Complex numbers

ْ‫ح‬َّ‫الر‬ ِ‫هللا‬ ِ‫م‬ْ‫س‬ِ‫ب‬
ْ‫ي‬ ِ‫ح‬َّ‫الر‬ ِ‫ن‬ ٰ‫م‬
ِ‫م‬
‫ہے‬ ‫مہربان‬ ‫واال‬ ‫کرنے‬ ‫رحم‬ ‫نہایت‬ ‫جو‬ ‫شروع‬ ‫نام‬ ‫کے‬ ‫ہللا‬

PRESENTED TO:
Mam Habiba Ishaq
(Lecturer at UOG)
Group G

1
Nayab Ali
20021519-014
Group Members
2
Numan Ali
20021519-137
3
Saqib
20021519-001

HISTORY OF COMPLEX NUMBERS
The history of mathematics shows that man started with the set of counting numbers and
invented, by stages the negative numbers, rational numbers and irrational numbers etc. and
collectively these numbers called real numbers , At that time, imaginary numbers were
poorly understood, and regarded by some as fictitious or useless much as zero .
Then in 16th century Rafael Bombelli 1st discover the imaginary
numbers.
Rafael Bombelli
Discovered by:

One way of viewing imaginary numbers is to consider a
standard number line, positively increasing in magnitude to the
right, and negatively increasing in magnitude to the left. At 0 on
this x-axis, a y-axis can be drawn with "positive" direction going
up; "positive" imaginary numbers then increase in magnitude
upwards, and "negative" imaginary numbers increase in
magnitude downwards. This vertical axis is often called the
"imaginary axis"
An illustration of the complex plane. The
imaginary numbers are on the vertical
coordinate axis
Geometric interpretation:
• The backbone of this new number system is
the imaginary unit, or the number “i".

Since square of positive as well as negative number is a
positive number , the square root of a negative number does
not exist in the realm of real numbers. Then here we use
imaginary numbers.
Complex Number:
In the 16th-century Italian mathematician Gerolamo Cardano was
credited with introducing complex numbers
Example:
−9
As we can see that this is not a real number
so ;
= 9 i
Gerolamo Cardano
=3 i

Example:
𝑥2+1=0
𝑥2=-1
x =± −1
−1 does not belong to the set
of real numbers. We, therefore,
foe convenience call it
imaginary number denoted by i
A complex number is a number that can be expressed in
the form a + b i, where a and b are real numbers,
and i represents the “imaginary unit”, satisfying the
equation
ⅈ²= -1 Because no real number satisfies this equation, i is
called an imaginary number. For the complex number a + b
i, a is called the real part and b is called the imaginary
part. The set of complex numbers is denoted by either of
the symbols ℂ or C.
A complex number can be visually
represented as a pair of
numbers (a, b) forming a vector on
a diagram called an Argand
diagram, representing the complex
plane. Re is the real axis, Im is the
imaginary axis, and i is the
“imaginary unit” that satisfies ⅈ²=
-1

OPERATIONS WITH COMPLEX NUMBERS:
1. Addition:
To add two complex numbers , add the real part to the real part and the imaginary part to the imaginary part.
(a+bi )+(c+di )=(a+c)+(b+d) i
• Example:
(2+7i )+(3−4i )
=(2+3)+(7+(−4)) i
=5+3i
2. Subtraction:
To subtract two complex numbers, subtract the real part from the real part and the imaginary part from the
imaginary part.
(a+bi )−(c+di )=(a−c)+(b−d) i
• Example :
(9+5i)−(4+7i)
=(9−4)+(5−7)i
=5−2i

3.Multiplication:
To multiply two complex numbers, use the FOIL method and combine like terms .
(a+bi)(c+di)=ac+adi +bci +bd i²
=ac+(ad+bc)i −bd (Remember i² = −1)
=(ac−bd)+(ad+bc) i
• Example 3:
(3+2i)(5+6i)
=15+18i+10i+12i ²
=15+28i −12
=3+28i

• 4.Division:
To divide two comlex numbers, multiply the numerator and denominator by the complex conjugate , expand and
simplify. Then, write the final answer in standard form.
• Example:
3 + 2i
4 − 5i
=
3 + 2i
4 − 5i
×
4+ 5i
4 + 5i
=
12 + 15i + 8i + 10𝑖2
16 + 25
=
2 + 23i
41
=
2
41
+
23i
41
Conjugate of a complex number:
Let z=x+yi is a complex number then 𝒛=x-yi is conjugate of complex number.
NOTE: conjugate of complex number is also a complex number.

Properties of Complex Numbers:
Properties of complex numbers are given below;
Addition laws:
• Closure :
The sum of two complex numbers is , by definition , a complex number. Hence, the set of
complex numbers is closed under addition.
Example :
(5+ i2) + (0+3i )
= 5 + i(2 + 3)
= 5 + i5
• Commutative property :
For two complex numbers z 1 = a + ib and z2 = c + id
z1 + z2 = (a + ib) +(c +id) = (a + c) + i( b + d)
z2 + z1 = (c + id) +(a + ib)= (c + a) + i(d + b)
But we know that, a + c = c + a and b + d = d + b
∴ z1 + z2 = z2 + z1
8-2i=8-2i
(3 + 2i) + (5 - 4i) = (5 - 4i) + (3 + 2i)
Example:

• Associative Property :
Consider three complex numbers,
z1 = a + ib , z2 = c + id and z3 = e + if
(z1 + z2 )+ z3 = z1 +(z 2 + z 3 )
(a + ib + c + id ) + (e + if) = (a + ib) + ( c +id + e + if)
[(a + c) + i( b +d)] + (e + if) = (a + ib) +[(c + e) + i( d +f)]
(a + c + e ) + i(b + d + f ) = ( a + c + e) + i(b + d + f)
Example:
(7 + 5i) + [(3 + 2i) + (5 - 4i)] = [(7 + 5i) + (3 + 2i)] + (5 - 4i)
(7 + 5i) +(8-2i)=(10+7i)+ (5 - 4i)
(15+3i)=(15+3i)

Additive Identity:
Let a + ib be the identity for addition. Then
(x + iy) + (a + ib) = x + iy
⇒ (x + a) + i( y + b) = x + iy
⇒ x + a = x and y + b = y
⇒ a = 0 and b = 0
Hence, the additive identity is the complex number 0 + i0 , written simply as 0.
Example:
( 5 + i7)
So, addition of 0 in question;
0+( 5 + i7) =( 5 + i7) +0=( 5 + i7)
Additive Inverse:
z = a + ib so its additive inverse will be -z which -(a + ib) = - a – ib
Example:
- 5 + i7 .
z = -5 + i7 so additive inverse will be -z
so -z = - (z)
= - ( -5 + i7)
= 5 - i7

Multiplication Laws:
Closure :
The product of two complex numbers is , by definition , a complex number. Hence, the set of
complex numbers is closed under multiplication.
Example :
(3+2i)(5+6i)
=15+18i+10i+12i²
=15+28i−12
=3+28i
Commutative property :
For two complex numbers z1 = a + ib and z2 = c + id , we have
z1 . z2 = (a + ib)(c + id) = (ac -bd) + i(ad + bc) (since i2 = -1)
And
z2 . z1 = (c + id)(a + ib) = (ca-bd) + i(cb + da)
But a, b, c , d are real numbers, so, ac - bd = ca - db and ad + bc = cb + da
Example :
(3+2i)(5+6i)=(5+6i) (3+2i)
15+18i+10i+12i ²= 15+18i+10i+12i²
15+28i−12 = 15+28i−12
3+28i= 3+28i

• Distributive Law:
Distribution laws are given below:
Right distribution Law:
(z1 + z2 ). z3 = z1 . z3+ z2 . z3
Left distribution Law:
z1 . ( z2 + z3 ) = z1 . z2 + z1 . z3
Examples:
• (6 + 8i)(4 + 2i)
=6(4 + 2i) + 8i(4 + 2i)
=6(4) + 6(2i) + 8i(4) + 8i(2i)
=24 + 12i + 32i + 16i2
=24 + 44i + 16(-1)
=24 + 44i – 16
=8 + 44i

• Associative Property :
Consider the three complex numbers,
z1 = a + ib , z2 = c + id and z3 = e + if
(z1 . z2 ). z3 = [(a + ib).( c + id )] .(e + if)
=[(ac - bd)+ i(ad +bc)] . (e + if)
=(ac-bd). e + i(ad +bc)e + i(ac -bd)f + i2 (ad +bc).f
=(ace -bde - adf -bcf) + i(ade + bce + acf -bdf) -------------(1)
And
z1.(z2 . z3) = (a+ib).[(c +id).(e +if)]
= (ace - adf - bcf -bde) + i(acf + ade + bce -bdf) -----------(2)
Thus, from (1) and (2)
(z1 . z2 ). z3 = z1 .(z2 . z3 )
Multiplicatve Identity:
Let c + id be the multiplicative identity of a + ib. Then

• (a + ib)(c + id) = a + ib
⇒ (ac - bd) + i(ad + bc) = a + ib
⇒ ac - bd = a and ad + bc = b
ac - a = bd and bc - b = -ad
a(c - 1) = bd ----------------(1)
b(c - 1) = -ad ---------------(2)
Multiply equation (1) by a and equation (2) by b and then add
a2 (c - 1) = abd
b2 (c - 1) = -abd
------------------------------
(a2 + b2 )(c - 1) = 0
So, either a2 + b2 = 0 or c - 1 = 0
but a2 + b2 ≠ 0
So, c - 1 = 0 ⇒ c = 1
∴ d = 0
c + id = 1 + i0 = 1
Hence the multiplicative identity of the complex number is 1.
• Example:
• 1×( 5 + i7) =( 5 + i7) ×1=( 5 + i7)

• Multiplicative Inverse:
The complex number z* is called multiplicative inverse of z.
It is donated by 𝑧−1.
𝑧−1
= (
𝑎
𝑎2+𝑏2 ,
−𝑏
𝑎2+𝑏2)
Lets see ……..
• The multiplicative inverse of ‘z’ is ‘
1
𝑧
Let z = a + 𝑖𝑏
So Its Multiplicative Inverse is ;
1
𝑧
=
1
𝑎+𝑖𝑏
1
𝑧
=
1
𝑎+𝑖𝑏
×
𝑎−𝑖𝑏
𝑎−𝑖𝑏
1
𝑧
=
𝑎−𝑖𝑏
(𝑎+𝑖𝑏)(𝑎−𝑖𝑏)
1
𝑧
=
𝑎−𝑖𝑏

1
𝑧
=
𝑎−𝑖𝑏
1
𝑧
=
𝑎−𝑖𝑏
𝑎2−𝑏2𝑖2
1
𝑧
=
𝑎−𝑖𝑏
𝑎2−𝑏2(−1)
1
𝑧
=
𝑎−𝑖𝑏
𝑎2+𝑏2 ∵
1
𝑧
=𝑧−1
𝑧−1 =
𝑎−𝑖𝑏
𝑎2+𝑏2
So , (
𝑎
𝑎2+𝑏2 ,
−𝑏
𝑎2+𝑏2)
Example:
Multiplicative inverse of (-4,7) is:
= (
−4
(−4)2+72 ,
−7
(−4)2+72)
= (
−4
65
,
−7
65
)

Scalar Multiplication:
To multiply a complex number by a scalar, multiply the real part by the
scalar and multiply the imaginary part by the scalar:
c(a + bi) = ca + cbi
Example:
8(4 + 2i)
=32+12i
Hence:
⇒ (a+b)z=az+bz
⇒ a(z1 +z2 )=az1 +az2
⇒ (ab)z=a(bz)

Properties of conjugate of a complex number:
If z, z1 and z2 are complex number, then
𝒛𝟏 ∓ 𝒛𝟐 = 𝒛𝟏 ∓ 𝒛𝟐
Proof : Let 𝑧1 = 𝑎 + 𝑖𝑏 and 𝑧2 = 𝑐 + 𝑖𝑑
Then,
𝑧1 ∓ 𝑧2 = 𝑎 + 𝑖𝑏 ∓ 𝑐 + 𝑖𝑑
= 𝑎 ∓ 𝑐 + 𝑖𝑏 ∓ 𝑖𝑑
= 𝑎 ∓ 𝑐 − 𝑖(𝑏 ∓ 𝑑)
= 𝑎 − 𝑖𝑏 ∓ 𝑐 ∓ 𝑖𝑑
= 𝑎 − 𝑖𝑏 ∓ (𝑐 − 𝑖𝑑)
= 𝑧1 ∓ 𝑧2

𝐳𝟏. 𝐳𝟐 = 𝐳𝟏. 𝐳𝟐
Proof : Let 𝑧1 = 𝑎 + 𝜄𝑏 and 𝑧2 = 𝑐 + 𝜄𝑑
Then, 𝑧1. 𝑧2 = 𝑎 + 𝜄𝑏 . (𝑐 + 𝜄𝑑)
𝑎𝑐 + 𝑖𝑎𝑑 + 𝑖𝑏𝑐 + 𝑖2𝑏𝑑
𝑎𝑐 + 𝑖𝑎𝑑 + 𝑖𝑏𝑐 − 𝑏𝑑
𝑎𝑐 − 𝑏𝑑 + 𝑖(𝑎𝑑 + 𝑏𝑐)
𝑎𝑐 − 𝑏𝑑 − 𝑖 𝑎𝑑 + 𝑏𝑐
𝑎𝑐 + 𝑖2𝑏𝑑 − 𝑖𝑎𝑑 − 𝑖𝑏𝑐
𝑎𝑐 − 𝑖𝑏𝑐 + 𝑖2𝑏𝑑 − 𝑖𝑎𝑑
𝑐 𝑎 − 𝑖𝑏 − 𝑖𝑑(𝑎 − 𝑖𝑏)
𝑎 − 𝑖𝑏 . (𝑐 − 𝑖𝑑)
𝑧1. 𝑧2
(2)

𝒛𝟏
𝒛𝟐
=
𝒛𝟏
𝒛𝟐
Proof:
𝑧1
𝑧2
= 𝑧1.
1
𝑧2
Using multiplicative property of conjugate we have
𝑧1.
1
𝑧2
𝑧1
𝑧2
𝒛= z
Proof: let 𝑧 = 𝑎 + 𝑖𝑏
Then, 𝑧 = 𝑎 + 𝑖𝑏 = 𝑎 − 𝑖𝑏 = 𝑎 + 𝑖𝑏 = 𝑧

Sum as well as the Product of any two conjugate complex
numbers is a real number.
Let the two conjugate complex numbers are a+bι , a-bι
Sum of two conjugate complex numbers=(a+bι) + (a-bι)
=a + ιb + a -ι b
= 2a
NOW
Product:
Product of two conjugate complex numbers = (a+bι) (a-bι)
= ( a ) 2 - ( a b ι ) + ( a b ι ) - ( b ι ) 2
= ( a ) 2 - ( b ι ) 2
= a 2 - b 2 ι 2
= a 2 - b 2 ( - 1 )
= a 2 + b 2
Both 2a and a2 + b2 are real numbers.

‫جزا‬
‫ك‬
‫ﷲ‬

Complex numbers

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