Problem 30 - Intermediate Algebra

Question

Add or subtract. Write the sum or difference in the form \(a+b i .\) See Example 3 $$ (8-3 i)+(-8+3 i) $$

Step-by-Step Solution

Answer

The sum is 0.

Step 1: Identify Real Parts

Examine the expression \((8-3i) + (-8+3i)\). The real parts of the numbers are 8 and -8. Add these real parts together: \(8 + (-8) = 0\).

Step 2: Identify Imaginary Parts

Now, look at the imaginary parts, which are \(-3i\) and \(3i\). Add the imaginary parts: \(-3i + 3i = 0i\).

Step 3: Write the Result in Standard Form

Combine the results of the real and imaginary parts. Since both are zero, the sum of the expression is \(0 + 0i\). This simplifies to \(0\).

Key Concepts

Addition of Complex Numbers

When performing the addition of complex numbers, the aim is to combine both the real and imaginary parts separately. Complex numbers are typically expressed in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.

To add two complex numbers, such as \((x_1 + y_1i)\) and \((x_2 + y_2i)\), we must:

• Add the real parts: \(x_1 + x_2\)
• Add the imaginary parts: \(y_1i + y_2i\)

The result is a new complex number \((x_1 + x_2) + (y_1 + y_2)i\).

For example, in the exercise \((8-3i)+(-8+3i)\), adding the real parts \((8 + -8)\) gives 0, and adding the imaginary parts \((-3i + 3i)\) also results in 0i. Therefore, the complete sum of these complex numbers is simply \(0 + 0i\) or just \(0\). This process is straightforward, but ensuring each component is dealt with correctly is important.

Imaginary Numbers

Imaginary numbers are a fascinating concept in mathematics. They arise when we take the square root of a negative number, which is not possible with real numbers.

The unit imaginary number is \(i\), defined by the property \(i^2 = -1\).

In complex number expressions like \(a+bi\), the term \(bi\) represents the imaginary part. For example, in \((8 - 3i)\), \(\(-3i\)\) is imaginary.

Imaginary numbers are essential because they allow solutions to equations that have no real number solutions. They extend our number system to include all possible solutions to polynomial equations. In calculations involving addition, subtraction, multiplication, or division, imagine complex numbers as vectors; each can be handled separately from the real portion.

Remember, when adding or subtracting imaginary numbers like \(-3i + 3i\), the process involves treating \(i\) as a term that can simplify just like regular algebraic terms.

Real Numbers

Real numbers are the numbers you'll commonly encounter in day-to-day life. They include integers, fractions, and irrational numbers, like \( oot(2, 3)\).

In the context of complex numbers, the real part is the coefficient of \(i^0\) (since anything to the power of zero is 1). In a complex number expressed as \(a + bi\), \(a\) is the real number.

Knowing how to identify real numbers within complex expressions is vital. It allows you to separate them from imaginary parts effectively, facilitating various operations such as addition or subtraction. When combining complex numbers, like in \((8-3i)+(-8+3i)\), we first add the real components.

This method allows for a simplified solution strategy by dealing with real parts independently of imaginary portions. Thus in given problem, \(8 + -8\) results in 0, reflecting the fact that real numbers act identically in complex numbers as they do standalone, by simple addition.