Problem 22 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the addition or subtraction and write the result in standard form. $$(11-2 i)-(-3+6 i)$$

Step-by-Step Solution

Answer

The solution to the problem is \(14 - 8i\).

Step 1: Distribute the Negative

Firstly, distribute the negative sign from the second parenthesis to each term within that parenthesis. So the equation becomes: \[(11-2i) - (-3+6i) \rightarrow (11-2i) + (3-6i)\]

Step 2: Combine Similar Terms

Next, group the real numbers together (11 and 3) and the imaginary numbers together (-2i and -6i), then perform the addition: \[(11+3) + (-2i-6i) = 14 - 8i\]

Step 3: Write the result in Standard Form

The result in standard form is simply \(14 - 8i\) which is in the form \(a + bi\) where \(a\) is the real part and \(bi\) is the imaginary part

Key Concepts

Addition of Complex Numbers

Adding complex numbers is straightforward. Complex numbers consist of a real part and an imaginary part and are generally expressed as \(a + bi\). When adding two complex numbers, you need to add the real parts together and the imaginary parts together separately. For example, given two complex numbers \((a+bi)\) and \((c+di)\):

• Add the real parts: \(a + c\)
• Add the imaginary parts: \((bi + di)\)

So, the result becomes \((a+c) + (b+d)i\).
This keeps the real and imaginary parts clear and separate, making calculations much simpler.

Subtraction of Complex Numbers

Subtraction of complex numbers involves a similar process to addition, with one main difference: you subtract instead of add. Given two complex numbers, \((a + bi)\) and \((c + di)\), the subtraction \((a + bi) - (c + di)\) is performed as follows:

• Subtract the real parts: \(a - c\)
• Subtract the imaginary parts: \((bi - di)\)

This results in \((a-c) + (b-d)i\).
Make sure to distribute any negative signs if you're subtracting expressions with parentheses, as seen in expressions like \((11-2i)-(-3+6i)\).

Standard Form of Complex Numbers

The standard form of a complex number is a way to express it clearly and is written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. This format helps simplify further operations such as multiplication, division, and graphic representation on a plane.
To convert any complex expression to standard form, just ensure it follows the format \(a + bi\).
In the exercise solution, after the subtraction is performed, the result \(14 - 8i\) is already in standard form, showing it clearly as a combination of the real and imaginary parts.

Imaginary Numbers

Imaginary numbers are special numbers used in math to extend our number system. They are based on the imaginary unit \(i\), where \(i^2 = -1\).
Imaginary numbers are useful in solving equations that do not have real solutions, particularly in certain fields of engineering and physics.
An imaginary number looks like \(bi\) where \(b\) is a real number.
In our exercise, both \(-2i\) and \(6i\) are imaginary numbers, and combining them resulted in \(-8i\), showcasing how real and imaginary components can interact in complex arithmetic.