Complex Numbers - JEE Important Topic

The concept of complex numbers and the practice of complex numbers were first introduced by Hero of Alexandria, a Greek mathematician, in the first century who was trying to calculate a negative number’s square root. But he changed the number to positive from negative and considered the numeric root value.

However, the real definition of complex numbers was introduced by Gerolamo Cardano, an Italian mathematician in the 16th century. He found that while finding the negative roots of a quadratic polynomial and cubic expression. Complex numbers applications can be seen in fluid dynamics, signal processing, scientific research, vibration analysis, and quantum mechanics. 

What are Complex Numbers?

In simple words, a complex number is the total of a real number as well as an imaginary number. In general, a complex number can be described in the form of $a+\iota b$ and is normally represented by z. It may be noted that, here, a as well as b is the actual number. The value of ‘a’ is called the real part and is denoted by ‘Re(z)’. On the other hand, the value of ‘b’ is called the imaginary part and is generally denoted by ‘Im(z)’. Furthermore, $\iota b$  can also be considered as an imaginary number.

As per the experts, complex numbers are like the building blocks of intricate Maths, for example, algebra, and can be applied in different aspects of real life, mainly in electromagnetism and electronics. 

A complex number is one that can be represented as a + ib, where ‘a’ stands for the real component and ‘b’ for the imaginary part; I is the imaginary unit, and it can be defined as solution of ${{x}^{2}}=-1$ or as the square root of -1. 

Important terms of complex number are listed below:

• Complex conjugate: Conjugate pairs of complex numbers are in the form a + bi  and a - bi. If we multiply the conjugate pair of complex numbers, then we get the result as ${{a}^{2}}+{{b}^{2}}$.

• Imaginary Number: The number which is in the form ki where the $i=\sqrt{-1}$ is an imaginary number.

• Imaginary Part: In any complex number given in the form a+bi, the term bi denotes the imaginary part of that complex number.

• Real Part: In any complex number given in the form a+bi, the term a denotes the real part of that complex number.

Properties of Conjugate, Modulus, and Argument

1. $\overline{{{z}_{1}}\pm {{z}_{2}}}={{\overline{z}}_{1}}\pm {{\overline{z}}_{2}}$

2. $\overline{{{z}_{1}}\cdot {{z}_{2}}}=\overline{{{z}_{1}}}\cdot \overline{{{z}_{2}}}$

3. $\overline{\left( \frac{{{z}_{1}}}{{{z}_{2}}} \right)}=\frac{\overline{{{z}_{1}}}}{\overline{{{z}_{2}}}}$

4. $\overline{{\bar{z}}}=z$

5. $\left| \left. {{z}_{1}}{{z}_{2}} \right| \right.=\left| \left. {{z}_{1}} \right| \right.\left| \left. {{z}_{2}} \right| \right.$

6. $\left| \left. \frac{{{z}_{1}}}{{{z}_{2}}} \right| \right.=\left( \frac{\left| \left. {{z}_{1}} \right| \right.}{\left| \left. {{z}_{2}} \right| \right.} \right)$

7. $\arg \left( \frac{{{z}_{1}}}{{{z}_{2}}} \right)=\arg \left( {{z}_{1}} \right)-\arg \left( {{z}_{2}} \right)$

8. $\arg \left( {{z}_{1}}{{z}_{2}} \right)=\arg \left( {{z}_{1}} \right)-\arg \left( {{z}_{2}} \right)$

Power of iota, $\iota $

In complex number formulas, the alphabet $\iota $ means the iota and represents the imaginary part of the complex number. If you have the value of ${{\iota }^{2}}=-1$, then this can be used to calculate the value of $\sqrt{-4}=\sqrt{4}{{\left( \sqrt{\iota } \right)}^{2}}=+2\iota $ .

• ${{\iota }^{2}}=\sqrt{-1}$

• ${{\iota }^{2}}=-1$

• ${{\iota }^{3}}=\iota .{{\iota }^{2}}=\iota \left( -1 \right)=-\iota $

• ${{\iota }^{4}}=\left( {{\iota }^{2}} \right)={{\left( -1 \right)}^{2}}=1$

• ${{\iota }^{4n}}=1$

• ${{\iota }^{4n+1}}=\iota $

• ${{\iota }^{4n+2}}=-1$

Algebra of Complex Numbers

Addition and Multiplication

To find accurate complex number solutions to a problem, you will have to understand the right method of multiplying and adding complex numbers.

As the complex numbers are a binomial, a numerical expression that has two different terms, arithmetic can be carried out in the same way as any binomial. You can combine the life terms as well as simplify. Let’s understand this with an example.

$\left( 5+3\iota \text{ }\!\!~\!\!\text{ } \right)+\left( 5-5\iota \text{ }\!\!~\!\!\text{ } \right)=\left( 5+5 \right)+\left( 3\iota -5\iota \text{ }\!\!~\!\!\text{ } \right)$

$=10-2\iota \text{ }\!\!~\!\!\text{  }\!\!~\!\!\text{  }\!\!~\!\!\text{ }$

The result is $10-2\iota $.

When it comes to multiplication, you can use the FOIL method to carry out a polynomial multiplication. The method is all about multiplying the first, then the outer, multiplying the inner and then the last. Once done, you need to add them all. Here is an example that will help you with this. 

$\left( 4-3\iota  \right)\left( 6+4\iota  \right)=\left( 4 \right)\left( 6 \right)+\left( 4 \right)\left( 4\iota  \right)+\left( -3\iota  \right)\left( 6 \right)+\left( -3\iota  \right)\left( 4\iota  \right)$

$=24+16\iota -18\iota -12{{\iota }^{2}}$

$=24-2\iota -12\left( -1 \right)$

$=24+12-2\iota $

$=36-2\iota $

Here ${{\iota }^{2}}$ means (-1) as I is the square root of -1.

Division of a Complex Number 

Well, the division of complex numbers is a little complicated, and you need to use conjugates. Speaking about complex conjugates, these are pairs of different complex numbers with different signs. For example, $\left( a+b\iota  \right)$ and $\left( a-b\iota  \right)$. When you multiply complex conjugate, it will cancel out the middle term. Let’s understand this with a simple example.

$\left( a+b\iota  \right)\left( a-b\iota  \right) = {{a}^{2}}-ab\iota +ab\iota +{{\left( b\iota  \right)}^{2}}$ 

If you simplify this, it will become ${{a}^{2}}-{{b}^{2}}{{\left( \iota  \right)}^{2}}={{a}^{2}}-{{b}^{2}}\left( -1 \right)$

Now, the final result will be ${{a}^{2}}+{{b}^{2}}$. 

While dividing any complex number, you need to determine the denominator’s conjugate, and then you will have to multiply the numerator as well as the denominator by the conjugate. 

Applications of Complex Numbers

Speaking about real-life applications of complex numbers, it can be used in different fields, such as engineering, computer science, Physics, and electronics. For example, complex numbers are used in fluid mechanics, an important area of Physics. Here, complex numbers are utilised to describe the possible flow in two dimensions. 

Conclusion

If you want to do better in the field of Mathematical Physics, then it is important for you to learn every single detail of a complex number. On the other hand, the complex number system is crucial as it is used as a convenient language to express wave functions without modifying the rules. You can learn more about the conjugate and identities of the complex numbers along with other complex numbers by going through the complex numbers exercises with answers pdf. And don’t forget to keep solving complex numbers exercises regularly to learn Mathematics effectively.