Real & Complex Numbers Flashcards
what are natural numbers
N={1,2,3,4,5,6,7,…..}
what are whole numbers?
W={0,1,2,3,4,5,6,7………}
Natural numbers are also called ____________?
Positive integers
the set of integers is denoted by ______?
Z
what are integers?
consist of +ve integers, 0, -ve integers
the set of irrational numbers is denoted by ______?
Q’
R=
Q U Q’
W
Z
Q
N
arrange in subset form.
N⊂ W⊂ Z⊂ Q
Q ∩ Q’ =
null
∩=common
Q=rational numbers
Q’= irrational numbers
if a prime number has a root, would it be rational or irrational?
irrational number
composite numbers?
numbers that have more than two factors.
how many types of decimal representations of rational numbers are there?
2 types
1. terminating decimal fractions
2. non-terminating and recurring decimal fractions
what are mixed recurring decimals?
the one in which at least one of the digits following the decimal point is not repeated, and subsequently, some digits are repeated.
-4.893728… is it an integer?
No. An integer is a number that is whole and positive or negative. It does not have a decimal or fractional part.
how can I convert recurring decimal into a rational number?
let it be x
multiply both sides with no. of repeating digits.
form 2 equations
subtract it
closure property? example? with explanation?
a + b equals R
ab equals R
commutative property? example? with explanation?
a+b=b+a
ab =ba
associative property? example? with explanation?
(a+b)+c= a+(b+c)
(ab)c = a(bc)
what is additive identity?
0
a + 0 = a
what is additive inverse?
what is it in real and complex numbers?
pair up to give 0
3 + (-3)= 0
property of real numbers with respect to addition.
pg 40
what is multiplicative identity?
1
a x 1= a
what is multiplicative inverse?
a 1/a
a x 1/a = 1
distributive property?
a * (b + c) = (a * b) + (a * c)
Multiplication Distributing Over Addition
Multiplication Distributing Over Subtraction
how many properties of equality of real numbers are there?
Explain?
7
Reflexive Property
Symmetric Property
Transitive Property
Additive Property
Multiplicative Property
cancellation property for addition
cancellation property for multiplication
symmetric property?
reflexive property?
how many properties of inequalities are there?
Explain?
5
Trichotomy
Transitive
Additive
Multiplicative
Multiplicative Inverse
n√a
label it.
explain it.
write it in exponential form.
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
difference between exponential and radical form?
how many properties of RADICALS are there?
5
how many laws of exponents we have?
7
what is inequalities?
Inequalities are mathematical expressions involving the symbols >, <, ≥ and ≤.
additive property of inequality of real numbers?
multiplicative inverse property of inequality of real numbers?
multiplicative property of inequality of real numbers?
______ of a real number is non-negative.
square
i =
i2=
√-1
-1
what is a pure imaginary number?
square root of a negative real number.
what is a complex number?
a no. of the form a+bi.
what is a complex number represented by?
z
what is C?
set of all complex numbers
in z, what is a and b denoted by?
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).
R ⊂ C ?
every real number is a complex number? T/F why?
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.
is 0 a complex number?
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.
make a flow chart of all the numbers?
page number 48
what is a-bi?
what is it denoted by?
complex conjugate of z
z bar
z bar bar=
z
A real number is a complex number with ___________ part of 0.
imaginary
prove that conjugate of a real number is the same real number?
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.
what is conjugate?
the conjugate of a number is another number with the same real part and the imaginary part with the opposite sign.
when are two complex numbers equal?
a + bi = c + di
if & only if a=c b=d
how many laws or properties of complex numbers?
3
z1=z1 (reflexive law)
z1=z2 then z2=z1 (symmetric law)
z1 = z2 and z2=z3 then z1= z3 (transitive law)
z1 + z2=
(a+c) + (b+d)i
the sum of 2 z = sum of corresponding real and imaginary parts.
