Intuitive Idea of Limits

A complex number is a number in the form of a +ib, where a and b are real numbers, and i is an indeterminate satisfying i² = −1.

a is on the real axis, and b is on the imaginary axis. Thus, complex number formed by this real and imaginary axis is given as Z=a+ib

We must also know a few properties of iota multiplication:

i²=-1

i³=-i

i4=1

Multiplication of Complex Numbers

Two complex numbers can be multiplied by a simple algebraic multiplication method. Let us take a simple example of the multiplication of complex numbers.

Eg:Z1=2+3i and Z2=1+5i

So, Z1*Z2 = (2+3i)* (1+5i)

=2+10i+3i+15i2

=2+13i-15 ( since i2=-1)

= -13+13i

Let us first learn about a few properties of multiplication of complex numbers.

Properties of Complex Numbers

Commutative Property of Complex Numbers

Multiplication of two complex numbers is said to be commutative when .Z2Z1=Z1Z2

Let us take an example to prove the commutative property of complex numbers.

Z1=1+i  and  Z2 =2+2i

Z1Z2= (1+i) (2+2i) = 2+2i+2i+2i2

=2+4i-2 = 0+4i…………………….(1)

Similarly;

Z2Z1= (2+2i) (1+i) = 2+2i+2i+2i2

= 2+4i-2

=0+4i…………………….(2)

Thus, 1 and 2 are similar. So complex numbers multiplication is commutative.

Associative Property of Complex Numbers

Multiplication of complex numbers is said to be associative where Z1(Z2Z3)=(Z1Z2)Z3

Let us take an example to prove complex numbers are associative.

Z1=2+3i

Z2=1+2i

Z3=1-i

Z2Z3  = (1+2i) (1-i)

 = 1-i+2i-2

= 1-i+2i-(-2) 

= 1-i+2i+2i2

=3+i

Z1(Z2Z3) = (2+3i) (3+i)

 = 6+2i+9i+3i2

= 6-3+11i

 = 3+11i……………………………………………………………(1)

Similarly,

Z1Z2= (2+3i) (1+2i)

 = 2+4i+3i+6i2 

=2-6+7i  

=  -4+7i

(Z1Z2)Z3= (-4+7i) (1-i)

= -4+4i+7i-7i2 

= 3+11i………………………………………………….(2)

Thus, 1 and 2 are equal. So complex multiplication is always associative.

Now let us see an important theorem of complex multiplication.

Complex Conjugate Root Theorem

Statement: The variable P is a polynomial with a+ib as one of its roots, then a-ib is also a root of the same polynomial P.

Multiplication of complex conjugate:

(a + ib)(a – ib) = a2 – (ib)2 = a2 – i2b2 = a2 + b2

Example:

(3 + i)(3 – i) = 32 – (i)2 = 32 – i2 = 9 + 1 = 10 = Square of Magnitude of 3 + i

The Multiplicative Inverse of a Complex Number

The multiplicative inverse of complex number z=1/z. We will rationalise the complex number first, and then we will try to solve it further by using the formula, a2−b2=(a−b)(a+b) to find the inverse of the complex number z.

Complete step-by-step solution:

In the question, we have been asked to find the multiplicative inverse of the complex number z. Before we proceed with the question, let us find out about complex numbers and the meaning of multiplicative inverse of complex numbers. Complex numbers are those numbers that can be expressed in the form of a+bi. The terms a and b represent real numbers. i is the solution of the equation x2=−1. Since there are no real numbers that satisfy this condition, it is termed as an imaginary number. The multiplicative inverse of complex numbers is simply the reciprocal of the number. In the question here we have z=4−3i. The multiplicative inverse of z is given by 1/z, so we get, 4−3i=1/4−3i.

Rationalising  1/(4−3i) by multiplying by, 4+3i, we get,

=4+3i(4-3i)(4+3i)

We know that (a−b)(a+b)=a2−b2, so in the denominator, we get,

=(4+3i)/((4)2−(3i)2)=4+3i/16−9i2

We know that i2=−1 from basic complex numbers, so we get,

=4+3i/(16−9(−1))=4+3i/16+9=4+3i/25

Splitting into the real and complex numbers, we get,

=4/25+3i /25

Therefore, the multiplicative inverse of the given complex number (4−3i)=(425+-3i25).

Note: Alternatively, we can also find the multiplicative inverse of any complex number z by using the direct formula of z−1=z/|z|2. We have z=3−4i, and we also have z=3+4i and |z|2=42+(−3)2=16+9=25. Substituting the values, we get, z−1=(425+-3i25)

Conclusion

Thus, knowing the properties of complex numbers is very important for the multiplication of complex numbers. And also, knowing the complex conjugate theorem is important while solving problems.