number
√ is called radical sign
n is called index
a is called radicand or base

(a)1/n

Question 29

difference between exponential and radical form?

Question 30

how many properties of RADICALS are there?

Answer 30

5

Question 31

how many laws of exponents we have?

Answer 31

7

Question 32

what is inequalities?

Answer 32

Inequalities are mathematical expressions involving the symbols >, <, ≥ and ≤.

Question 33

additive property of inequality of real numbers?

Question 33

multiplicative inverse property of inequality of real numbers?

Question 34

multiplicative property of inequality of real numbers?

Question 35

______ of a real number is non-negative.

Answer 35

square

Question 36

i =
i2=

Answer 36

√-1
-1

Question 37

what is a pure imaginary number?

Answer 37

square root of a negative real number.

Question 38

what is a complex number?

Answer 38

a no. of the form a+bi.

Question 39

what is a complex number represented by?

Answer 39

z

Question 40

what is C?

Answer 40

set of all complex numbers

Question 41

in z, what is a and b denoted by?

Answer 41

a= Re(z)
b=Im(z)
‘a’ is the real part of the complex number, denoted as Re(z), and ‘b’ is the imaginary part of the complex number, denoted as Im(z).

Question 42

R ⊂ C ?

Question 43

every real number is a complex number? T/F why?

Answer 43

True
a+0i where a is real number
This is because a real number can be expressed as a + 0i, where the imaginary part is 0.

Question 44

is 0 a complex number?

Answer 44

Yes, 0 is a complex number. 0 can be expressed as 0 + 0i, where the real part is 0 and the imaginary part is 0. Therefore, 0 is a complex number.

Question 45

make a flow chart of all the numbers?

Answer 45

page number 48

Question 46

what is a-bi?
what is it denoted by?

Answer 46

complex conjugate of z
z bar

Question 47

z bar bar=

Answer 47

z

Question 48

A real number is a complex number with ___________ part of 0.

Answer 48

imaginary

Question 49

prove that conjugate of a real number is the same real number?

Answer 49

Let x be a real number. Then, the conjugate of x is x - 0i. The real part of x - 0i is x, and the imaginary part of x - 0i is 0. Since the imaginary part of x - 0i is 0, the conjugate of x is simply x.

Question 50

what is conjugate?

Answer 50

the conjugate of a number is another number with the same real part and the imaginary part with the opposite sign.

Question 51

when are two complex numbers equal?

Answer 51

a + bi = c + di
if & only if a=c b=d

Question 52

how many laws or properties of complex numbers?

Answer 52

3
z1=z1 (reflexive law)
z1=z2 then z2=z1 (symmetric law)
z1 = z2 and z2=z3 then z1= z3 (transitive law)

Question 53

z1 + z2=

Answer 53

(a+c) + (b+d)i
the sum of 2 z = sum of corresponding real and imaginary parts.