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To square a complex number like \((1-3i)^2\), we must approach the problem similarly to squaring binomials in algebra. A complex number has the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. In our exercise, \(a = 1\) and \(b = -3\). The square of a complex number is found using:

• \((a + bi)^2 = a^2 + 2abi + (bi)^2\)

This process helps us break down the complex number into simpler parts to make the computation straightforward. Breaking the problem down into smaller parts, or terms, allows us to better understand how to reach the correct solution without mistakes. The outcome of squaring the complex number provides another complex number, which can be written as a simplified expression. This simplification step is crucial for clarity and accuracy in further mathematical work involving complex numbers.

The binomial theorem is a powerful tool that extends beyond real numbers and helps simplify operations with complex numbers. It provides a formula to expand expressions that are raised to a power, specifically in the form \((a + b)^n\). When n equals 2, as in our expression \((1-3i)^2\), the binomial theorem simplifies our task.

• For any complex number \((a + bi)^2\), it expands into \(a^2 + 2abi + (bi)^2\).

This expansion allows us to separately compute each term:

• The square of the real part, \(a^2\),
• The product of the real and imaginary parts multiplied by 2, \(2abi\),
• And the square of the imaginary part, \((bi)^2\)

Through these easy steps, the complexity of operations involving squared complex numbers is reduced. Understanding the binomial theorem's application to complex numbers enhances our ability to maneuver through algebraic expansions and simplifies various calculations.

Complex number arithmetic often involves operations similar to those with real numbers, but with added rules due to the imaginary unit \(i\), where \(i^2 = -1\). In our squared expression \((1-3i)^2\), we used arithmetic operations such as multiplication, addition, and subtraction.

• First, we calculated the square of the real part, \(1^2 = 1\).
• Next, the multiplication of the parts combined by two: \(2 \times 1 \times (-3)i = -6i\).
• The square of the imaginary part involved dealing with the imaginary unit: \((-3i)^2 = 9(i^2) = 9(-1) = -9\).

After each term is calculated, these results are combined:

• Add and simplify real terms to get: \(1 - 9 = -8\)
• Combine the imaginary part: \(-8 - 6i\)

These calculations allow us to combine the real and imaginary components into a simplified form. Mastery of complex number arithmetic is essential as it gives a comprehensive understanding of dealing with both parts of complex numbers effectively, ultimately producing results that can be applied in a broad range of mathematical problems.