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Math  /  Algebra

Questioni+i)2=−0.5\frac{i}{+i)^{2}}=-0.5

Studdy Solution

STEP 1

1. i i is the imaginary unit, defined as i=−1 i = \sqrt{-1} .
2. The expression involves complex number operations.
3. We need to simplify the expression and verify the equality.

STEP 2

1. Simplify the denominator (+i)2(+i)^2.
2. Simplify the fraction i(+i)2\frac{i}{(+i)^2}.
3. Verify if the simplified expression equals −0.5-0.5.

STEP 3

Simplify the denominator (+i)2(+i)^2:
(+i)2=i2(+i)^2 = i^2
Since i=−1 i = \sqrt{-1} , we know:
i2=−1i^2 = -1
Thus, the denominator simplifies to:
(+i)2=−1(+i)^2 = -1

STEP 4

Simplify the fraction i(+i)2\frac{i}{(+i)^2}:
Given the denominator from Step 1, we have:
i(+i)2=i−1\frac{i}{(+i)^2} = \frac{i}{-1}
This simplifies to:
i−1=−i\frac{i}{-1} = -i

STEP 5

Verify if the simplified expression equals −0.5-0.5:
We have simplified the expression to −i-i. To compare it with −0.5-0.5, consider the real and imaginary parts:
- The expression −i-i has a real part of 0 and an imaginary part of −1-1. - The number −0.5-0.5 is purely real with no imaginary part.
Since −i-i does not equal −0.5-0.5, the original equation i(+i)2=−0.5\frac{i}{(+i)^2} = -0.5 is false.
The equation i(+i)2=−0.5\frac{i}{(+i)^2} = -0.5 is not correct; the left side simplifies to −i-i, not −0.5-0.5.

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