Solved on Sep 21, 2023

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

STEP 1

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

STEP 2

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

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

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

STEP 1

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

STEP 2

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

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

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

STEP 1

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

STEP 2

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

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

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

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

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

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

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

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