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Chapter 2: Problem 22

Find the real and imaginary parts of the complex number. $$-3$$

Short Answer

Expert verified
The real part of the complex number is -3 and the imaginary part is 0.

Step by step solution

01

Identify the real part

The real part of a complex number is the number without the 'i'. Therefore, for the complex number -3, the real part is -3.
02

Identify the imaginary part

The imaginary part of a complex number is the number with the 'i'. However, there is no 'i' in the given complex number -3. Hence, the imaginary part is 0.

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Key Concepts

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

Real Part of a Complex Number
The real part of a complex number is an important component. Complex numbers are often expressed in the form of \(a + bi\). Here, \(a\) represents the real part. This is simply the portion of the complex number that does not involve 'i', the imaginary unit.
In real life and many mathematical problems, identifying the real part helps in understanding the essential value of the number without any imaginary factors.
  • Example: For the complex number \(-3 + 4i\), the real part is \(-3\).
  • In the case of \(-3\) (without 'i'), the entire number is its real part, which is \(-3\).
Remember, the real part maintains its value irrespective of the imaginary components.
Imaginary Part of a Complex Number
The imaginary part of a complex number gives the number its unique form. When you see an 'i' in a complex number, you're looking at the imaginary part. In the expression \(a + bi\), \(b\) is the imaginary coefficient, with 'i' representing the square root of \(-1\).
Identifying this part helps in complex operations like addition or multiplication involving complex numbers.
  • Example: In the complex number \(2 + 3i\), the imaginary part is \(3i\).
  • For a pure real number like \(-3\), there isn鈥檛 an 'i' component, thus the imaginary part is \(0\).
Recognizing when the imaginary part is zero is key in distinguishing between a real number and a complex number.
Complex Number Representation
Understanding complex number representation is fundamental in dealing with complex numbers. The typical form of a complex number \(a + bi\) elegantly shows both its real and imaginary parts.
The representation tells you all you need to know about a complex number at a glance.
  • In \(a + bi\), \(a\) is the real component, and \(b\) is the coefficient of 'i', the imaginary unit.
  • This representation is crucial for performing operations such as addition, subtraction, and even multiplication of complex numbers.
For example, if a complex number is denoted as \(5 - 7i\), it means the real part is \(5\) and the imaginary part is \(-7i\). Understanding this makes computations involving complex numbers much smoother and straightforward.

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Most popular questions from this chapter

Graph each quadratic function by finding a suitable viewing window with the help of the TABLE feature of a graphing utility. Also find the vertex of the associated parabola using the graphing utility. $$h(x)=x^{2}+5 x-20$$

Use the intersect feature of your graphing calculator to explore the real solution(s), if any, of \(x^{2}=x+k\) for \(k=0, k=-\frac{1}{4},\) and \(k=-3 .\) Also use the zero feature to explore the solution(s). Relate your observations to the quadratic formula.

In Exercises \(97-100,\) let \(f(t)=-t^{2}\) and \(g(x)=x^{2}-1\). Evaluate \((g \circ g)\left(\frac{2}{3}\right)\)

The production cost, in dollars, for \(x\) color brochures is \(C(x)=500+3 x .\) The fixed cost is \(\$ 500\) since that is the amount of money needed to start production even if no brochures are printed. (a) If the fixed cost is decreased by \(\$ 50,\) find the new cost function. (b) Graph both cost functions and interpret the effect of the decreased fixed cost.

A chartered bus company has the following price structure. A single bus ticket costs \(\$ 30 .\) For each additional ticket sold to a group of travelers, the price per ticket is reduced by \(\$ 0.50 .\) The reduced price applies to all the tickets sold to the group. (a) Calculate the total cost for one, two, and five tickets. (b) Using your calculations in part (a) as a guide, find a quadratic function that gives the total cost of the tickets. (c) How many tickets must be sold to maximize the revenue for the bus company?
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