Problem 3 - Intermediate Algebra

Question

Fill in the blanks. For the complex number \(2+5 i\), we call 2 the ____ part and 5 the ____ part.

Step-by-Step Solution

Answer

Real; imaginary.

Step 1: Identify the Real Part

In a complex number, it is commonly expressed in the form of \(a + bi\), where \(a\) represents the real number part and \(b\) represents the imaginary number part. For the complex number \(2+5i\), the number \(2\) is the real part.

Step 2: Identify the Imaginary Part

From the standard format \(a + bi\), the second part, \(5i\), represents the imaginary component of the complex number. The coefficient of \(i\), which is \(5\) in this case, is the imaginary part.

Key Concepts

Real Number

In mathematics, a real number is a value that represents a quantity along a continuous line, such as the number line. These numbers include all the familiar counting numbers along with fractions and irrationals such as decimals that do not repeat.

Real numbers can be positive, negative, or zero, and are incredibly important in everyday calculations and measurements.

• Positive real numbers represent values greater than zero, such as 3, 7.5, or π.
• Negative real numbers represent values less than zero, like -2 or -0.8.
• Zero is considered a real number too.

Real numbers can be expressed in decimal form or as fractions. In the context of complex numbers, the real number part, like the "2" in the complex number \(2+5i\), tells you the horizontal position of the number on the complex plane.

Imaginary Number

Imaginary numbers might sound 'imaginary,' but they are a fundamental concept in mathematics used to solve problems involving the square root of negative numbers. The imaginary unit, denoted as \(i\), is defined as the square root of -1, meaning \(i^2 = -1\).

When you multiply a real number by \(i\), it becomes an imaginary number.

• For example, \(3i\) is an imaginary number, resulting from the real number 3 multiplied by \(i\).
• The imaginary numbers form a perpendicular axis (imaginary axis) on the complex plane, allowing for two-dimensional representation of complex numbers.

In the complex number \(2+5i\), "5" is the imaginary part, indicating how far up or down you should go on the imaginary axis in the complex plane.

Standard Form of Complex Numbers

Complex numbers blend the real and imaginary numbers into a single expression, typically written in the form \(a + bi\). This is known as the standard form of complex numbers.

• "a" is the real part of the complex number.
• "b" is the coefficient of the imaginary part, coupled with the imaginary unit \(i\).

When you encounter a complex number like \(2+5i\), it is in standard form where 2 is the real component and 5 is the imaginary. Understanding this format is crucial for performing arithmetic with complex numbers, such as addition, subtraction, multiplication, and division. By knowing each part's meaning, you can determine operations like finding the conjugate or the modulus easily.Complex numbers provide a more comprehensive numerical system that incorporates solutions to more complex equations.