Solved on Sep 21, 2023

Choose all true statements about the imaginary unit ii: i2=1i^{2}=-1, i=1i=\sqrt{-1}, the solution of x2=1x^{2}=-1.

STEP 1

Assumptions1. The imaginary unit is denoted by ii. . By definition, ii is the square root of -1, i.e., i=1i=\sqrt{-1}.
3. The properties of ii are determined by its definition.

STEP 2

The first statement is i2=1i^{2}=-1. We can verify this statement by squaring the definition of ii.
i2=(1)2i^{2} = (\sqrt{-1})^{2}

STEP 3

Calculate the square of 1\sqrt{-1}.
i2=(1)2=1i^{2} = (\sqrt{-1})^{2} = -1So, the first statement i2=1i^{2}=-1 is true.

STEP 4

The second statement is i=1i=-1. This statement is not true by the definition of ii. The imaginary unit ii is not equal to -1.

STEP 5

The third statement is i=1i=\sqrt{-1}. This statement is true by the definition of ii.

STEP 6

The fourth statement is about the solution of the equation x2=1x^{2}=-1. We can solve this equation by taking the square root of both sides.
x2=1x=1x^{2}=-1 \Rightarrow x=\sqrt{-1}

STEP 7

Calculate the square root of -1.
x=1=ix=\sqrt{-1}=iSo, the solution of the equation x2=1x^{2}=-1 is x=ix=i and x=ix=-i. Therefore, the fourth statement is true.
In conclusion, the true statements about the imaginary unit are i2=1i^{2}=-1, i=1i=\sqrt{-1}, and the solution of x2=1x^{2}=-1.

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