Problem 78 - Elementary and Intermediate Algebra

Question

Simplify. Write the result in the form \(a+b i .\) \((2-3 i)+(4+5 i)\)

Step-by-Step Solution

Answer

6 + 2i

Step 1: Identify Like Terms

In the expression \((2-3i)+(4+5i)\), identify the real numbers and the imaginary terms. Real numbers are 2 and 4, and the imaginary numbers are -3i and 5i.

Step 2: Combine Real Terms

Add the real numbers together: \(2 + 4\). This equals 6.

Step 3: Combine Imaginary Terms

Add the imaginary terms: \(-3i + 5i\). This equals \(2i\).

Step 4: Compose the Result

Combine the sums of the real and imaginary parts to form a simplified expression: \(6 + 2i\).

Key Concepts

Imaginary Numbers

Imaginary numbers are a fascinating part of mathematics. They are numbers that, when squared, give a negative result. Regular or real numbers cannot achieve this because whenever you square a positive or a negative real number, the result is always positive. Imaginary numbers use the symbol "i," which is defined as the square root of -1. So, when you see an expression like \(-3i\), it's saying \(-3\) times the square root of \(-1\).
Imaginary numbers allow us to expand our number system beyond the real numbers. They are useful in solving equations that do not have solutions within the realm of real numbers. For example, the equation \(x^2 + 1 = 0\) doesn't have a real solution, but in the complex number system, we see that \(x = i\) and \(x = -i\) are solutions. Imaginary numbers combine with real numbers to form complex numbers, which provide a more complete understanding of various mathematical concepts.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They are fundamental in mathematics for expressing relationships and calculating values. When working with complex numbers, like in our exercise, these expressions often include both real numbers and imaginary numbers.
An important property of algebraic expressions is that they can be simplified or manipulated to make calculations easier. For example, in the exercise \((2-3i) + (4+5i)\), each term can be broken down:

• 2 is a real number.
• -3i is an imaginary number.
• 4 is a real number.
• 5i is an imaginary number.

To simplify this expression, we separately handle the real parts and the imaginary parts before combining them into a simpler form.

Simplification

Simplification in mathematics means reducing an expression to its most concise form, while keeping the value the same. It makes complex expressions easier to understand and work with. In our exercise, simplification involves combining like terms.
First, identify and group the real numbers and the imaginary numbers separately. Bringing like terms together helps in reducing complexity:

• Add the real numbers: \(2 + 4 = 6\).
• Add the imaginary numbers: \(-3i + 5i = 2i\).

After individually simplifying these groups of terms, we combine the results. In this case, the final simplified expression is \(6 + 2i\). This format \(a + bi\) expresses complex numbers clearly and is commonly used to make interpreting and handling complex calculations more straightforward.