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Algebraic Operations on Complex Numbers

 

In performing operations with complex numbers we can proceed as in the algebra of real numbers replacing i2 by -1 when it occurs.

1.   Addition of Complex Numbers

(a+bi) + (c+di) = (a+c) + (b+d)i 

 

Example:
z1 = 3 - 4i and z2 = -2 + 2i
z1 + z2 = (3 + (-2)) + (-4 + 2)i = 1 -2i

 

2.   Subtraction of Complex Numbers

(a+bi) - (c+di) = (a-c)+ (b-d)i

 

Example:
z1 = 3 - 4i and z2 = -2 + 2i
z1 - z2 = (3 - (-2)) - (-4 - 2)i = 5 +6i

 

3.   Multiplication of Complex Numbers

(a+bi)(c+di) = ac+bci+adi+bdi2

        = (ac-bd)+(ad+bc)i

 

Example:
z1 = 3 - 4i and z2 = -2 + 2i
z1 ∙ z2 = (3 ∙ (-2)) + (-4 ∙ (-2))i + (3 ∙ 2)i + (-4 ∙ 2)i2 = -6 + 14i -8i2

 

4.   Division of Complex Numbers

(a+bi)/(c+di)=(a+bi)/(c+di)×(c-di)/(c-di)
                    =(ac-adi+bci-bdi2)/(c2-d2i2)
                    =(ac+bd+(bc-ad)i)/(c2-d2i2)
                    =(ac+bd)/(c2+d2 )+(bc-ad)/(c2+d2) i

 

Example:
z1 = 3 - 4i and z2 = -2 + 2i
z1 / z2 = (3 ∙ (-2) + (-4 ∙ 2))/((-2)2 + 22) + (-4 ∙ (-2) - 3 ∙ 2)/((-2)2 + 22)i = -14/8 + (2/8)i = -7/4 + i/4

 

 

Inequalities in imaginary numbers are not defined. As for example z >0, 4 + zi < 2+4i are meaningless in complex numbers.

 

In real numbers, if a2 + b2=0 then a=b=0; however in complex numbers,

z12 + z22 does not imply z1 = z2 = 0

 

Complex conjugation: The complex conjugate of a + bi is a − bi. The importance of the complex conjugate lies in the fact that the product of a complex number and its conjugate is a real number.

 

(a + bi)(a − bi) = (a  + b)

Additive identity: Additive identity is the complex number that represents the notion of zero for addition of complex numbers is 0 + 0i.

 

(a + bi) + (0 + 0i) = a + bi

Multiplicative identity: Multiplicative identity is the complex number that represents the notion of one for multiplication of complex numbers is 1+ 0i.

 (a + bi)(1+ 0i) = a + bi

 

Example 1: Simplify (2√(-9) -3 )(3√(-16) - 1)

 

(2√(-9) -3 )(3√(-16) - 1)

=(2 · 3i - 3)(3 ·4i - 1)

=(6i - 3)(12i - 1)

= 72i2 – 36i – 6i + 3

= -69 – 42i


Example 2: Express in the form a + bj given


Example 3: Perform the division for the complex numbers
Solution: The given complex numbers are

We have to reduce the denominator as well as the numerator by multiplying the numerator and denominator both by the complex number 10+2i.



Use the formula (a+ b) (a-b) = a2- b2 to reduce the denominators. Use the formula (a+ b) 2= a2+2ab+ b2 to reduce the numerators.



Add and also square the terms to get the values.


In the complex numbers the value for i2 is -1.

Substitute the value of i2 in the above equation to reduce the denominators.




Example 4: Find the product of the complex numbers 1+ i and √ 3- i in polar form.

Solution: We can easily find that the polar form of 1+ i and √ 3- i are respectively,



We know that z1z2 = r1r2(cos12) + isin1 + θ2)). This formula says that to multiply two complex numbers we need to multiply the moduli and add the arguments. Therefore multiplying 1+ i and √ 3- i using the above rule, we will get:



Example 5: Simplify i64,002.

Solution: i64,002 = i64,000 + 2 = i4 · 16,000 + 2 = i2 = –1.



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