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Chapter 7: Problem 8

Decide whether each expression is equal to \(1,-1, i,\) or \(-i .\) $$-\sqrt{-1}$$

Short Answer

Expert verified
-i

Step by step solution

01

Recognize the expression

The given expression is \(-\sqrt{-1}\).
02

Identify \(\sqrt{-1}\)

Recall that \(\sqrt{-1} = i\).
03

Apply the sign

Given that \(\sqrt{-1} = i\), the expression can be rewritten as \(-i\).
04

Conclusion

The expression \(-\sqrt{-1}\) is equal to \(-i\).

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Key Concepts

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

imaginary unit
In mathematics, when we encounter the imaginary unit, we are dealing with a concept fundamental to complex numbers. The imaginary unit is represented by the symbol \( i \), and it holds the unique property of being the square root of negative one. This can be expressed as \(i = \sqrt{-1}\).
  • The imaginary unit helps us extend the concept of numbers beyond the real number line.
  • It is crucial in fields like engineering, physics, and signal processing.
  • Imaginary numbers, paired with real numbers, form complex numbers.
Together, these numbers allow for the solution of equations that have no real solutions, like \ x^2 + 1 = 0 \.
square root of negative one
The square root of negative one is a value that we cannot find on the real number line. This is because the square of any real number is always non-negative.
However, by introducing the imaginary unit \( i \), we can define the square root of negative one as \( i\ =\ \sqrt{-1} \).
  • This definition extends our number system to include complex numbers.
  • Complex numbers are formed as \( a + bi \, \) where \( a \) and \( b \) are real numbers.
  • It brings new solutions and understanding in mathematics and applied sciences.
Complex numbers open up a new dimension to problems that are otherwise unsolvable with real numbers alone.
negative imaginary number
A negative imaginary number is essentially the product of the imaginary unit \( i \) and a negative real number. Simply put, it is the negative sign applied to an imaginary number.
For example, in the exercise, we encounter the expression \(-\sqrt{-1} \, \) which simplifies to \( -i \).
  • Here, we recognized that the square root of negative one is \( i \).
  • Applying the negative sign, we get \( -i \), a negative imaginary number.
  • Negative imaginary numbers also play an important role in complex number arithmetic and have applications in various scientific disciplines.
Understanding negative imaginary numbers helps in fully grasping the broader concept of complex numbers, which model many real-world phenomena.

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Most popular questions from this chapter

Work each problem. Suppose someone claims that \(\sqrt[n]{a^{n}+b^{n}}\) must equal \(a+b,\) because when \(a=1\) and \(b=0,\) a true statement results: $$\sqrt[n]{a^{n}+b^{n}}=\sqrt[n]{1^{n}+0^{n}}=\sqrt[n]{1^{n}}=1=1+0=a+b$$ Explain why this is faulty reasoning.

Find the distance between each pair of points. \((x+y, y)\) and \((x-y, x)\)

Simplify each expression. Assume that all variables represent positive real numbers. $$ 4 m^{5 / 3}\left(m^{-2 / 3}-4 m^{-5 / 3}\right) $$

Rationalize each denominator. Assume that all variables represent positive real numbers. $$ -\sqrt{\frac{150 m^{5}}{n^{3}}} $$

Graph each circle. Identify the center and the radius. \((x-1)^{2}+(y+3)^{2}=16\)
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