Problem 49 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the complex conjugate of the complex number. Then multiply the number by its complex conjugate. $$-1+\sqrt{7} i$$

Step-by-Step Solution

Answer

The complex conjugate of the given complex number is \(-1-\sqrt{7}i\) and when the complex number is multiplied by its conjugate, the result is \(8\).

Step 1: Identify the Complex Number

The given complex number is \(-1+\sqrt{7}i\). It consists of two parts: a real part \(-1\) and an imaginary part \(\sqrt{7}\).

Step 2: Find the Complex Conjugate

To find the complex conjugate of \(-1+\sqrt{7}i\), just change the sign of the imaginary part, while keeping the real part the same. When the sign of \(\sqrt{7}i\) changes, the new complex conjugate becomes \(-1-\sqrt{7}i\).

Step 3: Multiply the Complex Number by its Conjugate

We multiply \(-1+\sqrt{7}i\) by its conjugate \(-1-\sqrt{7}i\) in the same way as normal multiplication, following the principle that the square of 'i' (\(i^2\)) is \(-1\) . The multiplication looks like this: \n\((-1+\sqrt{7}i) * (-1-\sqrt{7}i) = (-1)^2 - (\sqrt{7}i)^2 = 1 - (-7) = 8\).

Key Concepts

Understanding the Complex Conjugate

Complex numbers might seem intimidating initially, but they're quite intriguing once you get the hang of them. A complex number is pretty much like a combination of a real number and an imaginary number. Generally, it's written as

• \( a + bi \)

Here, \(a\) represents the real part, and \(bi\) denotes the imaginary part, where \(b\) is a real number and \(i\) is the imaginary unit.
Now, to find the complex conjugate, you only need to flip the sign of the imaginary part. Think of it like looking at a mirror version. So, for a complex number \(a+bi\), its complex conjugate is

• \(a-bi\).

This transformation is super handy when doing arithmetic with complex numbers because it helps to simplify them and makes certain problems much easier to solve. The complex conjugate can neutralize the imaginary part when you multiply them together. It's like balancing the equation to rid of the imaginary sections.

Exploring Real and Imaginary Parts

To grasp complex numbers entirely, it's essential to split them into two distinct parts: the real and the imaginary. The real part is the "normal" number you would ordinarily use in everyday math, like \( -1 \), as seen in our example. On the other hand, the imaginary part involves the letter \( i \), where \( i \) is defined as the square root of \( -1 \). This turns into numbers like \( \sqrt{7}i \), representing the imaginary part.
Now, why are these parts crucial? Well, in many mathematical situations, keeping these two sections separate helps unravel complex equations.

• The real and imaginary parts of a complex number are identified as the coefficient of the real number and the coefficient of the imaginary number respectively.
• They allow you to perform operations like addition, subtraction, and more by considering each part individually.

This division is central when it comes to computing combinations or simplifying equations, especially in scientific fields like engineering and physics.

Multiplication of Complex Numbers

Multiplication of complex numbers is a crucial operation and works similarly to the way you would multiply binomials. It's akin to using the distributive property but with a twist due to the inclusion of the imaginary unit \(i\).
Let's break it down. When multiplying two complex numbers, use the distributive law:

• \( (a+bi) \times (c+di) = ac + adi + bci + bdi^2 \).

Remember, since \( i^2 = -1 \), you must replace \( i^2 \) with \( -1 \) after expanding.

• This adjustment transforms the equation to: \[ ac - bd + (ad+bc)i \].

In our example, multiplying a complex number with its conjugate actually clears out the imaginary part. It’s a neat trick to verify your solution. We saw that

• \( (-1+\sqrt{7}i)(-1-\sqrt{7}i) \)

gives a purely real number after the operation—\(8\), no imaginary parts left. Multiplying with a conjugate is a classic method for simplifying complex expressions or finding magnitudes.