the polar form of 3+4i

Let's write

3 + 4 i

in polar form:

r \exp (i θ)

, where r is the modulus and θ is an argument

The complex number

a + i b

has modulus

\sqrt{a^{2} + b^{2}}

, so:

ⅰ

\mod (3 + 4 i)

\sqrt{3^{2} + 4^{2}}

\sqrt{3^{2} + 4^{2}}

ⓘ Calculate powers before multiplications, additions and subtractions. With a positive integer exponent, a power corresponds to a repeated product:

3^{2}

3 \times 3

9

4^{2}

4 \times 4

16

\sqrt{9 + 16}

ⓘ Parentheses are computed first:

9 + 16

25

\sqrt{25}

ⓘ The square root of a nonnegative real is defined as the nonnegative number whose square gives that real number:

\sqrt{25}

5

because

5^{2}

25

5

For nonzero

a + i b

, the argument is the directed angle from the positive real axis to the vector

(a, b)

; depending on the signs of a and b, it equals

0

π

\frac{π}{2}

- \frac{π}{2}

atan (\frac{b}{a})

atan (\frac{b}{a}) + π

atan (\frac{b}{a}) - π

ⅰ

\arg (3 + 4 i)

\arctan (\frac{4}{3})

\arctan (\frac{4}{3})

So the polar form is:

ⅰ

polar (3 + 4 i)

5 \exp (i \times \arctan (\frac{4}{3}))

▶Result:

5 \exp (i \times \arctan (\frac{4}{3}))

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