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What is a Complex Number?

Complex and Real Numbers

Have you ever wondered what lies beyond the world of real numbers? The answer is complex numbers! Complex numbers are numbers of the form a + bi, where a and b are real numbers, and i is an indeterminate satisfying i2 = −1. De Moivre’s Theorem is a powerful tool that allows us to solve complex equations and uncover the beauty of the complex number system. In this blog post, we’ll explore the fundamentals of complex numbers and uncover the secrets of De Moivre’s Theorem.


Complex Number Formulas and Identities

A complex number is a number of the form a + bi, where a and b are real numbers, and i is an indeterminate satisfying i2 = −1. For example, 2 + 3i is a complex number. This math reference article covers some elementary identities that allow you to easily identify and calculate relative problems on this topic.

  • Magnitude
  • Imaginary number
  • Argument
  • Conjugate
  • Complex number operations
  • Multiplication in polar form
  • Commutative law for addition and multiplication
  • Associative law for addition and multiplication and
  • De Moivre’s Theorem

What is a Complex Number?

A complex number is a number of the form a + bi, where a and b are real numbers, and i is an indeterminate satisfying i2 = −1 . Complex numbers were first discovered by mathematicians during the 16th century and since then have been used extensively in fields such as mathematics, engineering, physics, computer science and many more.

De Moivre’s theorem

De Moivre’s Theorem gives us insight into some very useful properties of complex numbers. This theorem states that for any complex number z and for any positive integer n, (z^n) = (r^n)(cos(nθ)+isin(nθ)), where r is the modulus and θ is the argument. Using this theorem we can easily calculate powers of complex numbers using only basic trigonometric identities.

In Taylor series expansion

Complex numbers have also been used to illustrate a variety of equations in calculus including Taylor Series expansion. Overall, complex numbers can help us understand various mathematical properties in ways that would otherwise be impossible to do with real numbers alone. With these tools at our disposal, we can continue to explore the beautiful world of complex numbers.

Magnitude, Conjugate and Imaginary

Magnitude, conjugate and imaginary number of complex numbers is given by, (r^n) = r*cos(nθ) + ir*sin(nθ). This theorem, also known as De Moivre’s theorem, states that any complex number can be written as the sum of its magnitude (r) multiplied by its conjugate, as well as its imaginary number (i). This theorem makes it easier to solve complex equations and use complex numbers for problems such as solving for angles of triangles.

Worked Example

4 + 3i  and 4 3i are known as complex numbers. Both solutions are of the form a + bi, a being termed the real part and bi the imaginary part. A complex number of the form a + bi is called a Cartesian complex number.

Find the magnitude of

magnitude of complex number

Step 1

The magnitude is defined by

2022 12 14 12h23 20

Step 2

Expression in Cartesian form by comparison

Cartesian form

Step 3

Using Pythagora’s theorem to resolve magnitude, where the real part =8, imaginary part = 12

pythagoras theorem
polar form

Step 4

Take the squares of both terms and expand

final answer of the magnitude of complex number
final answer


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