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Chapter 3: Problem 76

Plot the complex number. $$-2 i$$

Short Answer

Expert verified
The complex number \(-2i\) is plotted on the complex plane at the point \(0, -2\), i.e., two units below the origin on the negative y-axis.

Step by step solution

01

Understand Complex Numbers

A complex number has a real part and an imaginary part. It can be written in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part, 'i' being the imaginary unit with the property \(i^2 = -1\). This complex number can be represented as a point on a plane, known as the complex plane.
02

Identify Real and Imaginary Part

In the given complex number \(-2i\), there is no real part, so that would be '0'. The imaginary part is \(-2\).
03

Plot the Complex Number

Now plotting our complex number on the complex plane, the x-axis represents the real numbers and the y-axis represents the imaginary numbers. Since our real part is '0', and the imaginary part is \('-2'\), our point would lie on the negative y-axis, 2 units below the origin.

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Key Concepts

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

Complex Plane
The complex plane is a special coordinate system where we represent complex numbers visually. Think of it as similar to the regular x-y coordinate plane, but for complex numbers. Here, the x-axis represents the real part of a complex number, and the y-axis stands for the imaginary part.

  • A real number lies on the x-axis, where the imaginary part is zero.
  • An imaginary number is located on the y-axis, where the real part is zero.
  • A general complex number can be anywhere on the plane, given by its real and imaginary components.
The complex plane helps us visualize operations on complex numbers, such as addition, subtraction, and finding magnitudes. It becomes a map for moving around the complex world, where distances and angles play crucial roles.
Imaginary Unit
The imaginary unit is a mathematical concept denoted by 'i'. It is defined with the property that its square is \(i^2 = -1\).

Understanding 'i' is crucial because it allows us to extend our number system beyond the real numbers.

  • Real numbers are insufficient for solving all polynomial equations, specifically those with negative under square roots.
  • The introduction of 'i' makes it possible to express solutions involving square roots of negative numbers.
  • Every imaginary number is a real number multiplied by 'i'. For example, in the number \(-2i\), -2 is the real coefficient of the imaginary part.
Imaginary numbers, in combination with real numbers, form complex numbers. This expansion leads to a more complete number system useful in various mathematical and engineering fields.
Plotting Complex Numbers
Plotting complex numbers on the complex plane is a straightforward task. Each complex number corresponds to a specific point on this plane.

Here's how you do it:
  • Identify the real and imaginary parts of the complex number. For \(-2i\), the real part is 0, and the imaginary part is -2.
  • Starting at the origin, move along the x-axis according to the real part. If it's zero, stay at the origin on the x-axis.
  • Then, move parallel to the y-axis according to the imaginary part. A negative value indicates movement downwards.
  • The point where you end up is the graphical representation of the complex number.
For \(-2i\), plot the number by remaining at the origin on the x-axis and moving 2 units downwards on the y-axis. This gives you the location directly on the negative y-axis, representing \(-2i\) visually as a point.

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Most popular questions from this chapter

Write the function in the form \(f(x)=(x-k) q(x)+r\) for the given value of \(k\), and demonstrate that \(f(k)=r\). $$f(x)=x^{3}+3 x^{2}-7 x-6, \quad k=-\sqrt{2}$$

Use synthetic division to divide. Divisor \(x+2\) Dividend $$4 x^{3}-9 x+8 x^{2}-18$$

Use long division to divide. Divisor \(x^{2}-4\) Dividend $$x^{4}+2 x^{3}-3 x^{2}-8 x-4$$

Use synthetic division to divide. Divisor \(x+2\) Dividend $$6 x^{4}-15 x^{3}-11 x$$

Use long division to divide. Divisor \(3 x^{2}-2\) Dividend $$3 x^{3}-12 x^{2}-2 x+8$$
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