Multiply the numerator and denominator of <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mo>+</mo><mn>1</mn><mi>i</mi></mrow></mfrac></mstyle></math> by the conjugate of <math><mstyle displaystyle="true"><mn>1</mn><mo>+</mo><mn>1</mn><mi>i</mi></mstyle></math> to make the denominator real.

Combine.

Multiply <math><mstyle displaystyle="true"><mn>1</mn><mo>-</mo><mi>i</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Simplify the denominator.

Expand <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mn>1</mn><mi>i</mi><mo>)</mo></mrow><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>i</mi><mo>)</mo></mrow></mstyle></math> using the FOIL Method.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Simplify.

Multiply <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mi>i</mi></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mi>i</mi></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Use the power rule <math><mstyle displaystyle="true"><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></mstyle></math> to combine exponents.

Add <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn><mi>i</mi></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn><mi>i</mi></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Simplify each term.

Rewrite <math><mstyle displaystyle="true"><msup><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Split the fraction <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn><mo>-</mo><mi>i</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> into two fractions.

Move the negative in front of the fraction.

Apply the distributive property.

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

This is the trigonometric form of a complex number where <math><mstyle displaystyle="true"><mrow><mo>|</mo><mi>z</mi><mo>|</mo></mrow></mstyle></math> is the modulus and <math><mstyle displaystyle="true"><mi>θ</mi></mstyle></math> is the angle created on the complex plane.

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

Substitute the actual values of <math><mstyle displaystyle="true"><mi>a</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mi>b</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Apply the product rule to <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

One to any power is one.

Raise <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Use the power rule <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><mi>a</mi><mi>b</mi><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msup><msup><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msup></mstyle></math> to distribute the exponent.

Apply the product rule to <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Apply the product rule to <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><msup><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

One to any power is one.

Raise <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Combine the numerators over the common denominator.

Add <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factors.

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Rewrite <math><mstyle displaystyle="true"><msqrt><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></msqrt></mstyle></math> as <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>1</mn></msqrt></mrow><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle></math> .

Any root of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle></math> by <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle></math> .

Combine and simplify the denominator.

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle></math> .

Raise <math><mstyle displaystyle="true"><msqrt><mn>2</mn></msqrt></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><msqrt><mn>2</mn></msqrt></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Use the power rule <math><mstyle displaystyle="true"><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></mstyle></math> to combine exponents.

Rewrite <math><mstyle displaystyle="true"><msup><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Use <math><mstyle displaystyle="true"><mroot><mrow><msup><mrow><mi>a</mi></mrow><mrow><mi>x</mi></mrow></msup></mrow><mrow><mi>n</mi></mrow></mroot><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mfrac><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></msup></mstyle></math> to rewrite <math><mstyle displaystyle="true"><msqrt><mn>2</mn></msqrt></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>2</mn></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mstyle></math> .

Apply the power rule and multiply exponents, <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msup></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Evaluate the exponent.

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

Since inverse tangent of <math><mstyle displaystyle="true"><mfrac><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow><mrow><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfrac></mstyle></math> produces an angle in the second quadrant, the value of the angle is <math><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> .

Substitute the values of <math><mstyle displaystyle="true"><mi>θ</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mrow><mo>|</mo><mi>z</mi><mo>|</mo></mrow><mo>=</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Do you know how to Convert to Trigonometric Form -1/(1+i)? If not, you can write to our math experts in our application. The best solution for your task you can find above on this page.

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

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

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

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

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

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

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

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

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