Question

Simplify each expression. Then classify your results in the table below. a. \((-4+7 i)+(-4-7 i)\) b. \((2-6 i)-(-10+4 i)\) c. \((25+15 i)-(25-6 i)\) d. \((5+i)(8-i)\) e. \((17-3 i)+(-17-6 i)\) f. \((-1+2 i)(11-i)\) g. \((7+5 i)+(7-5 i)\) h. \((-3+6 i)-(-3-8 i)\)

Short answer

a. -8 (Real) b. \(12 - 10i\) (Mixed) c. \(21i\) (Imaginary) d. \(41 + 3i\) (Mixed) e. \(-9i\) (Imaginary) f. \(-9 + 23i\) (Mixed) g. 14 (Real) h. \(14i\) (Imaginary)

Step-by-step solution

Simplify Each Expression

Using the usual rules of arithmetic, calculate each expression. It is crucial to remember i denotes the imaginary unit. Hence i² equals -1. a. \((-4+7i)+(-4-7i) = -8 + 7i - 7i = -8\) b. \((2-6i) - (-10+4i) = 2 - 6i + 10 - 4i = 12 - 10i\) c. \((25+15i) - (25-6i) = 25 + 15i - 25 + 6i = 0 + 21i = 21i\) d. \((5+i)(8-i) = 5*8 + 5*(-i) + i*8 - i*i = 40 - 5i + 8i + 1 = 41 + 3i\) e. \((17-3i) + (-17-6i) = 17 - 3i - 17 - 6i = 0 - 9i = -9i\) f. \((-1+2i)(11-i) = -1*11 + (-1)*(-i) + 2i*11 - 2i*i = -11 + i + 22i + 2 = -9 + 23i\) g. \((7+5i)+(7-5i) = 7 + 5i + 7 - 5i = 14\) h. \((-3+6i) - (-3-8i) = -3 + 6i + 3 + 8i = 0 + 14i = 14i\)

Classify the Results

By nature, complex numbers have both real and imaginary parts. In this case, some results are purely real and some purely imaginary while some are mixed (having both a real and an imaginary part). a. -8 (Real) b. \(12 - 10i\) (Mixed) c. \(21i\) (Imaginary) d. \(41 + 3i\) (Mixed) e. \(-9i\) (Imaginary) f. \(-9 + 23i\) (Mixed) g. 14 (Real) h. \(14i\) (Imaginary)

Key concepts

Simplifying Complex Expressions

When simplifying complex expressions, we apply ordinary arithmetic rules while keeping in mind the special property of the imaginary unit, represented by the letter 'i'. This unit has a fundamental characteristic: its square, written as 'i²', is equal to -1.

Let's take the expression \( (-4+7i)+(-4-7i) \) as an example. To simplify it, combine like terms, which are the real numbers with real numbers, and the imaginary parts with imaginary parts. After canceling out the positive and negative imaginary parts, the expression simplifies to \( -8 \) which is a real number.

Understanding this principle is crucial for simplifying complex expressions properly. It's also important to recognize that complex expressions can be broken down into their real and imaginary components which can be manipulated independently before combining them back into their standard form, \( a + bi \) where \( a \) is the real part and \( b \) is the imaginary part.

Arithmetic Operations with Complex Numbers

Arithmetic operations with complex numbers come down to four basic types: addition, subtraction, multiplication, and division. Let's focus on addition and multiplication, as demonstrated in the given exercises.

For addition and subtraction, align the real parts and the imaginary parts and add or subtract them accordingly, just as we combine like terms in algebra. For example, \( (2-6i)-(10-4i) \) becomes \( 12-10i \) when simplified.

For multiplication, use the distributive law, also known as the FOIL method (First, Outer, Inner, Last). When multiplying \( (5+i)(8-i) \) the result is \( 41+3i \) after applying FOIL and simplifying any terms with \( i^2 \) into -1.

Each operation follows its unique rules, yet they all share a common focus on realigning complex numbers into a form that highlights both their real and imaginary components.

Classifying Complex Numbers

After simplifying complex expressions, we can classify the results based on the presence of real and imaginary components. A complex number is said to have a real part, an imaginary part, or both.

Here are three classifications:

• Purely Real: When there's no imaginary part, as shown by the simplified expression \( -8 \).
• Purely Imaginary: When there's no real part, like in \( 21i \) or \( -9i \).
• Mixed: When a number includes both real and imaginary parts, for instance, \( 41 + 3i \) or \( -9 + 23i \) are classified as mixed.

Correct classification is vital because it helps in anticipating the behavior of these numbers in further mathematical operations and applications, such as in complex number equations or when plotting them on the complex plane.