Problem 54 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the complex conjugate of the complex number. Then multiply the number by its complex conjugate. $$-8+\sqrt{15} i$$

Step-by-Step Solution

Answer

The complex conjugate of \(-8 + \sqrt{15} i\) is \(-8 - \sqrt{15} i\). Multiplication of these two numbers yields the value 49.

Step 1: Compute Complex Conjugate

The complex conjugate of \(-8 + \sqrt{15} i\) is \(-8 - \sqrt{15} i\).

Step 2: Multiply the Complex Number with its Conjugate

Multiplication of \(-8 + \sqrt{15} i\) and \(-8 - \sqrt{15} i\) will result into \((-8)^2 - (\sqrt{15} i)^2\) which is equal to \(64 - 15 = 49\).

Key Concepts

Complex Conjugate

A complex conjugate is a concept tied directly to complex numbers. When you have a complex number, such as \(a + bi\) (where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit), the complex conjugate is formed by changing the sign of the imaginary part. Thus, the complex conjugate of \(a + bi\) becomes \(a - bi\).
For example, given the complex number \(-8 + \sqrt{15} i\), its complex conjugate would be \(-8 - \sqrt{15} i\).

• Complex conjugates help simplify the process of dividing complex numbers.
• Multiplying a complex number by its conjugate always results in a real number. This is a key property that is used frequently in complex number operations.

The process becomes clearer when you see the multiplication with the given complex number and its conjugate.

Multiplication of Complex Numbers

Multiplying complex numbers involves using distributive properties much like regular algebra. For two complex numbers, say \((a + bi)\) and \((c + di)\), the product is calculated by expanding the expression:
\( (a + bi)(c + di) = ac + adi + bci + bdi^2 \). Note that since \(i^2 = -1\), the expression \(bdi^2\) actually becomes \(-bd\). This means the entire expression simplifies to \(ac - bd + (ad + bc)i\).

• This highlights an important point: notice how multiplying a complex number by its conjugate \((a + bi)(a - bi)\) eliminates the imaginary part, since \(bci\) and \(-bci\) cancel each other out.
• The result is a real number \(a^2 + b^2\), which simplifies the computation and brings down the complexity involved with imaginary units.

This property is why in the given solution, multiplying \(-8 + \sqrt{15} i\) by its conjugate results in the simple expression \((-8)^2 - (\sqrt{15})^2 = 64 - 15 = 49\). The resulting real number 49 confirms the property neatly.

Imaginary Unit

The imaginary unit is denoted with the symbol \(i\) and is beneficial in expressing numbers that have no real square root. In the context of complex numbers, \(i\) serves as the basic unit of imaginary numbers.
By definition, \(i\) is the square root of \(-1\), or mathematically represented as \(i^2 = -1\). This definition is vital and heavily used during complex number operations.

• Working with the imaginary unit requires remembering its square is \(-1\), and it heavily influences the behavior of products involving complex numbers.
• As seen in the process of multiplying complex numbers, the expression \(adi + bdi^2\) leads to terms that help simplify complex expressions back to real numbers.

Thus, the imaginary unit not only allows for expressing complex numbers, but it is also key to understanding the algebraic manipulation involved in complex number arithmetic.