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(2-5i)^2

2 min read Jun 16, 2024
(2-5i)^2

Squaring Complex Numbers: A Look at (2 - 5i)^2

Complex numbers, often represented as a + bi where 'a' and 'b' are real numbers and 'i' is the imaginary unit (√-1), are fascinating mathematical objects. Let's explore the squaring of a complex number, specifically (2 - 5i)^2.

Understanding the Process

Squaring a complex number means multiplying it by itself. Therefore, (2 - 5i)^2 is the same as:

(2 - 5i) * (2 - 5i)

To expand this, we can apply the distributive property (also known as FOIL method):

  • First: 2 * 2 = 4
  • Outer: 2 * (-5i) = -10i
  • Inner: (-5i) * 2 = -10i
  • Last: (-5i) * (-5i) = 25i^2

Now, we combine the terms:

4 - 10i - 10i + 25i^2

Remember that i^2 = -1, so we substitute:

4 - 10i - 10i + 25(-1)

Simplifying further:

4 - 10i - 10i - 25 = -21 - 20i

Result

Therefore, (2 - 5i)^2 = -21 - 20i.

This result highlights the unique nature of complex number multiplication. While squaring a real number always results in a positive value, squaring a complex number can yield a complex number with both a real and an imaginary component.

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