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Chapter 1: Problem 14

Find the real and imaginary parts of the complex number. $$i \sqrt{3}$$

Short Answer

Expert verified
Real part: 0, Imaginary part: \(\sqrt{3}\).

Step by step solution

01

Identify the Real Part

The real part of a complex number is the coefficient (or constant) without the imaginary unit \(i\). In the expression \(i \sqrt{3}\), there is no separate real number coefficient. Hence, the real part is 0.
02

Identify the Imaginary Part

The imaginary part of a complex number is the coefficient of the imaginary unit \(i\). In the given expression \(i \sqrt{3}\), the coefficient of \(i\) is \(\sqrt{3}\). Therefore, the imaginary part is \(\sqrt{3}\).

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Key Concepts

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

Real Part
In the realm of complex numbers, the real part refers to the component of the number that does not involve the imaginary unit, often denoted as "i". A complex number is generally expressed in the form \(a + bi\), where \(a\) is the real part. In the exercise you encountered, the complex number was \(i \sqrt{3}\). If we were to write it in the standard form \(a + bi\), it would appear as \(0 + i\sqrt{3}\). Here, you can see that there is no actual number accompanying the real part鈥攊t's essentially zero.
  • The real part is simply the number standing alone without "i".
  • If no such number exists, as in our case, the real part is zero.
  • Think of the real part as the "normal" number component in a complex expression.
This can help in visualizing where the real number sits alongside its imaginary counterpart, even when it's just not there, like zero in this example.
Imaginary Part
Understanding the imaginary part of a complex number could appear challenging at first. Yet, it's fairly straightforward. Complex numbers take the form \(a + bi\), with \(b\) as the imaginary part, which is directly tied to the imaginary unit "i". In other words, the imaginary part is the number in front of "i". In our example, \(i \sqrt{3}\), this imaginary part is represented by \(\sqrt{3}\).
  • The imaginary part multiplies with the imaginary unit "i" to complete its expression.
  • In our exercise, this was the \(\sqrt{3}\) accompanying the "i".
  • If "i" is missing, the imaginary part is essentially zero.
Grasping this concept will assist you in identifying and working with those parts of a complex number that contribute to its 'imaginary' nature.
Imaginary Unit
The imaginary unit, commonly represented as "i", is a foundational concept in complex numbers. It's noteworthy because it defies basic arithmetic rules. Essentially, "i" is defined as the square root of -1. This means that when you square "i", you get -1, or mathematically, \(i^2 = -1\).
  • It's a tool for extending numbers into the complex field, allowing for solutions that aren't possible in the real numbers alone.
  • In complex expressions, the imaginary unit "i" helps denote the presence of an imaginary part.
  • It's uniquely companioned with real numbers to form a full complex number, noted as \(a + bi\).
By learning about "i", you'll appreciate how it enables mathematicians and engineers to solve equations that go beyond real-only possibilities. Using "i" allows one to navigate both theoretical and practical mathematical contexts extensively.

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