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.
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
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
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.
Key Takeaway
Even when dealing with imaginary numbers, mathematical expressions combine like terms efficiently:
- Real parts stay real: 3 + 8 = 11
- Imaginary parts combine numerically: (-6 + 4)i = -2i
So, (3 + 8) + (-6i + 4i) = 11 - 2i
Understanding this process empowers anyone mastering algebra and complex numbers—key tools for advanced science and technology fields.
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If you enjoyed simplifying this expression, explore more about how real and imaginary numbers work together in complex analysis or electrical engineering applications!
