Problem 37 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The result of the multiplication (1+i)(3-2i) is 5 + i.

Step 1: Distribute

First, distribute each term in the first complex number with each term in the second complex number. This means that (1 * 3) + (1 * -2i) + (i * 3) + (i * -2i). This simplifies to 3 - 2i + 3i - 2i^2.

Step 2: Simplify

Next, simplify this result. According to the definition of \(i\), remember that \(i^2 = -1\). Substituting -1 in for \(i^2\) in our expression from Step 1 gives us 3 - 2i + 3i + 2.

Step 3: Combine like terms

Finally, combine like terms: the real numbers (3 and 2) and the complex numbers (-2i and +3i) to get the result 5 + i as the product of the two given complex numbers.

Key Concepts

Complex Number Operations

Complex numbers are numbers that have both real and imaginary parts. When performing operations on complex numbers, such as addition, subtraction, multiplication, or division, make sure to work with both parts separately. For example, in the expression

• (1 + i)(3 - 2i),

we apply the distributive property: multiplying each part of the first complex number with each part of the second. This gives us:

• Distribute each element:
  • 1 * 3 = 3,
  • 1 * (-2i) = -2i,
  • i * 3 = 3i,
  • i * (-2i) = -2i^2.

After distribution, we arrive at the expression: 3 - 2i + 3i - 2i^2.
Next, it's important to simplify by using the properties of the imaginary unit (i). In complex number operations, pay special attention to how terms interact, especially when involving powers of i.

Standard Form

In complex numbers, the standard form is a way of expressing results that makes them easy to read and use in calculations. A complex number is in standard form if it is expressed as:

• a + bi,

where 'a' represents the real part and 'b' represents the imaginary part.
This standardization helps in comparing and performing further operations on complex numbers. Consider once we've multiplied and simplified our terms: 3 - 2i + 3i + 2.
Combine like terms by adding or subtracting separately the real parts (3 + 2) and the imaginary parts (-2i + 3i):

• Real part: 3 + 2 = 5,
• Imaginary part: -2i + 3i = i.

Thus, the final result in standard form is: 5 + i.
Writing complex numbers in standard form ensures clarity and precision in mathematical operations.

Imaginary Unit

The imaginary unit, denoted as i, is a fundamental concept in complex numbers that helps us extend the real number system.
By definition, i is the square root of -1. This results in the crucial property that:

• i^2 = -1.

Understanding this property is key when working with complex numbers, as it impacts how we simplify expressions. For instance, when multiplying (1 + i)(3 - 2i), we end up with the term -2i^2.
Using the property i^2 = -1, we substitute:

• -2i^2 = -2(-1) = 2.

This allows for the expression to be further simplified from 3 - 2i + 3i - 2i^2 to 3 - 2i + 3i + 2. Recognizing the imaginary unit's properties helps in correctly performing operations and ensuring all terms are simplified properly. The imaginary unit allows for elegant and powerful solutions in complex number arithmetic.