Find each quotient. Write answers in standard form. -5 / i

검증된 단계별 안내

Recall that the problem is to find the quotient of \(\frac{-5}{i}\) and write the answer in standard form, which means expressing the result as a complex number in the form \(a + bi\), where \(a\) and \(b\) are real numbers.

To simplify the expression \(\frac{-5}{i}\), multiply both the numerator and the denominator by the complex conjugate of the denominator. Since the denominator is \(i\), its conjugate is \(-i\). So multiply numerator and denominator by \(-i\):

\[\frac{-5}{i} \times \frac{-i}{-i} = \frac{-5 \times (-i)}{i \times (-i)}\]

Simplify the numerator and denominator separately. The numerator becomes \(-5 \times (-i) = 5i\). The denominator is \(i \times (-i) = -i^2\). Recall that \(i^2 = -1\), so the denominator simplifies to \(-(-1) = 1\).

Now, write the simplified expression as \(\frac{5i}{1} = 5i\). In standard form, this is \(0 + 5i\), where the real part \(a = 0\) and the imaginary part \(b = 5\).

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주요 개념

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Complex Numbers and the Imaginary Unit

Complex numbers consist of a real part and an imaginary part, expressed as a + bi, where i is the imaginary unit with the property i² = -1. Understanding i is essential for manipulating expressions involving imaginary numbers.

Division of Complex Numbers

Dividing complex numbers often requires multiplying numerator and denominator by the conjugate of the denominator to eliminate imaginary terms from the denominator, resulting in a complex number in standard form a + bi.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers. Writing answers in this form means separating the real and imaginary parts clearly, which is the goal when simplifying quotients involving i.