Problem 63 - Intermediate Algebra

Question

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (1-i)(1+i) $$

Step-by-Step Solution

Answer

The result is \(2 + 0i\).

Step 1: Recognize a Pattern

When we multiply conjugate pairs of complex numbers, such as \((1-i)(1+i)\), we can apply the formula for the difference of squares: \(a^2 - b^2\). Here, \(a = 1\) and \(b = i\).

Step 2: Apply the Difference of Squares

Using the formula \((1-i)(1+i) = 1^2 - i^2\), we need to calculate the square of each term. Thus, \(1^2 = 1\) and \(i^2 = -1\) since \(i^2 = -1\) by definition of the imaginary unit \(i\).

Step 3: Substitute Squares into Formula

Substitute the squares into the difference of squares formula to get \(1 - (-1)\). Simplify this to \(1 + 1\).

Step 4: Simplify to Final Form

Calculate \(1 + 1\) to get \(2\). Since there is no imaginary part, the result in form \(a + bi\) is \(2 + 0i\).

Key Concepts

Conjugate Pairs

Conjugate pairs are a unique type of complex numbers. A conjugate of a complex number is formed by changing the sign of the imaginary part, while keeping the real part unchanged. For instance, if you have a complex number in the form of \(a + bi\), its conjugate will be \(a - bi\). Conjugates are essential in simplifying complex number expressions, especially when dividing complex numbers or rationalizing denominators.

An important property of conjugates is that when a complex number and its conjugate are multiplied, the result is a real number. This process eliminates the imaginary parts. In our exercise,

• We have the pair
  • \((1 - i)\)
  • \((1 + i)\)

Conjugate pairs like these follow the form of the difference of squares when multiplied, resulting in a real number.

Difference of Squares

The difference of squares is a fundamental algebraic principle. This formula is used to simplify expressions that involve squaring and subtracting terms. It is expressed as

• \(a^2 - b^2\)

where \(a\) and \(b\) are any numbers or expressions. This formula is especially useful in simplifying expressions involving conjugate pairs.

When applied to our example

• \((1-i)(1+i)\),
• we see that \(a = 1\) and \(b = i\).

Using the difference of squares, we calculate

• \(1^2 - i^2\).

Remember that \(i^2 = -1\), so our expression simplifies to

• \(1 - (-1) = 1 + 1 = 2\).

By using this technique, complex conjugate pairs can be swiftly simplified to real numbers.

Imaginary Unit

The imaginary unit, represented as \(i\), is a number whose square is -1. It is the cornerstone of complex numbers, which are expressions in the form of \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit.

The unique property of \(i\) is that it allows us to express the square root of negative numbers, which is impossible within the realm of real numbers alone. This makes it essential in advanced mathematics and various fields such as engineering and physics.

In the context of our exercise, understanding that \(i^2 = -1\) is crucial. This property is used in the difference of squares to transform complex conjugate pairs into simpler, real results. By substituting \(i^2 = -1\), we eliminate the imaginary component when manipulating expressions, leading to simplified results like \(2\) in our example.

Operations with Complex Numbers

When dealing with complex numbers, a set of operations must be mastered: addition, subtraction, multiplication, and division. Just like with real numbers, these operations follow specific rules.

**Addition and Subtraction:**

• Combine like terms (real parts move together, imaginary parts mesh together).
• For example, \((3 + 2i) + (1 + 4i) = 4 + 6i\).

**Multiplication:**

• Apply the distributive property (also called FOIL when dealing with binomials).
• For complex numbers, remember \(i^2 = -1\) and adjust terms accordingly.
• In our exercise, we used
  • \((1-i)(1+i)\) which utilized both multiplication and the difference of squares concept.

**Division:**

• Involves multiplying by the conjugate to rationalize denominators.
• This process eliminates imaginary numbers from the denominator, yielding a simpler real result.

Each operation relies on understanding the nature of complex numbers and using key properties to simplify expressions. Mastery of these concepts allows for efficient and accurate manipulation of complex numbers.