Powers of the Imaginary Unit 2

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Year 12

Complex Numbers

Powers of the Imaginary Unit

Powers of the Imaginary Unit 2

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Question 1 of 4

Simplify

$i^{2006}$ $i^(2006)$

1. $- 1$ $-1$
2. $- i$ $-i$
3. $i$ $i$
4. $1$ $1$

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Imaginary numbers have the properties

i^{0} = 1

$i^0=1$ ,

i^{1} = i

$i^1=i$ ,

i^{2} = - 1

$i^2=-1$ and

i^{3} = - i

$i^3=-i$ .

Use the formula

i^{4 n + r}

$i^(4n+r)$ to simplify the powers of the imaginary unit.

	$i^{2006}$ $i^(2006)$	Divide the power $2006$ $2006$ by $4$ $4$ . This gives $501$ $501$ with a remainder of $2$ $2$ .
$=$ $=$	$i^{4 \times 501 + 2}$ $i^(4times501+2)$	Rewrite using the formula $i^{4 n + r}$ $i^(4n+r)$ where the $n = 501$ $n=501$ and the $r = 2$ $r=2$ .
$=$ $=$	$i^{501 + 2}$ $i^(501+\color(red)(2))$	Simplify

		$i^{501 + 2}$ $i^(501+\color(red)(2))$	The $2$ $2$ means that this number is the same as $i^{2} = - 1$ $i^2=-1$ .
	$=$ $=$	$- 1$ $-1$

- 1

$-1$

Question 2 of 4

Simplify

$i^{995}$ $i^(995)$

1. $0$ $0$
2. $- i$ $-i$
3. $i$ $i$
4. $1$ $1$

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Imaginary numbers have the properties

i^{0} = 1

$i^0=1$ ,

i^{1} = i

$i^1=i$ ,

i^{2} = - 1

$i^2=-1$ and

i^{3} = - i

$i^3=-i$ .

Use the formula

i^{4 n + r}

$i^(4n+r)$ to simplify the powers of the imaginary unit.

	$i^{995}$ $i^(995)$	Divide the power $995$ $995$ by $4$ $4$ . This gives $248$ $248$ with a remainder of $3$ $3$ .
$=$ $=$	$i^{4 \times 248 + 3}$ $i^(4times248+3)$	Rewrite using the formula $i^{4 n + r}$ $i^(4n+r)$ where the $n = 248$ $n=248$ and the $r = 3$ $r=3$ .
$=$ $=$	$i^{992 + 3}$ $i^(992+\color(red)(3))$	Simplify

		$i^{992 + 3}$ $i^(992+\color(red)(3))$	The $3$ $3$ means that this number is the same as $i^{3} = - i$ $i^3=-i$ .
	$=$ $=$	$- i$ $-i$

- i

$-i$

Question 3 of 4

Simplify

$i^{222, 002}$ $i^(222,002)$

1. $i$ $i$
2. $- 1$ $-1$
3. $1$ $1$
4. $- i$ $-i$

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Imaginary numbers have the properties

i^{0} = 1

$i^0=1$ ,

i^{1} = i

$i^1=i$ ,

i^{2} = - 1

$i^2=-1$ and

i^{3} = - i

$i^3=-i$ .

Use the formula

i^{4 n + r}

$i^(4n+r)$ to simplify the powers of the imaginary unit.

	$i^{222, 002}$ $i^(222,002)$	Divide the power $222, 002$ $222,002$ by $4$ $4$ . This gives $55, 500$ $55,500$ with a remainder of $2$ $2$ .
$=$ $=$	$i^{4 \times 55, 500 + 2}$ $i^(4times55,500+2)$	Rewrite using the formula $i^{4 n + r}$ $i^(4n+r)$ where the $n = 55, 500$ $n=55,500$ and the $r = 2$ $r=2$ .
$=$ $=$	$i^{222, 000 + 2}$ $i^(222,000+\color(red)(2))$	Simplify

		$i^{222, 000 + 2}$ $i^(222,000+\color(red)(2))$	The $2$ $2$ means that this number is the same as $i^{2} = - 1$ $i^2=-1$ .
	$=$ $=$	$- 1$ $-1$

- 1

$-1$

Question 4 of 4

Simplify

$i^{780}$ $i^(780)$

1. $- i$ $-i$
2. $i$ $i$
3. $0$ $0$
4. $1$ $1$

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Imaginary numbers have the properties

i^{0} = 1

$i^0=1$ ,

i^{1} = i

$i^1=i$ ,

i^{2} = - 1

$i^2=-1$ and

i^{3} = - i

$i^3=-i$ .

Use the formula

i^{4 n + r}

$i^(4n+r)$ to simplify the powers of the imaginary unit.

	$i^{780}$ $i^(780)$	Divide the power $780$ $780$ by $4$ $4$ . This gives $195$ $195$ with a remainder of $0$ $0$ .
$=$ $=$	$i^{4 \times 195 + 0}$ $i^(4times195+0)$	Rewrite using the formula $i^{4 n + r}$ $i^(4n+r)$ where the $n = 195$ $n=195$ and the $r = 0$ $r=0$ .
$=$ $=$	$i^{780 + 0}$ $i^(780+\color(red)(0))$	Simplify

		$i^{780 + 0}$ $i^(780+\color(red)(0))$	The $0$ $0$ means that this number is the same as $i^{0} = 1$ $i^0=1$ .
	$=$ $=$	$1$ $1$

1

$1$

Powers of the Imaginary Unit 2

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