Imaginary numbers, also known as complex numbers, are a fundamental concept in mathematics. They are defined as numbers that involve the imaginary unit, denoted by the symbol “$i$,” which is defined as the square root of $-1$. The Imaginary number is often represented in the form $a + bi$, where “$a$” and “$b$” are real numbers, and “$i$” represents the imaginary unit.

## The origin of imaginary numbers

The concept of imaginary numbers was first introduced by mathematicians in the 16th century, but it was met with skepticism and resistance due to its seemingly paradoxical nature. However, over time, mathematicians realized the significance and utility of imaginary numbers in solving complex equations and representing certain mathematical phenomena.

## Key properties of the imaginary number

One of the key properties of the imaginary number is their ability to extend the real number system to include solutions to equations that cannot be expressed using only real numbers. For example, the equation $x^2 + 1 = 0$ has no real solutions, but it can be solved by introducing the imaginary number. In this case, the solution is $x = \pm i$, where “$i$” represents the imaginary unit.

Imaginary complex numbers also play a crucial role in various branches of mathematics, such as complex analysis, quantum mechanics, and electrical engineering. They are used to represent and analyze oscillatory phenomena, such as alternating currents and electromagnetic waves.

## Imaginary Numbers Rules

- The initial principle governing the imaginary number $i$ is expressed as $i^2=-1$ in mathematical discourse.
- The solutions to the quadratic equation $x^2+1=0$ are the imaginary and complex numbers $i$ and $-i$.
- In a more general context, let $a$, $b$, and $c$ be real numbers such that $a$ is not equal to zero, and the discriminant $\Delta$ defined as $b^2-4ac$ is less than zero. In this case, the quadratic equation $ax^2+bx+c=0$ has two complex solutions, which can be expressed as follows: $$ x_1=\frac{-b+i\sqrt{-\Delta}}{2a}\quad\text{and}\quad x_2=\frac{-b-i\sqrt{-\Delta}}{2a}.$$
- For any natural number $n$, $i^{2n}=(-1)^n$ and $i^{2n+1}=(-1)^ni.$

## Exercises in complex numbers

Exercise 1: Demonstrate that for all $x\in \mathbb{R}$, the inequality $|e^{ix}-1|\le |x|$ holds. |
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For any $x\in \mathbb{R}$, the expression $e^{ix}-1$ can be rewritten as $e^{\frac{ix}{2}} (e^{\frac{ix}{2}}-e^{\frac{-ix}{2}})$. This can be further simplified to $e^{\frac{ix}{2}} 2\sin\left( \frac{x}{2}\right)$. Since $|\sin(y)|\le |y|$ for all $y\in \mathbb{R}$, we can conclude that $|e^{ix}-1|=2 \left| \sin\left( \frac{x}{2}\right)\right|\le 2 \left| \frac{x}{2}\right|=|x|$. |

Exercise 2: Determine the modulus and argument of the complex numbers \begin{align*}z_1=\frac{\sqrt{6}-i\sqrt{2}}{2},\quad z_2=e^{e^{i\beta}},\quad \beta\in\mathbb{R}.\end{align*} |
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It is known that $\cos(\pi/6)=\sqrt{3}/2$ and $\sin(\pi/6)=1/2$. Therefore, we have \begin{align*} z_1=\sqrt{2}\left( \frac{\sqrt{3}}{2}-\frac{i}{2}\right)= \sqrt{2}\left(\cos(\frac{\pi}{6})-i \sin(\frac{\pi}{6}) \right)= \sqrt{2}e^{-i\frac{pi}{6}}.\end{align*} Thus, the modulus of $z_1$ is $|z_1|=\sqrt{2}$ and the argument is $\arg(z_1)=-\frac{\pi}{6}$. On the other hand, we have \begin{align*} z_2= e^{\cos(\beta)+i\sin(\beta)}= e^{\cos(\beta)} e^{i\sin(\beta)}.\end{align*} Therefore, $|z_2|=e^{\cos(\beta)}$ and $\arg(z_2)=\sin(\beta)$. |

Exercise 3: Determine the algebraic form of the following complex numbers: \begin{align*}
z_1=(5+i5)^4, \quad z_2= \left(\frac{1+i}{1+i\sqrt{3}}\right)^{40}.
\end{align*} |
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The expression $5+i5=3\sqrt{2}(\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2})=3\sqrt{2} e^{i\frac{\pi}{4}}$ can be written as follows. Let $z_1$ be the complex number $5+i5$. Then, we have $z_1=3\sqrt{2} e^{i\frac{\pi}{4}}$. Using this result, we can calculate $z_1$ raised to the power of 6 and multiplied by $e^{i\frac{3\pi}{2}}$. Thus, we obtain $z_1^6 e^{i\frac{3\pi}{2}}=-i5800$. Similarly, we have $1+i=\sqrt{2}e^{i\frac{\pi}{4}}$ and $1+i\sqrt{3}=2 e^{i\frac{\pi}{3}}$. Let $z_2$ be the complex number $1+i\sqrt{3}$. Then, we can calculate $z_2$ raised to the power of 10 and multiplied by $e^{i\frac{40 \pi}{3}}$. This gives us $z_2=\frac{\sqrt{3}}{8}-i\frac{1}{8}$. |

## Conclusion

In conclusion, the imaginary number is an essential concept in mathematics, providing a powerful tool for solving complex equations and representing various mathematical phenomena. Despite their initial skepticism, they have become an integral part of mathematical theory and find applications in numerous fields.

## Q&A for imaginary numbers

Q1: Can you provide an example of how imaginary numbers are used in solving equations? |
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A1: Certainly! Consider the equation $x^2+4=0$. Using imaginary numbers, we find that $\pm 2i,$ demonstrating their role in solving equations with no real solutions. |

Q2: Are imaginary numbers used in practical applications outside of mathematics? |
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A2: Yes, the imaginary number has practical applications in fields such as electrical engineering, quantum mechanics, signal processing, and more. It simplifies complex calculations in these disciplines. |

Q3: Can you take the square root of a negative real number? |
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A3: No, the square root of a negative real number is undefined in the realm of real numbers. Imaginary numbers, represented by ‘i,’ were introduced to address this limitation. |