Problem 33 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operation and write the result in standard form. $$(2+3 i)(1-i)$$

Step-by-Step Solution

Answer

The result of the multiplication of the given complex numbers is \(-1 + i\).

Step 1: Distribute

Start by distributing within the brackets to multiply every term: \((2+3i)(1) + (2+3i)(-i)\). This simplifies to \(2+3i - 2i - 3i^2\).

Step 2: Substitute \(i^2\)

Since \(i^2\) equals -1, the equation becomes \(2+3i -2i + 3(-1)\) which simplifies to \(2+i -3\).

Step 3: Combine Like Terms

At this step, combine real terms with real terms and imaginary terms with imaginary terms. This yields a result of \(-1+i\).

Key Concepts

Standard Form

In complex numbers, standard form refers to expressing the number as a combination of real and imaginary parts. It is written as \(a + bi\), where \(a\) represents the real part and \(b\) represents the coefficient of the imaginary part \(i\). In our original exercise, the solution ends up in standard form as \(-1 + i\).

• **Real Part**: The real part, \(a\), can be any real number. In the final result, the real part is \(-1\).
• **Imaginary Part**: This refers to \(b\), found in the term \(bi\). Here, \(b\) is the coefficient of \(i\), which is 1 in our solution.

Standard form simplifies understanding complex numbers and operations on them, as well as displaying them in a consistent way for comparison and further mathematical manipulation.

Imaginary Numbers

Imaginary numbers arise from the concept of taking the square root of negative numbers. The imaginary unit \(i\) is defined by the property that \(i^2 = -1\). This property becomes particularly handy when performing arithmetic with complex numbers.

• **Basic Definition**: \(i = \sqrt{-1}\), but more often used is \(i^2 = -1\), allowing simplification of expressions involving higher powers of \(i\).
• **Operations**: When working with imaginary numbers, treat them as terms to be combined with like terms, much as you would with algebraic variables.

In the original problem, noticing that \(-3i^2\) converts to 3 because \(i^2\) is \(-1\) is crucial to simplifying our solution. This illustrates how imaginary numbers can transform expressions in complex arithmetic.

Distributive Property

The distributive property is a fundamental principle in algebra that applies to expressions involving multiplication over addition. It states that \(a(b + c) = ab + ac\). In working with complex numbers, this same property allows us to expand products like \((2 + 3i)(1 - i)\).

• **Expanding Expressions**: To distribute, multiply each term in the first pair of parentheses by each term in the second, as seen in our problem: \((2 + 3i) \cdot 1 + (2 + 3i) \cdot (-i)\).
• **Resulting Terms**: This multiplication breaks down into simpler parts: \(2 + 3i - 2i - 3i^2\). Each portion represents an application of the distributive property.

Using this property ensures that all parts of an equation are calculated accurately, a crucial step that transitions complicated products into more manageable pieces for further arithmetic handling.