(1+2i)^2

Jasonbradley

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Jun 16, 2024

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Squaring Complex Numbers: (1+2i)^2

In mathematics, complex numbers are often represented in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1 (i.e., i² = -1).

This article will explore how to square the complex number (1 + 2i).

Understanding the Process

Squaring a complex number essentially means multiplying it by itself:

(1 + 2i)² = (1 + 2i) * (1 + 2i)

To expand this, we can use the distributive property (also known as FOIL - First, Outer, Inner, Last):

1. First: 1 * 1 = 1
2. Outer: 1 * 2i = 2i
3. Inner: 2i * 1 = 2i
4. Last: 2i * 2i = 4i²

Adding these terms together gives us:

1 + 2i + 2i + 4i²

Simplifying the Expression

Remember that i² = -1. Substituting this into our expression:

1 + 2i + 2i + 4(-1)

Simplifying further:

1 + 2i + 2i - 4

Combining like terms:

(1 - 4) + (2 + 2)i

The Result

Finally, we get:

**(1 + 2i)² = ** -3 + 4i

Therefore, the square of the complex number (1 + 2i) is -3 + 4i.