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# Imaginary Number|Definition & Meaning

**Definition**

**Numbers** that **produce** a **negative **result **when** **squared** are called **imaginary numbers**. Imaginary numbers can be **computed** by **taking** the **square root** of **negative numbers** without a definite value.

Even though it **doesn’t exist** on the **number line** and can’t be quantified, imaginary numbers exist and have **mathematical applications**.

**Representation of Imaginary Number**

An imaginary number is** represented** by the **greek letter “i,”** which denotes the **square root** of **negative unity**. **Generally**, imaginary numbers are **written** as the **product** of some **real number** with the **iota** $i$ or simply i.

When combined with a** real number a** and an **imaginary number bi, a + bi** **forms** a **complex number**, in which the **real part a** is called the **real part** of the **complex number**, and the imaginary part bi is called the **imaginary part**.

The imaginary unit is typically **denoted** by **j** **instead of** **i** in many engineering and control systems problems **since i** is mainly used to **represent** **electric current.**

**Imaginary Unit**

Consider the **equation** shown below. The **solution** of this **quadratic equation** can be **generalized** by taking the **square root** on **both sides** which gives:

x^{2} = -1

\[\mathsf{\sqrt{x^2}=\sqrt{-1}}\]

x = $\mathsf{\sqrt{-1}}$

$\sqrt{-1}$ equals the **imaginary unit** we **call iota** (i, $i$ or sometimes j):

i = $\mathsf{\sqrt{-1}}$

**Pure Imaginary Number **

It is **possible** to** create** **infinitely** many **pure imaginary numbers** by **taking multiples** of this **imaginary unit** i. We can better understand **how these numbers relate to real numbers** by taking the squares of these numbers. Using the number 4i as an example, let’s **square it**. We can square 4i just as we would expect given the **properties of integer exponents**.

Squaring 4i:

(4i)^{2} = 16i^{2}

Based on **arithmetic**, i^{2} = -1. The **value of i** is therefore equal to the **square root of −1**. A **simple condition** for a **purely imaginary** number is that its **real part** is **equal** to **zero.**

**Value of Imaginary Unit **

Here we will discuss the **different values** of **imaginary units**.

In the particular case of the **quadratic equation**: z^{2} = x, where **x** is a **known value**. **One way** of presenting its **solution** is as z = $\mathsf{\sqrt{x}}$. To calculate some imaginary numbers, **we must follow** the following **rules**:

- The value of imaginary unit
**i = $\mathsf{\sqrt{-1}}$** - The value of imaginary number
**i**^{2}is -1 - The value of imaginary number
**i**^{3}is -i - The value of an imaginary number having
**N**power**i**^{4N}is 1

**Graphical Interpretation of Imaginary Number**

**Imaginary numbers** can be **arranged perpendicular** to the **real axis** because they** lie on** the **vertical axis** of the **complex number** plane. You can **visualize** imaginary numbers **by considering** a **number line** with **positive magnitudes** **increasing rightward** and **negative magnitudes** **increasing leftward**. Imaginary numbers **increase** in magnitude **upwards** when they are drawn on the** x-axis**, while imaginary numbers vary **downwards** when they are drawn on the **y-axis**.

The **“imaginary axis”** is sometimes **referred** to as this **vertical axis** and the **horizontal axis** is known as the **“real axis”**.Suppose we want to **plot** a purely **imaginary number** 2+2i then it will be 2** units** in the** vertical direction** and 2 in the **horizontal direction**, this concept is illustrated in the figure below.

**Application of Imaginary Numbers**

**In Calculus and Signal Processing**

Imaginary numbers, sometimes called complex numbers, are applied to **real-life situations** like q**uadratic equations** and **electricity**. Equations that are **not touching** the** x-axis** are called **imaginary numbers** in **quadratic planes**. **Advanced calculus** often **makes use** of imaginary numbers. As well as **cellular** and **wireless technology**, imaginary numbers can also be applied to **signal processing**, a process that is useful in radar technology as well. In essence, imaginary numbers are used when measuring **sine or cosine waves**.

**In Electrical Engineering**

In electrical engineering, specifically in electronics that use **alternating current (AC)**, imaginary numbers are especially useful. As a **sine wave changes** from **positive to negative**, **AC** electricity **changes** from **positive to negative**. Due to the fact that AC currents may not match properly on the waves, combining them can be very problematic. **Calculating AC electricity** and avoiding **electrocution** are** made easier** by **using imaginary currents** and real numbers.

**Pairing Imaginary Numbers With Arithmetic Operations **

**Addition**

When** two numbers** involving imaginary numbers are being **added** to each other then the **rule of thumb** is that we have to **add** the **real part to the real** and the** imaginary part to the imaginary**.

**Example: **(x+iy) + (a+ib) = (x+a) + i(y+b)

**Subtraction**

The subtraction process follows the same rule as for addition **real** part should be **subtracted** **from the real** and the **imaginary** part **from** the **imaginary**.

**Example: **(x+iy) – (a+bi) = (x-a) + i(y-b)

**Multiplication**

The **rule of thumb** for **multiplication** is that the **product of the real part** (xa) should be **subtracted** from the **product of the imaginary part** (yb). This term as a whole will be **considered as the real part** of the result.

The **imaginary term** in the result will be equal to the **sum** of the **product **of the** first term’s imaginary part **with the** second term’s real part **(ya) and the **product **of the **first term’s real part **with the **second term’s imaginary part **(xb)**.**

Take a look at the following **example** to understand this:

(x+iy) (a+ib)=(x+iy)a+ (x+iy)ib

(x+iy)(a+ib)=(xa+iya+ixb+i^{2}yb)

(x+iy)(a+ib)=(xa+iya+ixb-yb)

(x+iy)(a+ib) = (xa-yb) + i(ya+xb)

**Division**

For division, we will **multiply** the** conjugate **with both the **numerator** and **denominator** (the conjugate is basically the denominator with the **imaginary part sign reversed**). The conjugate of $a+bi$ is $a-bi$. In conjugate, since the **sign** of the imaginary part is **reversed,** the division process is illustrated below.

\[\mathsf{\frac{(x+iy)}{(a+ib)}=\dfrac{(x+iy).(a-ib)}{(a+ib).(a-ib)}}\]

\[\mathsf{\frac{(x+iy)}{(a+ib)}=\dfrac{(xa-yb)+i(ya+xb)}{(a^{2}-(ib)^{2})}}\]

\[\mathsf{\frac{(x+iy)}{(a+ib)}=\dfrac{(xa-yb)+i(ya+xb)}{(a^{2}+b^{2})}}\]

**An Example of Operations on Imaginary Numbers**

Show the **multiplication, division, and plot** of mathematical terms involving **imaginary numbers**. Consider the two imaginary numbers: 3+i, 2+i2.

**Solution**

**Multiplication**

(3+i)(2+2i) = (3+i)2 +(3+i)i2

(3+i)(2+2i) = (6+i2+6i+2i^{2})

(3+i)(2+2i) = (6+2i+6i-2)

(3+i)(2+2i) = (4+8i)

**Division**

\[\mathsf{\dfrac{(3+i)}{(2+i2)}=\dfrac{(3+i).(2-i2)}{(2+i2).(2-i2)}}\]

\[\mathsf{\dfrac{(3+i)}{(2+i2)}=\dfrac{(4+8i)}{(2^{2}-(i2)^{2})}}\]

\[\mathsf{\dfrac{(3+i)}{(2+i2)}=\dfrac{(4+8i)}{(4+4)}}\]

\[\mathsf{\dfrac{(3+i)}{(2+i2)}=\dfrac{(4+8i)}{(8)}}\]

**Plot**

**3+i: **We can clearly see it has to move **3 units in the real axis,** which is the horizontal axis, and **one unit in the imaginary axis,** which is the vertical axis.

**2+2i: **We can clearly see it has to move **2 units in the real axis,** which is the horizontal axis, and **2 units in the imaginary axis,** which is the vertical axis.

*All mathematical drawings and images were created with GeoGebra.*