Understanding the Equation: a² + b² = (12 + 16i) − (10 + 6i) = 2 + 10i

In this article, we explore the fascinating mathematics behind the equation a² + b² = (12 + 16i) − (10 + 6i) = 2 + 10i, explaining how complex numbers interact in algebraic operations and confirming that the calculation is mathematically accurate. Whether you're a student learning complex arithmetic or a teacher seeking a clear example, this guide breaks down each step to help build a solid foundation.

Understanding the Context

What Does a² + b² Mean in Complex Numbers?

The expression a² + b² involves complex numbers—numbers expressed in the form x + yi, where x and y are real numbers and i is the imaginary unit with the property that i² = −1. When calculating a² + b², especially when a and b are complex, we perform the arithmetic using standard rules, combining real and imaginary parts accordingly.

Step-by-Step Calculation

a^2 + b^2 = (12 + 16i) - (10 + 6i) = 2 + 10i

Key Insights

We start with the right-hand side of the equation:

(12 + 16i) − (10 + 6i)
This simplifies by subtracting the real and imaginary components separately:

  • Real part: 12 − 10 = 2
  • Imaginary part: 16i − 6i = 10i

Thus:
(12 + 16i) − (10 + 6i) = 2 + 10i

Now, the equation becomes:
a² + b² = 2 + 10i

Continue Reading

New area: \(12 \times 12 = 144\) cm²
Percentage increase: \(\left(\frac{144 - 64}{64}\right) \times 100 = 125\%\).
A bank offers an annual interest rate of 5% compounded annually. If you invest $1000, calculate the amount after 3 years.

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Final Thoughts

This confirms that finding a² + b² equals the complex number 2 + 10i, which lies in the complex plane with a real part of 2 and an imaginary part of 10.

Why Solving Equations Like This Matters

Understanding how to work with such expressions is valuable in many fields:

  • Engineering: Simulating systems involving oscillatory or wave-like behavior often uses complex numbers.
  • Physics: Quantum mechanics and electromagnetism rely on complex math for wave functions and field calculations.
  • Computer Graphics: Transformations with complex numbers help create rotations and scaling.

The equation a² + b² in complex numbers illustrates the principle of preserving the magnitude and phase during algebraic manipulations—a fundamental skill in advanced math and its applications.

How to Solve Complex Equations Like This

To solve equations involving a² + b² with complex values:

  1. Combine real parts and imaginary parts before squaring.
  2. Apply the identity: (x + yi)² = x² − y² + 2xyi.
  3. Perform squaring separately for each term, then add.
  4. Combine results, separating real and imaginary components.