What is I?

As the name suggests, imaginary numbers are not real, as they involve the square root of a negative number. A non-zero real number multiplied by the imaginary unit ‘i’ (which is also called iota), where i = √(-1), results in an imaginary number. 
 

Let’s understand this better by squaring some real numbers:

(- 4)² = -4 × -4 = 16

(8²)  = 8 × 8 = 64

(1.5)² = 1.5 × 1.5 = 2.25

In the above examples, were there any negative numbers? No. This indicates that any real number's square is always positive. Therefore, we need imaginary numbers to produce a negative square. We frequently encounter the square root of negative numbers in mathematics, particularly when applying the quadratic formula to solve quadratic equations. It is necessary to use imaginary numbers in these situations. Let’s now take a look at some imaginary numbers:

√(-4) = √(-1) · √4 = i (2) = 2i
√(-3) = √(-1) · √3 = i √3
 

The real number system is extended to include complex and imaginary numbers. Any number that includes the square root of a negative number is considered an imaginary number. They are of the form bi, where b \(\neq\) 0. The fundamental imaginary unit, represented by 𝑖, is the square root of −1.

This indicates that 𝑖² = −1. On the other hand, a complex number includes both imaginary and real numbers. They are of the form a + bi, where a is the real part and b is the imaginary part.

Students can perform addition and subtraction by combining like terms, real with real, imaginary with imaginary.

Example for addition:

\((3 + 2i) + (1 + 4i) = (3 + 1) + (2i + 4i) = 4 + 6i \)

Example for subtraction:

\((5 + 7i) - (2 + 3i) = (5 - 2) + (7i - 3i) = 3 + 4i \)

Multiplication uses the distributive property: each term of the first complex number is multiplied by each term of the second. Remember, i² = -1

\( (2 + 3i)(4 + i) = 2 \times 4 + 2 \times i + 3i \times 4 + 3i \times i \)

                        \(= 8 + 2i + 12i + 3i^2\)

                        \(= 8 + 14i + 3(-1) = 5 + 14i\)

For complex numbers, dividing complex numbers involves multiplying the numerator and denominator by the conjugate of the denominator to remove its imaginary part.

For example, divide: \( \frac{4 + 2i}{1 - i} \)

\( =\frac{4 + 2i}{1 - i} \)

\(= \frac{(4 + 2i)(1 + i)}{(1 - i)(1 + i)} \)

\(= \frac{4 + 4i + 2i + 2i^2}{1 - i^2} \)

\(= \frac{4 + 6i + 2(-1)}{1 - (-1)} \)

\(= \frac{2 + 6i}{2} = 1 + 3i\)

The “plane” in complex numbers refers to the complex plane, also called the Argand plane. This graph shows both the real and imaginary parts of a complex number: the horizontal axis represents real numbers, and the vertical axis represents imaginary numbers. A pure imaginary number, like 3i or −2i, lies on the vertical axis since its real part is zero.

For example, consider the complex number 3+4i. You move 3 units along the X-axis (real part) and 4 units along the Y-axis (imaginary part), plotting the point (3, 4). This point can be visualized as a vector from the origin (0, 0) to (3, 4). Dashed lines often show the movement along each axis. Such visualizations help in understanding operations like addition, subtraction, or finding the modulus; here, the modulus is \(|3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5.\)

In mathematics, 𝑖 is the imaginary unit, defined by \(i^2= -1\). It was introduced because no real number squared gives a negative result. This allows us to work with complex numbers of the form a+bi, where a and b are real.

For example, in 2+3i, 2 is the real part and 3𝑖 is the imaginary part. Equations like \(x^2+1=0\) this can now be solved, giving x = i, and x = -i

The basic definition of the imaginary unit, 𝑖, where 𝑖= -1, is used to calculate imaginary numbers. The property 𝑖² = −1 is added to standard arithmetic rules when calculating with imaginary numbers. The square root of −9, for instance, can be expressed as follows: -9 = 9 (-1) = 9  -1 = 3𝑖. 

The imaginary unit i has a unique property:

 \(i^2=−1\)  From this, higher powers of i follow a repeating pattern:

 𝑖¹ = 𝑖,

𝑖² = −1,

𝑖³ = −𝑖,

and 𝑖⁴ = 1

This cycle repeats every four powers. In general, for any positive integer 𝑛:

\(i^n = i^{n \bmod 4} \)

For example: calculate i²⁷:

\(27 \div 4 = 6 \text{ remainder } 3 \implies i^{27} = i^3 = -i \)

Solving problems involving imaginary numbers can get tricky. That’s why it is important to have a few tricks up our sleeves. Take a look at some handy tips and tricks given below to stay ahead of the curve.

• First and foremost, remember that 𝑖2 = −1; this identity is crucial for expression simplification. 

• Use the repeating pattern 𝑖, −1, −𝑖, 1, whenever working with powers of 𝑖.

• This pattern repeats every four powers. To prevent confusion, keep the real and imaginary parts of addition and subtraction distinct.

• To remove the imaginary unit from the denominator, first find the conjugate of the denominator. Then, multiply both the numerator and the denominator by the conjugate.

• Using these techniques improves the accuracy and ease of solving problems involving imaginary numbers.
   

Imaginary numbers, represented by i, extend the real number system and allow us to solve equations and model problems that cannot be addressed with only real numbers.

1. Electrical Engineering: Imaginary numbers are used in alternating current (AC) circuit analysis to represent the voltages and currents as complex numbers, making calculations easier.
   
    
2. Quantum Physics: Complex numbers describe wave functions of particles, helping physicists predict probabilities of particle behaviors.
   
    
3. Signal Processing: In audio, image, and communication technologies, i helps to process the signals using the Fourier transforms and other techniques.
   
    
4. Computer Graphics: Imaginary numbers simplify the rotations and transformations in 2D and 3D modeling for animations and simulations.
   
    
5. Control Systems & Robotics: Engineers use i to model and analyze system behaviors, such as stability and response, in robotics or automated systems.