Question

\((-i)^{-1}\) id purely real

Short answer

\((-i)^{-1} = i\) is not purely real, it is purely imaginary.

Step-by-step solution

Understanding the Problem

We need to determine if the expression \((-i)^{-1}\) is purely real. An expression is purely real if it has no imaginary part.

Recall the Definition of Negative Exponents

The expression \((-i)^{-1}\) refers to the reciprocal of \(-i\). So, our task is to find the reciprocal of \(-i\).

Find the Reciprocal of \(-i\)

To find the reciprocal of a complex number like \(-i\), we multiply by its conjugate divided by its modulus squared. The conjugate of \(-i\) is \(i\), and its modulus squared is \((-i)(i) = 1\). So, the reciprocal of \(-i\) is \(\frac{i}{1} = i\).

Evaluate if the Result is Purely Real

The resulting expression for \((-i)^{-1}\) is \(i\). This is not purely real, because it has an imaginary part of 1.

Key concepts

Reciprocal of Complex Number

The reciprocal of a complex number is essentially its inverse. A complex number can be written in the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit, defined as \(i^2 = -1\). Finding the reciprocal involves a few steps to ensure that both the real and imaginary parts switch places with a proper balance.
To find the reciprocal of a complex number like \(-i\), one needs to multiply the numerator and the denominator by the complex conjugate of the number. The complex conjugate of a number is obtained by changing the sign of the imaginary part. For \(-i\), the conjugate is \(i\).

• First, calculate the modulus of \(-i\), which is \((-i)(i) = 1\).
• This means the reciprocal of \(-i\) is obtained by multiplying with its conjugate and dividing by the modulus squared: \(\frac{i}{1} = i\).

Thus, we find that the reciprocal of \(-i\) is \(i\). This solution shows the replacement of position between real and imaginary components if considered in reciprocal form.

Purely Real Numbers

A purely real number is one without any imaginary component. A complex number is purely real if it is of the form \(a + 0i\), meaning the imaginary part \(b\) equals zero. When dealing with complex numbers, determining whether an expression is purely real requires analyzing if the imaginary portion is completely eliminated.
In the exercise solution, after finding the reciprocal \(i\) of \(-i\), we check its form. Since \(i\) can be written as \(0 + 1i\), where the real part is \(0\) and the imaginary part is \(1\), it is clearly not purely real.

• Real numbers can be directly plotted on the numeric line where only non-complex numbers exist.
• The absence of an \(i\) term (or having it equal to zero) signifies the number being purely real.

It's important to note that even though a complex number might simplify through operations, recognizing whether it retains an imaginary component is key to understanding its nature when determining purity in terms of real numbers.

Complex Conjugate

The complex conjugate of a complex number is a clever tool that changes the sign of the imaginary part while keeping the real part unchanged. For a number \(a + bi\), the conjugate is \(a - bi\), and when both multiply, the result is always a non-negative real number.
Using the conjugate simplifies calculations, especially when forming reciprocals or simplifying division among complex numbers. In finding the reciprocal of \(-i\), the use of the complex conjugate \(i\) was crucial.

• The multiplication of a complex number by its conjugate results directly in its modulus squared:
• For \(-i\), multiplying by \(i\): \((-i)(i) = 1\).
• This results in an opportunity to simplify otherwise complex algebraic fractions into real-valued functions.

Understanding how to leverage the complex conjugate offers considerable advantages in reducing complexity and enhancing clarity in mathematical operations involving complex numbers.