Problem 28 - College Algebra with Modeling and Visualization

Question

Write the expression in standard form. $$ (7+i)-(-8+5 i) $$

Step-by-Step Solution

Answer

The expression in standard form is \( 15 - 4i \).

Step 1: Understand the expression

The given expression is \((7+i)-(-8+5i)\). We need to simplify this expression by removing the brackets and combining like terms.

Step 2: Distribute the negative sign

Distribute the negative sign to the terms inside the parenthesis: \( (7+i) + 8 - 5i \). This changes the signs of the terms in the second parenthesis.

Step 3: Combine like terms

Group the real numbers together and the imaginary numbers together: \( (7 + 8) + (i - 5i) \).

Step 4: Simplify the expression

Add the real number terms and the imaginary number terms: \( 15 - 4i \). This is the expression in standard form.

Key Concepts

Standard Form

In the world of complex numbers, the standard form is the way of expressing complex numbers in an organized manner. A complex number is often written as \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) represents the 'real part' and \(bi\) represents the 'imaginary part' of the complex number.

The 'standard form' allows for a neat and clear expression of complex numbers, making it easier to perform arithmetic operations. When you're given an expression like \((7+i)-(-8+5i)\), writing it in standard form helps simplify and solve the problem effectively.

• First, distribute any negative signs and simplify terms.
• Next, group like terms together.
• Finally, write the expression as \(a + bi\), summing up the real and imaginary parts separately.

This process ensures that the complex number is expressed in a clean and simple manner, for easier handling in mathematical operations.

Imaginary Numbers

Imaginary numbers are an intriguing aspect of complex numbers. They involve the unit imaginary number \(i\), which is defined as the square root of \(-1\), i.e., \(i^2 = -1\). Unlike real numbers, imaginary numbers allow for the square root of negative numbers, expanding the scope of numbers we can work with in mathematics.

The role of imaginary numbers is crucial in many fields, from engineering to quantum physics. In a complex number written in standard form as \(a + bi\), the imaginary component is represented by \(bi\). Here, \(b\) is a real number coefficient multiplying \(i\).

When working with expressions like \((7+i) - (-8+5i)\), the handling of imaginary numbers is just as vital as dealing with real numbers. They are combined just like regular algebraic terms:

• Identify the imaginary parts of the terms.
• Combine and simplify them as needed, while keeping the \(i\) notation intact.

This method preserves the integrity of the complex number system while allowing for straightforward arithmetic.

Real Numbers

Real numbers are the foundational blocks of not just complex numbers, but most of the mathematics we use daily. They include all the numbers on the number line, encompassing both positive and negative numbers, zero, and fractions. In complex numbers, such as those expressed in the form \(a + bi\), \(a\) is the real number part.

When performing operations on complex expressions like \((7+i)-(-8+5i)\), identifying and managing real numbers is a key step. Real numbers in the expression are combined just like regular arithmetic:

• Locate all real parts of each term.
• Add or subtract these real numbers separately from the imaginary components.
• Summing these gives the real part of the standard form expression.

Understanding how real numbers interact within complex numbers is crucial for simplifying expressions into their standard form, ensuring clarity and precision in mathematical analysis.