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Chapter 3: Problem 2

For the complex number \(5+2 i\), the imaginary part is and the real part is

Short Answer

Expert verified
The real part of the complex number \(5 + 2i\) is 5, and the imaginary part is 2.

Step by step solution

01

Identify the Complex Number

The complex number given is \(5 + 2i\).
02

Determine the Real Part

The real part of a complex number is the number without the 'i'. Here, the number before the plus sign, which is 5, is the real part.
03

Determine the Imaginary Part

The imaginary part of a complex number is the number with the 'i'. Here, the number after the plus sign, which is 2, is the imaginary part.

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Key Concepts

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

Understanding the Real Part of a Complex Number
The real part of a complex number is the component that does not involve the imaginary unit, denoted by 'i'.
In any complex number of the form \(a + bi\), where \(a\) and \(b\) are real numbers, \(a\) represents the real part.
For example, in the complex number \(5 + 2i\), the real part is 5.
  • The real part is straightforward and easy to identify.
  • It's just the coefficient without the imaginary unit 'i'.
  • Real parts can be positive, negative, or zero.
Real parts help in comparing complex numbers and are crucial for further operations.
They hold the position along the horizontal axis (real axis) in a complex plane.
Recognizing the Imaginary Part of a Complex Number
The imaginary part of a complex number involves the imaginary unit 'i', which is defined by the property \(i^2 = -1\).
In complex numbers of the form \(a + bi\), \(b\) is the imaginary part.In our example, the complex number \(5 + 2i\) has 2 as its imaginary part because it is the coefficient of 'i'.
The 'i' does not count as part of the numerical value; it indicates the presence of the imaginary unit.
  • This imaginary component is crucial for calculations within complex number arithmetic.
  • Even though it's called "imaginary," it plays an essential role in mathematics, physics, and engineering.
  • The imaginary part is plotted on the vertical axis (imaginary axis) in a complex plane.
Understanding the imaginary part lets you visualize and work with complex numbers more effectively.
Applying Algebra to Complex Numbers
Algebra with complex numbers involves combining and manipulating their real and imaginary parts.
Unlike real numbers, complex numbers are added, subtracted, multiplied, and often even divided using both their parts. For instance:

  • Addition: \((a + bi) + (c + di) = (a + c) + (b + d)i\)
  • Subtraction: \((a + bi) - (c + di) = (a - c) + (b - d)i\)
  • Multiplication: \((a + bi)(c + di) = (ac - bd) + (ad + bc)i\)
When multiplying complex numbers, remember the special property \(i^2 = -1\).
This transforms a seeming imaginary product into a real one in calculations.
Furthermore, algebraic operations on complex numbers usually result in another complex number.
This solution set is rich and diverse, offering insights into many fields beyond pure maths.

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Most popular questions from this chapter

Determine whether the given value of \(x\) is a solution to the equation. \(x^3-6=2 x^2+9-3 x ; x=-5\)

Solve the equation. Check your solution(s). \(x^2-4=-11\)

Solve the equation. Check your solution(s). \(x^2+9=0\)

Write the quadratic function in vertex form. Then identify the vertex. \(f(x)=x^2-3 x+4\)

MAKING AN ARGUMENT You claim the system of inequalities below, where \(a\) and \(b\) are real numbers, has no solution. Your friend claims the system will always have at least one solution. Who is correct? Explain. $$ \begin{aligned} &y<(x+a)^2 \\ &y<(x+b)^2 \end{aligned} $$
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