B9 - Exponential form

Euler’s definition

How can an exponential represent $-1$? On the unit circle, the point at angle $\pi$ has coordinates $(-1,0)$, so its complex number is $\cos\pi+\mathrm{i}\sin\pi=-1$.

We give the unit-circle number at angle $\theta$ the compact name $\mathrm{e}^{\mathrm{i}\theta}$. Here $\theta$ is a real angle in radians. The imaginary exponent describes a rotation; it is not a positive real exponential value.

For example, $\mathrm{e}^{\mathrm{i}\pi/2}=\mathrm{i}$, $\mathrm{e}^{\mathrm{i}\pi}=-1$ and $\mathrm{e}^{2\pi\mathrm{i}}=1$. The last equality reflects a full turn, not a failure of exponent notation.

[DIAGRAM: asset_name: Exponentials around the unit circle; asset_slug: aqa7367-exponential-circle; caption: The number e to the i theta lies on the unit circle at angle theta; description: An equal-scale Argand diagram shows the unit circle, the points 1, i, minus 1 and minus i, and a blue radius to e to the i pi over 3 at coordinates one half and root three over two. A dashed horizontal and vertical projection shows cosine pi over three and sine pi over three. A right-hand panel gives Euler’s definition, modulus one, and the equality of angles differing by a full turn.]

The coordinates show two important facts:

Different angles can therefore name the same number. In particular, $\mathrm{e}^{\mathrm{i}\pi}=\mathrm{e}^{-\mathrm{i}\pi}$. If you use the principal interval $-\pi<\arg z\le\pi$, the principal argument of $-1$ is $\pi$.

Use the unit circle to read a new angle.

Converting forms and combining arguments

A positive factor in front of the exponential gives the distance from the origin. Thus $2\mathrm{e}^{\mathrm{i}\pi/6}$ and $2(\cos(\pi/6)+\mathrm{i}\sin(\pi/6))$ both mean $\sqrt3+\mathrm{i}$.

For $z=x+\mathrm{i}y\ne0$, find $r=\sqrt{x^2+y^2}>0$, then choose the correct quadrant for an argument $\theta$. The three forms are

The familiar polar product rule now looks like an index law: multiplying adds angles. Dividing must undo that addition, so it subtracts the denominator’s angle.

Why do the laws work? Expanding $(\cos\alpha+\mathrm{i}\sin\alpha)(\cos\beta+\mathrm{i}\sin\beta)$ gives the cosine and sine addition formulae. The exponential notation packages that existing result.

Keep both numbers fixed and change multiplication to division. Only the radius operation and angle operation change.

Reduce an angle by a multiple of $2\pi$ only when a particular argument interval is required. The number itself is already correct before that reduction.

Now change the numerator’s angle and complete the same conversion.

Radius, arguments and zero

The radius in the standard exponential form is the modulus, so it is positive for a nonzero number. A minus sign belongs in the direction rather than in the modulus.

For example, $-2\mathrm{e}^{\mathrm{i}\pi/6}$ is a valid expression, but its leading coefficient is not its modulus. Turn through another half-turn to write its standard form:

Both sides have modulus $2$. Their coordinates are $-\sqrt3-\mathrm{i}$, whose principal argument is $-5\pi/6$.

Zero is different. Since $|\mathrm{e}^{\mathrm{i}\theta}|=1$, no real $\theta$ makes $\mathrm{e}^{\mathrm{i}\theta}=0$. Multiplying by zero does give $0\mathrm{e}^{\mathrm{i}\theta}=0$, for every $\theta$, but those arbitrary angles are not arguments of zero. The zero vector has no direction.

These conditions also explain why you may divide by $r\mathrm{e}^{\mathrm{i}\theta}$ when $r>0$: neither factor is zero. For a zero numerator and a nonzero denominator, the quotient is simply zero; there is no numerator argument to subtract.

The opening puzzle now has an exact answer: $\mathrm{e}^{\mathrm{i}\pi}=-1$ because Euler’s definition gives $\cos\pi=-1$ and $\sin\pi=0$. A unit-circle exponential can be negative real while its modulus remains positive.

Distinguish a correct algebraic equality from an invalid claim about an argument.

Quick Recap

• $\mathrm{e}^{\mathrm{i}\theta}=\cos\theta+\mathrm{i}\sin\theta$, with $\theta$ in radians.
• For $z\ne0$, the standard form is $r\mathrm{e}^{\mathrm{i}\theta}$, where $r=|z|>0$.
• Products add arguments; quotients subtract them and require a nonzero denominator.
• Arguments differing by $2k\pi$ represent the same number. Zero has no argument.

An exponential represents $-1$ when its imaginary exponent describes a half-turn: $\mathrm{e}^{\mathrm{i}\pi}=-1$.