Problem 46 - Precalculus Student Solutions Manual 5th

Question

Find each sum or difference. Write the answer in standard form. $$(-3+2 i)-(-4+2 i)$$

Step-by-Step Solution

Answer

1

Step 1: Identify the numbers

The given expression is \( (-3+2i)-(-4+2i) \). Identify each complex number. Here, \( -3 + 2i \) is the first complex number and \( -4 + 2i \) is the second complex number.

Step 2: Distribute the subtraction

Distribute the subtraction across the second complex number: \( (-3+2i)-(-4+2i) = -3+2i -(-4) - 2i \). This simplifies to \( -3 + 2i + 4 - 2i \).

Step 3: Combine like terms

Combine the real parts and the imaginary parts separately: \( -3 + 4 \) and \( 2i - 2i \). This simplifies to \( 1 + 0i \).

Step 4: Simplify the expression

The resulting expression is \( 1 + 0i \). In standard form, this is \( 1 \) since the imaginary part is zero.

Key Concepts

complex numbers

Complex numbers are an extension of real numbers and consist of two parts: a real part and an imaginary part. For example, in the complex number \( -3 + 2i \), \-3\is the real part, and \( 2i \)is the imaginary part. Complex numbers are usually denoted as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \), called the imaginary unit, is defined by the property that \( i^2 = -1 \). This makes it a useful way to extend the number line to two dimensions, incorporating both real and imaginary parts.

standard form

The standard form of a complex number is expressed as \( a + bi \) where \( a \) is the real part and \( b \) is the coefficient of the imaginary part. For example, the expression \( -3 + 2i \) is already in standard form. Standard form helps us easily identify the real and imaginary components, and it’s the usual way to write and work with complex numbers in mathematics. When performing operations like addition or subtraction on complex numbers, it’s essential to keep the numbers in this format so that the operations are straightforward.

imaginary numbers

Imaginary numbers are numbers that can be written as a real number multiplied by the imaginary unit \( i \). The imaginary unit \( i \) has the defining property \( i^2 = -1 \). These numbers are crucial when dealing with square roots of negative numbers, which do not have real solutions. For example, in the complex number \( -3 + 2i \), \( 2i \) is the imaginary part. Imaginary numbers allow us to extend our number system to solve equations that do not have real solutions.

real numbers

Real numbers are the numbers we are most familiar with, including integers, fractions, and irrational numbers. They can be positive, negative, or zero. When a complex number contains no imaginary part, it’s purely real. For instance, \( 1 + 0i \) simplifies to \( 1 \), a real number. In the exercise provided, the final answer \( 1 + 0i \) is simplified to \( 1 \), highlighting that the imaginary part is zero, leaving us with a real number as the result.