Convert to Trigonometric Form -1/(1+i)

$- \frac{1}{1 + i}$

Multiply the numerator and denominator of $\frac{1}{1 + 1 i}$ by the conjugate of $1 + 1 i$ to make the denominator real.

$- (\frac{1}{1 + 1 i} \cdot \frac{1 - i}{1 - i})$

Multiply.

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Combine.

$- \frac{1 (1 - i)}{(1 + 1 i) (1 - i)}$

Multiply $1 - i$ by $1$ .

$- \frac{1 - i}{(1 + 1 i) (1 - i)}$

Simplify the denominator.

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Expand $(1 + 1 i) (1 - i)$ using the FOIL Method.

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Apply the distributive property.

$- \frac{1 - i}{1 (1 - i) + 1 i (1 - i)}$

Apply the distributive property.

$- \frac{1 - i}{1 \cdot 1 + 1 (- i) + 1 i (1 - i)}$

Apply the distributive property.

$- \frac{1 - i}{1 \cdot 1 + 1 (- i) + 1 i \cdot 1 + 1 i (- i)}$

Simplify.

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Multiply $1$ by $1$ .

$- \frac{1 - i}{1 + 1 (- i) + 1 i \cdot 1 + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$- \frac{1 - i}{1 - 1 i + 1 i \cdot 1 + 1 i (- i)}$

Multiply $1$ by $1$ .

$- \frac{1 - i}{1 - 1 i + 1 i + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$- \frac{1 - i}{1 - 1 i + 1 i - i i}$

Raise $i$ to the power of $1$ .

$- \frac{1 - i}{1 - 1 i + 1 i - (i^{1} i)}$

Raise $i$ to the power of $1$ .

$- \frac{1 - i}{1 - 1 i + 1 i - (i^{1} i^{1})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{1 - i}{1 - 1 i + 1 i - i^{1 + 1}}$

Add $1$ and $1$ .

$- \frac{1 - i}{1 - 1 i + 1 i - i^{2}}$

Add $- 1 i$ and $1 i$ .

$- \frac{1 - i}{1 + 0 - i^{2}}$

Add $1$ and $0$ .

$- \frac{1 - i}{1 - i^{2}}$

Simplify each term.

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Rewrite $i^{2}$ as $- 1$ .

$- \frac{1 - i}{1 - - 1}$

Multiply $- 1$ by $- 1$ .

$- \frac{1 - i}{1 + 1}$

Add $1$ and $1$ .

$- \frac{1 - i}{2}$

Split the fraction $\frac{1 - i}{2}$ into two fractions.

$- (\frac{1}{2} + \frac{- i}{2})$

Move the negative in front of the fraction.

$- (\frac{1}{2} - \frac{i}{2})$

Apply the distributive property.

$- \frac{1}{2} - - \frac{i}{2}$

Multiply $- - \frac{i}{2}$ .

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Multiply $- 1$ by $- 1$ .

$- \frac{1}{2} + 1 \frac{i}{2}$

Multiply $\frac{i}{2}$ by $1$ .

$- \frac{1}{2} + \frac{i}{2}$

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Substitute the actual values of $a = - \frac{1}{2}$ and $b = \frac{1}{2}$ .

$| z | = \sqrt{{(\frac{1}{2})}^{2} + {(- \frac{1}{2})}^{2}}$

Find $| z |$ .

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Apply the product rule to $\frac{1}{2}$ .

$| z | = \sqrt{\frac{1^{2}}{2^{2}} + {(- \frac{1}{2})}^{2}}$

One to any power is one.

$| z | = \sqrt{\frac{1}{2^{2}} + {(- \frac{1}{2})}^{2}}$

Raise $2$ to the power of $2$ .

$| z | = \sqrt{\frac{1}{4} + {(- \frac{1}{2})}^{2}}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Apply the product rule to $- \frac{1}{2}$ .

$| z | = \sqrt{\frac{1}{4} + {(- 1)}^{2} {(\frac{1}{2})}^{2}}$

Apply the product rule to $\frac{1}{2}$ .

$| z | = \sqrt{\frac{1}{4} + {(- 1)}^{2} (\frac{1^{2}}{2^{2}})}$

Raise $- 1$ to the power of $2$ .

$| z | = \sqrt{\frac{1}{4} + 1 (\frac{1^{2}}{2^{2}})}$

Multiply $\frac{1^{2}}{2^{2}}$ by $1$ .

$| z | = \sqrt{\frac{1}{4} + \frac{1^{2}}{2^{2}}}$

One to any power is one.

$| z | = \sqrt{\frac{1}{4} + \frac{1}{2^{2}}}$

Raise $2$ to the power of $2$ .

$| z | = \sqrt{\frac{1}{4} + \frac{1}{4}}$

Combine the numerators over the common denominator.

$| z | = \sqrt{\frac{1 + 1}{4}}$

Add $1$ and $1$ .

$| z | = \sqrt{\frac{2}{4}}$

Cancel the common factor of $2$ and $4$ .

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Factor $2$ out of $2$ .

$| z | = \sqrt{\frac{2 (1)}{4}}$

Cancel the common factors.

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Factor $2$ out of $4$ .

$| z | = \sqrt{\frac{2 \cdot 1}{2 \cdot 2}}$

Cancel the common factor.

$| z | = \sqrt{\frac{2 \cdot 1}{2 \cdot 2}}$

Rewrite the expression.

$| z | = \sqrt{\frac{1}{2}}$

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$| z | = \frac{\sqrt{1}}{\sqrt{2}}$

Any root of $1$ is $1$ .

$| z | = \frac{1}{\sqrt{2}}$

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$| z | = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

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Multiply $\frac{1}{\sqrt{2}}$ and $\frac{\sqrt{2}}{\sqrt{2}}$ .

$| z | = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$| z | = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$| z | = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$| z | = \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$| z | = \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$| z | = \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$| z | = \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$| z | = \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$| z | = \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Rewrite the expression.

$| z | = \frac{\sqrt{2}}{2}$

Evaluate the exponent.

$| z | = \frac{\sqrt{2}}{2}$

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \arctan (\frac{\frac{1}{2}}{- \frac{1}{2}})$

Since inverse tangent of $\frac{\frac{1}{2}}{- \frac{1}{2}}$ produces an angle in the second quadrant, the value of the angle is $\frac{3 π}{4}$ .

$θ = \frac{3 π}{4}$

Substitute the values of $θ = \frac{3 π}{4}$ and $| z | = \frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2 (\cos (\frac{3 π}{4}) + i \sin (\frac{3 π}{4}))}$

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Name

Name	one billion seven hundred eighty-three million eight hundred thirteen thousand two hundred sixty-three

Interesting facts

1783813263 has 8 divisors, whose sum is 2378688000
The reverse of 1783813263 is 3623183871
Previous prime number is 10399

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 42
Digital Root 6

Name

Name	one billion sixty-two million eight hundred sixty-seven thousand one hundred twenty-eight

Interesting facts

1062867128 has 32 divisors, whose sum is 3588178824
The reverse of 1062867128 is 8217682601
Previous prime number is 4013

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 41
Digital Root 5

Name

Name	one billion thirty-two million four hundred nine thousand two hundred twenty-seven

Interesting facts

1032409227 has 4 divisors, whose sum is 1376545640
The reverse of 1032409227 is 7229042301
Previous prime number is 3

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 30
Digital Root 3

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