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Chapter 4: Problem 1

In Exercises 1-6, plot the complex number and find its absolute value. $$ -7 i $$

Short Answer

Expert verified
The complex number -7i is represented by the point (0,-7) in the complex plane and its absolute value is 7.

Step by step solution

01

Identify real and imaginary parts

On the complex plane, the x-axis is considered as the real axis and the y-axis is considered as the imaginary axis. The given complex number is \(-7i\), which means it has 0 as its real part (nothing before 'i', i.e., there is no 'x' value) and -7 as its imaginary part (i.e., the 'y' value).
02

Plot the complex number

Plot the point (0, -7) on the complex plane. Since the real part is 0, we start at the origin and since the imaginary part is -7, we move 7 units down along the imaginary axis. This gives us the point representing the complex number -7i.
03

Calculate the absolute value

The absolute value of a complex number \(a+bi\) is given by \(\sqrt{a^2 + b^2}\). Hence, the absolute value of \(-7i\) (which equals to \(0 - 7i\)) is \(\sqrt{0^2 + (-7)^2} = \sqrt{49} = 7.

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Key Concepts

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

Understanding Absolute Value of Complex Numbers
Complex numbers often involve taking the absolute value, which might initially seem perplexing. The absolute value of a complex number is its distance from the origin on the complex plane.
To calculate it, use the formula \( \sqrt{a^2 + b^2} \) where "a" is the real part and "bi" is the imaginary part. For example, in the calculation for the number \(-7i\), "a" is 0 and "bi" is -7. Thus,
  • The calculation becomes \( \sqrt{0^2 + (-7)^2} \)
  • This simplifies to \( \sqrt{49} \), which equals 7.
This means that even though \(-7i\) is purely imaginary with no real part, its absolute value is still a tangible distance, making these abstract numbers easier to grasp.
Identifying the Imaginary Part of Complex Numbers
The imaginary part of a complex number is the coefficient paired with the imaginary unit 'i'. In complex numbers, such as \(-7i\), this part drives its location on the vertical axis of a complex plane.
For the complex number \(-7i\):
  • The real part is 0 (since there is no additional number).
  • The imaginary part is -7, represented by -7i.
This imaginary part informs us how far and in which direction (up or down) from the origin the number is positioned on the y-axis of the complex plane. Identifying this part is critical to plotting and understanding the behavior of the complex number.
Navigating the Complex Plane
The complex plane, also known as the Argand plane, helps visualize complex numbers by representing them as points or vectors. This plane utilizes:
  • The horizontal axis as the real axis.
  • The vertical axis as the imaginary axis.
When plotting \(-7i\), the real part is 0, so the point starts at the origin and lies entirely along the imaginary axis through -7.
  • You move from the origin downward to the point (0, -7).
  • This visual representation aids in grasping how complex numbers behave in their own unique space.
Understanding this plane is fundamental to visualizing and interpreting complex numbers effectively, allowing one to go beyond mere numerical expressions to gain insights into their practical application on this two-dimensional plane.

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Most popular questions from this chapter

In Exercises 1-24, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$ (-1+i)^{10} $$

In Exercises 97-100, solve the equation and check your solution. $$ 4(5 x-6)-3(6 x+1)=0 $$

In Exercises 89-96, use the dot product to find the magnitude of \(u\). $$ \mathbf{u}=16 \mathbf{i}+4 \mathbf{j} $$

Prove that the complex conjugate of the sum of two complex numbers \(a_{1}+b_{1} i\) and \(a_{2}+b_{2} i\) is the sum of their complex conjugates.

In Exercises 75-82, simplify the complex number and write it in standard form. $$ \frac{1}{(2 i)^{3}} $$
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