# Imaginary Numbers

Imaginary numbers are numbers that can be written as a multiple of the imaginary unit i, where i is defined as the square root of -1. Examples include 2i, −3i, and 5i.

### History and Naming

The concept of imaginary numbers was first introduced in the 16th century by Italian mathematician Rafael Bombelli. Initially, the notion was met with skepticism and confusion. They got their name because they were deemed “impossible” and “imaginary,” as they couldn’t be represented on the number line of real numbers.

### Complex Numbers and Notation

A complex number is a combination of a real number and an imaginary number. Write a complex number in the form a+bi, where a and b are real numbers. For example, 3+4i is a complex number.

### Importance and Practical Applications

Imaginary numbers have numerous practical applications. Electrical engineers use them for analyzing and designing electrical circuits (AC circuit analysis), while control engineers use them for stability analysis of control systems. In physics, they find use in quantum mechanics and other fields to describe and predict the behavior of physical systems. Math uses imaginary numbers in the Mandelbrot equation for fractals and for quadratic equation solutions.

For example, electrical engineering calculates the impedance of a circuit using imaginary numbers. The impedance Z is Z = R + jX. Here, R is the resistance, X is the reactance, and j is the imaginary number. Engineering uses j in place of i because i typically represents current.

### Basic Arithmetic Operations

For the most part, arithmetic involving imaginary numbers work like regular algebra:

#### Addition and Subtraction

Combine like terms to add or subtract imaginary numbers. In other words, deal with the real and imaginary portions separately.

(a+bi) + (c+di ) = (a+c) + (b+d)i

#### Example:

(2+3i) + (4+5i) = 6+8i

#### Multiplication

To multiply imaginary numbers, use the distributive property and remember that i2=−1.

(a+bi)(c+di) = (acbd) + (ad+bc)i

#### Example:

(3+2i)(1+4i) = (3−8) + (12+2)i = −5+14i

#### Division

To divide, multiply by the conjugate and simplify.

(a + bi) ​/ (c + di)= (a+bi)(cdi)​ / (c2+d2)

#### Example:

(2+3i​) / (1 – 4i) = (2 + 3i)(1 + 4i) / (1 + 16) = (-10 + 11i)/17​

### Equality Fallacy with Square Roots of Negative Numbers

However, take care when working with square roots of negative numbers and imaginary numbers. These numbers do not always obey the same arithmetic rules as real numbers, leading to potential confusion and errors if not handled correctly.

One typical mistake occurs with the square roots of negative numbers and the equation √a​×√b​=√(a×b)​, which holds for nonnegative real numbers a and b but does not always hold when dealing with negative numbers.

#### Example:

Let’s consider the equation √−1×√-1​.

Use the principal square root definition, √−1​=i.

So, you might incorrectly assume:

√-1 x √-1 = √(-1 x -1) = √1 = 1

But in reality:

√-1 x √-1 = i x i = i2 = -1

### Interesting Property: The Cycle of i

Imaginary numbers exhibit a unique cycling property:

• i = √−1​
• i2 = −1
• i3 = −i
• i4 = 1

And then it repeats:

• i5 = √−1 or i
• i6 = −1,…

This property helps simplify expressions involving higher powers of i.

### Is Zero an Imaginary Number?

Zero is an imaginary number and it is also a real number. (It is the only number that is both real and imaginary.) This is because imaginary numbers are numbers that can be written in the form bi where b is a real number and i is the imaginary unit with the property that i2=−1. When b=0, bi=0, which applies to both the real numbers and the imaginary numbers.

The set of imaginary numbers is actually a subset of the set of complex numbers, where a complex number is written in the form a+bi, with a and b as real numbers. In the special case where a=0 and b=0, a+bi is equal to 0, making 0 a complex number as well.

In summary:

• 0 is an imaginary number because it can be expressed as 0⋅i where b=0.
• 0 is a real number
• 0 is also a complex number, in the form 0+0i.

### References

• Giaquinta, Mariano; Modica, Giuseppe (2004). Mathematical Analysis: Approximation and Discrete Processes (illustrated ed.). Springer Science & Business Media. ISBN 978-0-8176-4337-9.
• Ingard, K.U. (1988). Fundamentals of Waves and Oscillations. Cambridge University Press. ISBN 0-521-33957-X.
• Kuipers, J. B. (1999). Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality. Princeton University Press. ISBN 0-691-10298-8.
• Martinez, Albert A. (2006). Negative Math: How Mathematical Rules Can Be Positively Bent. Princeton: Princeton University Press. ISBN 0-691-12309-8.
• Nahin, Paul (1998). An Imaginary Tale: the Story of the Square Root of −1. Princeton: Princeton University Press. ISBN 0-691-02795-1.