COMPLEX NUMBER NOTES FOR NDA

Complex Number

Complex numbers are helpful in finding the square root of negative numbers. The concept of complex numbers was first referred to in the 1st century by a greek mathematician, Hero of Alexandria when he tried to find the square root of a negative number. But he merely changed the negative into positive and simply took the numeric root value. Further, the real identity of a complex number was defined in the 16th century by Italian mathematician Gerolamo Cardano, in the process of finding the negative roots of cubic and quadratic polynomial expressions.

Complex numbers have applications in many scientific research, signal processing, electromagnetism, fluid dynamics, quantum mechanics, and vibration analysis. Here we can understand the definition, terminology, visualization of complex numbers, properties, and operations of complex numbers.

We know that the square of a real number is always non-negative e.g. (4)^2 = 16 and (– 4)^2 = 16. Therefore, square root of 16 is ± 4. What about the square root of a negative number? It is clear that a negative number can not have a real square root. So we need to extend the system of real numbers to a system in which we can find out the square roots of negative numbers. Euler (1707 - 1783) was the first mathematician to introduce the symbol i (iota) for positive square root of – 1 i.e., i = √−1 .

Imaginary numbers
Square root of a negative number is called an imaginary number.,
for example, √-2, √-3.

Two complex numbers z 1 = a + ib and z 2 = c + id are equal if a = c and b = d.

Example 1 If 4x + i(3x – y) = 3 + i (– 6), where x and y are real numbers, then find the values of x and y.

Solution : We have 4x + i (3x – y) = 3 + i (–6) ... (1)

Equating the real and the imaginary parts of (1),

we get 4x = 3, 3x – y = – 6,

which, on solving simultaneously,

give x = 3/4 and y = 33/4

Addition of two complex numbers

Let z1 = a + ib and z2 = c + id be any two complex numbers.

Then, the sum z 1 + z 2 is defined as follows: z 1 + z2 = (a + c) + i (b + d), which is again a complex number.

For example, (2 + i3) + (– 6 +i5) = (2 – 6) + i (3 + 5) = – 4 + i 8

The addition of complex numbers satisfy the following properties:

(i) The closure law The sum of two complex numbers is a complex number, i.e., z 1 + z 2 is a complex number for all complex numbers z 1 and z 2 .

(ii) The commutative law For any two complex numbers z 1 and z 2 , z 1 + z 2 = z 2 + z1

(iii) The associative law For any three complex numbers z 1 , z 2 , z 3 , (z 1 + z 2 ) + z 3 = z 1 + (z 2 + z 3 ).

(iv) The existence of additive identity There exists the complex number 0 + i 0 (denoted as 0), called the additive identity or the zero complex number, such that, for every complex number z, z + 0 = z. (v) The existence of additive inverse To every complex number z = a + ib, we have the complex number – a + i(– b) (denoted as – z), called the additive inverse or negative of z. We observe that z + (–z) = 0 (the additive identity).

Difference of two complex numbers

Given any two complex numbers z 1 and z 2 , the difference z 1 – z 2 is defined as follows: z 1 – z 2 = z 1 + (– z2 ). For example, (6 + 3i) – (2 – i) = (6 + 3i) + (– 2 + i ) = 4 + 4i and (2 – i) – (6 + 3i) = (2 – i) + ( – 6 – 3i) = – 4 – 4i

TOPIC IN COMPLEX NUMBER :-
#basic properties,
#modulus,
#argument,
#cube roots of unity.