Question

1) Compare and contrast real vs. complex numbers. State at least 3 total facts that are similarities or differences.

Answer 1

Certainly! Here are three comparisons between real and complex numbers:

### 1. Definition:
- **Real Numbers**: Real numbers include all the numbers on the number line, encompassing rational numbers (like 1, 0.5, -3) and irrational numbers (like √2, π).
- **Complex Numbers**: Complex numbers extend real numbers to include imaginary units. A complex number is expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit defined as \(i^2 = -1\).

### 2. Number Line vs. Plane:
- **Real Numbers**: Real numbers can be visually represented as points on a one-dimensional number line.
- **Complex Numbers**: Complex numbers are represented as points in a two-dimensional plane, known as the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.

### 3. Arithmetic Operations:
- **Real Numbers**: Arithmetic operations (addition, subtraction, multiplication, and division) among real numbers always yield real numbers.
- **Complex Numbers**: When performing arithmetic operations on complex numbers, the results can yield complex numbers, even when operations involve only real numbers. For example, the sum of \(2 + 3i\) and \(1 + 4i\) is \(3 + 7i\), which is still a complex number.

These points illustrate key similarities and differences between real and complex numbers, emphasizing their definitions, representations, and properties in arithmetic.