Problem 48 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the complex conjugate of the complex number. Then multiply the number by its complex conjugate. $$3+5 i$$

Step-by-Step Solution

Answer

The complex conjugate of \(3+5i\) is \(3-5i\). When the number is multiplied by its conjugate, the result is 34.

Step 1: Identify the Complex Number

The given complex number is \(3+5i\). Here, real part (a) is 3 and imaginary part (b) is 5.

Step 2: Write the Complex Conjugate

The conjugate of a complex number \(a+bi\) is \(a-bi\). So, the conjugate of the given number \(3+5i\) is \(3-5i\).

Step 3: Multiply the Number by its Conjugate

To multiply a complex number \(a+bi\) with its conjugate \(a-bi\), we use the rule (x+y)(x-y) = x^2 - y^2. Hence, \((3+5i)(3-5i) = 3^2 - (5i)^2 = 9 - (-25) = 9+25 = 34\).

Key Concepts

Understanding Complex Numbers

A complex number can be thought of as an extension of the real numbers we are accustomed to. It's essentially a pair of numbers, composed of a real part and an imaginary part. The imaginary part is a multiple of the imaginary unit, denoted by 'i', which is defined by the property that i^2 = -1.

For example, consider the complex number 3 + 5i. It has a real part of 3 and an imaginary part of 5. We often represent complex numbers in the form a + bi, where a stands for the real part and b for the imaginary part. This notation is quite helpful in visualizing and performing algebraic operations with complex numbers.

Multiplying Complex Numbers

Multiplying complex numbers might seem daunting, but it follows the same principles of distribution that apply to real numbers. When you have two complex numbers in the form a + bi and c + di, their product is found by multiplying each part of the first complex number by each part of the second, then combining like terms.

Here's how the multiplication works out, step by step:

• Multiply the real parts: ac.
• Multiply the real part of the first by the imaginary part of the second: adi.
• Multiply the imaginary part of the first by the real part of the second: bci.
• Multiply the imaginary parts: bdi^2 (Remember, i^2 = -1).

Finally, combine the real terms and the imaginary terms to get the product (ac - bd) + (ad + bc)i. It's the same process you'd use with polynomials—just with an extra rule for i^2.

Algebraic Operations with Complex Numbers

Complex numbers can be added, subtracted, multiplied, and divided just like real numbers, with some additional twists due to the presence of the imaginary unit 'i'. For example, when adding complex numbers, you only add the real parts with real parts and the imaginary parts with imaginary parts. The result is another complex number.

Similarly, when subtracting complex numbers, subtract the real part of the second number from the real part of the first and do the same with the imaginary parts. The tricky part comes with multiplication and division, where you need to take into account the rule for i^2, as illustrated in the multiplication of a complex number by its conjugate.

Remember, the conjugate of a complex number a + bi is a - bi. Multiplying a complex number by its conjugate is a special case that results in a real number. This is because the product of (a + bi)(a - bi) is a^2 - b^2i^2, which simplifies to a^2 + b^2 since i^2 = -1. This operation is particularly useful for rationalizing denominators in complex number fractions.