Problem 36 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the complex conjugate of the complex number. Then multiply the number by its complex conjugate. $$-2+4 i$$

Step-by-Step Solution

Answer

The complex conjugate of \(-2+4i\) is \(-2-4i\), and by multiplying \(-2+4i\) and its conjugate, the result is \(-12\).

Step 1: Find the Complex Conjugate

The complex conjugate of a complex number \(a+bi\) is \(a-bi\). So the conjugate of \(-2+4i\) is \(-2-4i\).

Step 2: Multiply the number by its conjugate

Multiplying \(-2+4i\) by its conjugate \(-2-4i\) using the multiplication rule for complex numbers \( (a+bi)(c+di) = a*c-b*d + (ad+bc)i \), we have \(-2*(-2)- 4*4 + (4*(-2) -2*(-4))i = 4-16+ 0i= -12+0i\). Since the imaginary part is zero, the result is a real number \(-12\).

Key Concepts

Complex Conjugate

In the fascinating world of complex numbers, the concept of a complex conjugate plays a crucial role. A complex number is generally represented in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The complex conjugate of this number is expressed as \(a - bi\). What does this mean, and why is it important?

Essentially, a complex conjugate involves changing the sign of the imaginary part of a complex number. For example, for the complex number \(-2 + 4i\), its conjugate is \(-2 - 4i\). Both these forms differ only in the sign of their imaginary components.

The idea of a complex conjugate helps us specifically when we multiply complex numbers or divide them. Using conjugates often simplifies these calculations, bringing out the real number components. This becomes vital in processes such as finding the modulus of a complex number or simplifying complex fractions.

Multiplication of Complex Numbers

Multiplying complex numbers might appear a bit tricky at first, but it follows straightforward rules just like multiplying binomials. You are usually dealing with two complex numbers in the form \((a + bi)(c + di)\). Here's what to remember:

• Multiply like you would two binomials: \(a \times c, b \times di\), and so forth.
• Carol: Combine the results of the real parts and the imaginary parts separately.
• Be attentive with powers of \(i\), remembering that \(i^2 = -1\).

Let's look at the example using the numbers \(-2 + 4i\) and its conjugate \(-2 - 4i\).

Using the rule \((a+bi)(c+di) = ac - bd + (ad+bc)i\):

\(-2\times -2 - 4\times 4 + (-2\times -4 +4\times -2)i\), simplifying gives \(4 - 16 + 0i = -12 + 0i\).

The multiplication shows that combining a number with its complex conjugate yields a real number by eliminating the imaginary component, offering an insightful property of these mathematical constructs.

Imaginary Unit

The concept of the imaginary unit, often denoted as \(i\), is a defining feature of complex numbers. By definition, \(i\) is the square root of \(-1\). This means that any time you see \(i\), you're dealing with something that doesn't fit neatly on the number line of real numbers. The imaginary unit is not 'real' in the everyday sense, but it importantly allows us to express and work with quantities that would otherwise be impossible to solve in traditional real number terms.

Key properties include:

• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

These cyclic properties are useful when performing algebraic operations with complex numbers. They help to simplify expressions, especially in multiplication. Understanding and utilizing \(i\) is fundamental in electronics, signal processing, and many areas of physics, where phenomena are best described using complex numbers. Thus, although imaginary, \(i\) plays a very real role in how we approach and solve modern mathematical problems.