Understanding the Calculation: (3 + 8) + (-6i + 4i) Explained

Math problems combining real and imaginary numbers can feel complex at first, but simplifying them step by step makes them easy to grasp. Today, we’ll break down the expression (3 + 8) + (-6i + 4i)—a blend of real numbers and imaginary numbers—and explain how to solve it with clarity.

Understanding the Context

What Is the Expression?

The expression (3 + 8) + (-6i + 4i) involves both real parts (numbers without imaginary units) and imaginary parts (numbers multiplied by the imaginary unit i). In algebra, it's common to combine like terms separately.

Step 1: Combine Real Numbers

= (3 + 8) + (-6i + 4i)

Key Insights

Start with the first part:
3 + 8
These are simple real numbers:
= 11

Step 2: Combine Imaginary Parts

Next, work on:
(-6i + 4i)
Here, both terms have the same imaginary unit i, so we can add the coefficients directly:
= (-6 + 4)i
= -2i

Continue Reading

\Delta A = A_2 - A_1 = 181.5\sqrt{3} - 150\sqrt{3} = 31.5\sqrt{3} \text{ cm}^2
Thus, the area of the hexagon increases by:
\boxed{31.5\sqrt{3} \text{ cm}^2}

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Final Thoughts

Step 3: Add the Results

Now combine both simplified parts:
11 + (-2i)
Or simply:
= 11 - 2i

This is the final simplified form—a complex number with a real part 11 and an imaginary part -2i.

Why Does This Matter in Math and Science?

Complex numbers are essential in engineering, physics, and computer science. Combining real and imaginary components correctly allows professionals to model waves, vibrations, electrical currents, and more accurately. Understanding simple operations like (3 + 8) + (-6i + 4i) builds a strong foundation for working with complex arithmetic.

Summary

  • (3 + 8) = 11 (real numbers)
  • (-6i + 4i) = -2i (pure imaginary)
  • Final result: 11 - 2i

Combining real and imaginary terms follows the same logic as adding simple real numbers—just remember to keep the imaginary unit i consistent and combine coefficients carefully.