Divide (1+i)/(1-i)

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

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

Multiply $i$ by $1$ .

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

Multiply $i$ by $1$ .

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

Multiply $i i$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Subtract $1$ from $1$ .

$\frac{0 + i + i}{(1 - i) (1 + i)}$

Add $0$ and $i$ .

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

Add $i$ and $i$ .

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

Simplify the denominator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $1$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Subtract $i$ from $1 i$ .

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

Add $1$ and $0$ .

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

Simplify each term.

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

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

Multiply $- 1$ by $- 1$ .

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

Add $1$ and $1$ .

$\frac{2 i}{2}$

Step 3

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 i}{2}$

Divide $i$ by $1$ .

$i$

Divide (1+i)/(1-i)

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