Problem 67 - A Graphical Approach to Precalculus with Limits

Question

Multiply as indicated. Write each product in standand form. $$i(3-4 i)(3+4 i)$$

Step-by-Step Solution

Answer

The product is in standard form: 0 + 25i.

Step 1: Express complex conjugates multiplication

First, observe that the expression \((3 - 4i)(3 + 4i)\) is a multiplication of conjugates. Recall that \((a - bi)(a + bi) = a^2 + b^2\). So, set \(a = 3\) and \(b = 4\). We can apply the formula here.

Step 2: Calculate the multiplication of conjugates

Using the formula, calculate \((3^2 + 4^2)\). This becomes \(9 + 16 = 25\). Thus, \((3 - 4i)(3 + 4i) = 25\).

Step 3: Multiply the result with i

Now, multiply the result \(25\) by \(i\). This yields \(25i\).

Step 4: Write the result in standard form

The standard form for complex numbers is \(a + bi\). Hence, \(25i\) can be written as \(0 + 25i\).

Key Concepts

Complex Conjugates

The concept of complex conjugates is helpful in simplifying expressions, such as the one in the original exercise. A complex conjugate is a pair of complex numbers. If your complex number is in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, its complex conjugate will be \(a - bi\).

When you multiply a complex number by its conjugate, the result is always a real number. This happens because the imaginary parts cancel each other out. In mathematical terms, this is expressed as \((a+bi)(a-bi) = a^2 + b^2\).

• To find the conjugate of \(3 - 4i\), you switch the sign of the imaginary part, giving \(3 + 4i\).
• Multiplying \(3 - 4i\) and \(3 + 4i\), the result is a real number: \(25\).

Standard Form

Complex numbers are often expressed in what's known as standard form. This form is \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

Standard form is useful for easily identifying the real and imaginary components of a complex number. It is particularly important when you need to perform operations like addition, subtraction, or in this case, multiplication.

• In the original exercise, the expression \(i(3-4i)(3+4i)\) initially results in \(25i\).
• Since \(25i\) lacks a real component, it can be written in standard form as \(0 + 25i\).

Imaginary Unit

The imaginary unit, represented as \(i\), is one of the fundamental elements of complex numbers. It is defined by the property that \(i^2 = -1\). This unique attribute is what allows complex numbers to express quantities that real numbers alone cannot.

In the original exercise, \(i\) plays a pivotal role. It acts as a coefficient to our result after multiplying complex conjugates. When \(25\) is multiplied by \(i\), we are using the fundamental property of \(i\) to convert our real result into an imaginary number.

• The operation transitions \(25\) into \(25i\), illustrating \(i\)'s ability to represent a purely imaginary number.
• This showcases the significance of \(i\) in expanding the scope of mathematical operations beyond just real numbers.