Problem 3 - Elementary and 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

2 is the real part and 5 is the imaginary part.

Step 1: Identify Real Part

A complex number is written in the form \(a + bi\) where \(a\) is the real part and \(b\) is the imaginary part. For the complex number \(2+5i\), identify the real part as 2.

Step 2: Identify Imaginary Part

In the same expression \(a + bi\), the imaginary part is the coefficient of \(i\), which is \(b\). For \(2+5i\), the imaginary part is 5.

Key Concepts

Real Part

In mathematics, when dealing with complex numbers, it's essential to understand the concept of the "Real Part". A complex number is usually expressed in the form \(a + bi\). Here, the real part is denoted by \(a\), which is simply a real number without any imaginary component.

• The real part represents the horizontal position on the complex plane.
• It is the component that does not involve \(i\), the imaginary unit.
• The real part can be any real number, positive, negative, or zero.

In our example, the complex number is \(2 + 5i\), where 2 is the real part. This is the portion of the complex number that corresponds directly to a number on the real number line. Thus, there are no imaginary components to worry about in this part, and calculations involving only the real part follow the standard rules of arithmetic.

Imaginary Part

The imaginary part of a complex number pairs with the real part to form a complete complex number, taking the form \(a + bi\). It is represented as \(bi\), where \(i\) is the imaginary unit.

• \(i\), the imaginary unit, is defined by the property \(i^2 = -1\).
• The imaginary part is the coefficient \(b\) in the term \(bi\).
• This part of the complex number describes its vertical position on the complex plane.

In the complex number \(2 + 5i\), the imaginary part is 5. This means its influence is in the multiplication with \(i\), contributing to the vertical displacement on the complex plane. Although called imaginary, these parts are just as essential in mathematical operations and can lead to tangible results when combined with their real counterparts.

Coefficient of i

Understanding the "Coefficient of i" is key when navigating complex numbers. In the format \(a + bi\), \(b\) takes on the role of the coefficient of \(i\). This is important because:

• The coefficient signifies how many units of \(i\) are present in the complex number.
• It directly indicates the magnitude of the imaginary component before \(i\) transforms it into something orthogonal to the real component.
• The coefficient can also be a positive or negative real number, representing the direction of the imaginary part on the complex plane.

For \(2 + 5i\), the coefficient of \(i\) is 5, meaning the complex number has an imaginary component scaled by 5 units in the direction determined by \(i\). This component is what gives complexity its richness, allowing for representations and calculations in two dimensions, which is unlike standard real numbers typically confined to a linear path.