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Chapter 1: Problem 35

Determine the real and imaginary parts of the complex number. $$3-7 i$$

Short Answer

Expert verified
Real part: 3, Imaginary part: -7

Step by step solution

01

Identify the complex number

The complex number provided is \(3 - 7i\). A complex number is generally written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.
02

Determine the real part

The real part of the complex number corresponds to the coefficient of the non-imaginary component. In this case, the real part is the number \(3\).
03

Determine the imaginary part

The imaginary part of the complex number corresponds to the coefficient of the imaginary component (including the sign). In this complex number, the imaginary part is \(-7\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real Part
In a complex number, such as the one provided in the exercise, the real part is represented by the variable typically denoted as 'a' in the general form of a complex number, which is given as \( a + bi \). It is the coefficient that stands alone without the imaginary unit 'i'.
For the complex number \(3 - 7i\), the real part is simply the constant term without any 'i'. Therefore, in this example, the real part is 3. Understanding the real part of a complex number is crucial as it helps in identifying the position of the number on the real axis when plotting in the complex plane.
Imaginary Part
The imaginary part of a complex number is the coefficient of the imaginary component. This coefficient is paired with the imaginary unit 'i'. In the general form \( a + bi \), 'b' denotes the imaginary part. It is important to note that the imaginary part includes its sign (positive or negative).
In the provided complex number \(3 - 7i\), the imaginary part is the number that multiplies the imaginary unit 'i'. Hence, the imaginary part is -7. This value indicates how far the number extends in the direction of the imaginary axis on the complex plane.
It鈥檚 essential to distinguish between the imaginary part and the imaginary unit itself, to avoid confusion in mathematical operations involving complex numbers.
Imaginary Unit
The imaginary unit is a fundamental concept in complex numbers. It is denoted by 'i' and is defined by the property \( i^2 = -1 \). This unique property allows for the extension of the real number system to include complex numbers.
The imaginary unit 'i' is what enables a complex number to have an imaginary part. For instance, in the complex number \(3 - 7i\), the '-7' is multiplied by 'i', making it the imaginary part.
Using 'i', we can represent numbers that involve the square roots of negative numbers, which are not possible in the set of real numbers alone. This broadens the scope of solutions for many mathematical equations.
In summary, the imaginary unit is pivotal in complex number theory, providing a new dimension and enriching the number system.

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