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Chapter 2: Problem 3

Fill in the blanks. \(A\)_______ _______number has the form \(a+b i,\) where \(a=0, b \neq 0.\)

Short Answer

Expert verified
The completed sentence is: A pure imaginary number has the form \(a+bi,\) where \(a=0, b鈮�0\).

Step by step solution

01

Identify the type of number

It is known from theory, that a number of the form \(0 + bi\), where \(b 鈮� 0\) is called a pure imaginary number.
02

Fill in the blanks

Fill in the blanks with 'pure imaginary' to get 'A pure imaginary number has the form \(a+bi,\) where \(a=0, b鈮�0\).'

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Key Concepts

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

Complex Numbers
Complex numbers are an essential part of mathematics, enabling us to perform calculations that involve the square root of negative numbers. They are written in the form of \(a + bi\), where \(a\) and \(b\) are real numbers.
The component \(a\) is the real part, and \(bi\) is the imaginary part. Complex numbers allow for an extended understanding of numbers beyond the real number line.
  • Examples of complex numbers include \(3 + 4i\) and \(-1 + 6i\).
  • When the real part \(a = 0\), the complex number becomes a pure imaginary number.
  • Complex numbers can be plotted on a plane called the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.
Complex numbers are used widely in engineering, physics, and applied mathematics, making them a crucial concept to grasp.
Imaginary Unit
The imaginary unit is a mathematical concept representing the square root of \(-1\). It is denoted by the symbol \(i\).
In the context of complex numbers, \(i\) is used to create the imaginary part of a complex number, as in \(bi\).
  • By definition, \(i^2 = -1\).
  • An important property is that powers of \(i\) cycle every four terms: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then it repeats.
  • Because \(i\) helps handle operations involving the square roots of negatives, it extends the possible solutions for equations in algebra.
The imaginary unit is a fundamental concept that helps bridge the gap between real and complex number operations, expanding the realm of solvable math problems.
Mathematical Notation
Mathematics relies heavily on notation to represent numbers, operations, and relations in a concise manner. Mathematical notation allows for the easy communication of complex ideas.
For complex numbers and imaginary units, specific notation is used to illustrate these concepts clearly.
  • \(a + bi\) is the standard notation for complex numbers, showing the real and imaginary parts clearly.
  • \(i\) represents the imaginary unit, essential for denoting the square root of \(-1\).
  • Using notation, mathematical expressions become easier to manipulate and calculate, as in solving quadratic equations using complex numbers.
Understanding mathematical notation is key to navigating complex equations and concepts swiftly. It provides a universally understood language for mathematicians and students alike.

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Most popular questions from this chapter

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