A complex number z is an ordered pair (x, y) of real numbers x and y.
z ≡ (x, y) ≡ x + iy
Re z = x (Real part), Im z = y (imaginary part) and i2 = -1 i.e., i = √-1 (imaginary unit)
(a) x + iy = a + ib ⇔ x = a and y = b
(b) For z = x + iy,
if x = 0 then z = iy (pure imaginary)
if y = 0 theb z = x (pure real)
Addition of complex numbers:
Let z1 = x1 + iy1 and z2 = x2 + iy2, then
z1 + z2 = (x1 + x2) + i(y1 + y2)
Multiplication of complex numbers:
z1z2 = (x1x2 - y1y2) + i(x1y2 + x2y1)
Subtraction of complex numbers:
z1 - z2 = (x1 - x2) + i(y1 - y2)
Division of complex numbers:
z = z1/z2 = x + iy, where x = (x1x2 + y1y2)/ (x22 + y22)
y = (x1y2 + x2y1)/(x22 + y22) , z2 ≠ 0
Pactical rule:
z = z1/z2 = (x1 + iy1)/(x2 + iy2)
= [(x1 + iy1)/(x2 - iy2)] × [(x2 + iy2)/(x2 + iy2)]
= x + iy.
Polar Form of Complex Numbers:
Let (x, y) be the co-ordinate in cartesian co-ordinate system and (r, θ) be the co-ordinates in polar co-ordinates, then
x = r cosθ, y = r sinθ
so, z = x + iy = r(cosθ + i sinθ)
Here r = |z| = √(x2 + y2) = √zz̅ (absolute value/modulus of z)
θ = arg z =tan-1 y/x (argument of z)
Triangular inequality: |z1 + z2| ≤ |z1| + |z2|
Generalized triangle inequality: |z1 + z2 + ... + zn| ≤ |z1| + |z2| + ... + |zn|.
Numbers
Counting Numbers (Natural numbers): 1, 2, 3, 4, ...
Even numbers: Divisible by 2.
Old numbers: Not divisible by 2.
Prime numbers: Number greater than one and whose only divisors are one and number itself. i.e., 2, 3, 5, 7, 11, 13, ...
Composite numbers: 4, 6, 8, 9, 10, 12, ...
Fundamental Theorems of Arithmetic:
Principle of induction: If ℤ is a set of integers such that
(a) 1 ∈ ℤ
(b) n ∈ ℤ ⇒ n + 1 ∈ ℤ
(c) All integers ≥ ∈ ℤ
Well ordering principle: If A is a non-empty set of positive integers, then A contains a smallest member.
Divisibility: d/n (d divides n) ⇒ n = cd, for some c.
Common divisor: If d/a and d/b then d is common divisor of a and b.
Greatest common divisor: If d/a and d/b and for every e/a and e/b ⇒ e/d then d is greatest common divisor of a and b, denoted by (a, b) = d.
Relative Prime: a and b are relative prime if (a, b) = 1.
Prime number: An integer n is prime if n>1 and if the only positive divisors of n are 1 and n.
Composite number: If n>1 and n is not prime, then n is composite number.
z ≡ (x, y) ≡ x + iy
Re z = x (Real part), Im z = y (imaginary part) and i2 = -1 i.e., i = √-1 (imaginary unit)
(a) x + iy = a + ib ⇔ x = a and y = b
(b) For z = x + iy,
if x = 0 then z = iy (pure imaginary)
if y = 0 theb z = x (pure real)
Addition of complex numbers:
Let z1 = x1 + iy1 and z2 = x2 + iy2, then
z1 + z2 = (x1 + x2) + i(y1 + y2)
Multiplication of complex numbers:
z1z2 = (x1x2 - y1y2) + i(x1y2 + x2y1)
Subtraction of complex numbers:
z1 - z2 = (x1 - x2) + i(y1 - y2)
Division of complex numbers:
z = z1/z2 = x + iy, where x = (x1x2 + y1y2)/ (x22 + y22)
y = (x1y2 + x2y1)/(x22 + y22) , z2 ≠ 0
Pactical rule:
z = z1/z2 = (x1 + iy1)/(x2 + iy2)
= [(x1 + iy1)/(x2 - iy2)] × [(x2 + iy2)/(x2 + iy2)]
= x + iy.
Polar Form of Complex Numbers:
Let (x, y) be the co-ordinate in cartesian co-ordinate system and (r, θ) be the co-ordinates in polar co-ordinates, then
x = r cosθ, y = r sinθ
so, z = x + iy = r(cosθ + i sinθ)
Here r = |z| = √(x2 + y2) = √zz̅ (absolute value/modulus of z)
θ = arg z =tan-1 y/x (argument of z)
Triangular inequality: |z1 + z2| ≤ |z1| + |z2|
Generalized triangle inequality: |z1 + z2 + ... + zn| ≤ |z1| + |z2| + ... + |zn|.
Numbers
Counting Numbers (Natural numbers): 1, 2, 3, 4, ...
Even numbers: Divisible by 2.
Old numbers: Not divisible by 2.
Prime numbers: Number greater than one and whose only divisors are one and number itself. i.e., 2, 3, 5, 7, 11, 13, ...
Composite numbers: 4, 6, 8, 9, 10, 12, ...
Fundamental Theorems of Arithmetic:
Principle of induction: If ℤ is a set of integers such that
(a) 1 ∈ ℤ
(b) n ∈ ℤ ⇒ n + 1 ∈ ℤ
(c) All integers ≥ ∈ ℤ
Well ordering principle: If A is a non-empty set of positive integers, then A contains a smallest member.
Divisibility: d/n (d divides n) ⇒ n = cd, for some c.
Common divisor: If d/a and d/b then d is common divisor of a and b.
Greatest common divisor: If d/a and d/b and for every e/a and e/b ⇒ e/d then d is greatest common divisor of a and b, denoted by (a, b) = d.
Relative Prime: a and b are relative prime if (a, b) = 1.
Prime number: An integer n is prime if n>1 and if the only positive divisors of n are 1 and n.
Composite number: If n>1 and n is not prime, then n is composite number.
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