Problem 22 - Elementary Algebra

Question

\((3-4 i)-(10-9 i)\)

Step-by-Step Solution

Answer

-7 + 5i

Step 1: Identify the complex numbers

The expression \( (3-4i) - (10-9i) \) involves two complex numbers: \( 3-4i \) and \( 10-9i \).

Step 2: Distribute the subtraction

Apply the subtraction operation to each part of the complex numbers: real parts with real parts, and imaginary parts with imaginary parts. This gives \( (3-4i) - (10-9i) = 3 - 10 + (-4i - (-9i)) \).

Step 3: Simplify the real parts

Subtract the real parts: \( 3 - 10 = -7 \).

Step 4: Simplify the imaginary parts

Subtract the imaginary parts: \( -4i - (-9i) \). Since subtracting a negative is the same as adding, this becomes \( -4i + 9i = 5i \).

Step 5: Combine and write the result

Combine the simplified real and imaginary parts to get the final result: \(-7 + 5i \).

Key Concepts

Complex Numbers

A complex number is a number that combines both a real part and an imaginary part. The format is generally written as \(a + bi\), where \(a\) represents the real part and \(b\) represents the imaginary part. The imaginary part is always accompanied by the imaginary unit \(i\), which is defined as the square root of -1. In other words, \(i^2 = -1\). For example, in the complex number \(3 - 4i\), 3 is the real part and -4 is the imaginary part. Complex numbers are useful in many fields, including engineering and physics.

Imaginary Numbers

Imaginary numbers are numbers that, when squared, give a negative result. This is actually not possible with regular (real) numbers, as squaring a real number always results in a non-negative number. To solve this problem, mathematicians created the imaginary unit \(i\), defined by \(i^2 = -1\). Imaginary numbers are crucial for solving equations that do not have real solutions. For instance, the equation \(x^2 + 1 = 0\) does not have a solution among the real numbers, but it does in the complex plane: \(x = \pm i\). Imaginary numbers are also pivotal in complex number arithmetic, including both addition and subtraction.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It helps in expressing mathematical relationships and solving equations. In algebra, you often simplify expressions or solve for unknowns. Complex number subtraction, as seen in the exercise \((3-4i) - (10-9i)\), involves applying algebraic principles. First, distribute the subtraction over both the real and imaginary parts, then simplify each part separately. Combining the real parts, you get \(3 - 10 = -7\), and for the imaginary parts, you get \(-4i - (-9i)\), which simplifies to \(-4i + 9i = 5i\). Hence, the result is \(-7 + 5i\).