Functions that commute

At the bottom of this post is a one-liner that I use in my classes the first time I present a theorem where two functions are permitted to commute. At many layers of the mathematics curriculum, students learn about that various functions can essentially commute with each other. In other words, the order in which the operations is performed doesn’t affect the final answer. Here’s a partial list off the top of my head:

  1. Arithmetic/Algebra: a \cdot (b + c) = a \cdot b + a \cdot c. This of course is commonly called the distributive property (and not the commutative property), but the essential idea is that the same answer is obtained whether the multiplications are performed first or if the addition is performed first.
  2. Algebra: If a,b > 0, then \sqrt{ab} = \sqrt{a} \sqrt{b}.
  3. Algebra: If a,b > 0 and x is any real number, then (ab)^x = a^x b^x.
  4. Precalculus: \displaystyle \sum_{i=1}^n (a_i+b_i) = \displaystyle \sum_{i=1}^n a_i + \sum_{i=1}^n b_i.
  5. Precalculus: \displaystyle \sum_{i=1}^n c a_i = c \displaystyle \sum_{i=1}^n a_i.
  6. Calculus: If f is continuous at an interior point c, then \displaystyle \lim_{x \to c} f(x) = f(c).
  7. Calculus: If f and g are differentiable, then (f+g)' = f' + g'.
  8. Calculus: If f is differentiable and c is a constant, then (cf)' = cf'.
  9. Calculus: If f and g are integrable, then \int (f+g) = \int f + \int g.
  10. Calculus: If f is integrable and c is a constant, then \int cf = c \int f.
  11. Calculus: If f: \mathbb{R}^2 \to \mathbb{R} is integrable, \iint f(x,y) dx dy = \iint f(x,y) dy dx.
  12. Calculus: For most differentiable function f: \mathbb{R}^2 \to \mathbb{R} that arise in practice, \displaystyle \frac{\partial^2 f}{\partial x \partial y} = \displaystyle \frac{\partial^2 f}{\partial y \partial x}.
  13. Probability: If X and Y are random variables, then E(X+Y) = E(X) + E(Y).
  14. Probability: If X is a random variable and c is a constant, then E(cX) = c E(X).
  15. Probability: If X and Y are independent random variables, then E(XY) = E(X) E(Y).
  16. Probability: If X and Y are independent random variables, then \hbox{Var}(X+Y) = \hbox{Var}(X) + \hbox{Var}(Y).
  17. Set theory: If A, B, and C are sets, then A \cup (B \cap C) = (A \cup B) \cap (A \cup C).
  18. Set theory: If A, B, and C are sets, then A \cap (B \cup C) = (A \cap B) \cup (A \cap C).

However, there are plenty of instances when two functions do not commute. Most of these, of course, are common mistakes that students make when they first encounter these concepts. Here’s a partial list off the top of my head. (For all of these, the inequality sign means that the two sides do not have to be equal… though there may be special cases when equality happens to happen.)

  1. Algebra: (a+b)^x \ne a^x + b^x if x \ne 1. Important special cases are x = 2, x = 1/2, and x = -1.
  2. Algebra/Precalculus: \log_b(x+y) = \log_b x + \log_b y. I call this the third classic blunder.
  3. Precalculus: (f \circ g)(x) \ne (g \circ f)(x).
  4. Precalculus: \sin(x+y) \ne \sin x + \sin y, \cos(x+y) \ne \cos x + \cos y, etc.
  5. Precalculus: \displaystyle \sum_{i=1}^n (a_i b_i) \ne \displaystyle \left(\sum_{i=1}^n a_i \right) \left( \sum_{i=1}^n b_i \right).
  6. Calculus: (fg)' \ne f' \cdot g'.
  7. Calculus \left( \displaystyle \frac{f}{g} \right)' \ne \displaystyle \frac{f'}{g'}
  8. Calculus: \int fg \ne \left( \int f \right) \left( \int g \right).
  9. Probability: If X and Y are dependent random variables, then E(XY) \ne E(X) E(Y).
  10. Probability: If X and Y are dependent random variables, then \hbox{Var}(X+Y) \ne \hbox{Var}(X) + \hbox{Var}(Y).

All this to say, it’s a big deal when two functions commute, because this doesn’t happen all the time.

green lineI wish I could remember the speaker’s name, but I heard the following one-liner at a state mathematics conference many years ago, and I’ve used it to great effect in my classes ever since. Whenever I present a property where two functions commute, I’ll say, “In other words, the order of operations does not matter. This is a big deal, because, in real life, the order of operations usually is important. For example, this morning, you probably got dressed and then went outside. The order was important.”

 

Calculators and complex numbers (Part 24)

In this series of posts, I explore properties of complex numbers that explain some surprising answers to exponential and logarithmic problems using a calculator (see video at the bottom of this post). These posts form the basis for a sequence of lectures given to my future secondary teachers.

To begin, we recall that the trigonometric form of a complex number z = a+bi is

z = r(\cos \theta + i \sin \theta) = r e^{i \theta}

where r = |z| = \sqrt{a^2 + b^2} and \tan \theta = b/a, with \theta in the appropriate quadrant. As noted before, this is analogous to converting from rectangular coordinates to polar coordinates.

Theorem. If z = x + i y, where x and y are real numbers, then

e^z = e^x (\cos y + i \sin y)

Definition. Let z = r e^{i \theta} be a complex number so that -\pi < \theta \le \theta. Then we define

\log z = \ln r + i \theta.

Definition. Let z and w be complex numbers so that z \ne 0. Then we define

z^w = e^{w \log z}

Technical point: for the latter two definitions, these are the principal values of the functions. In complex analysis, these are usually considered multiply-defined functions. But I’m not going to worry about this technicality here and will only consider the principal values.

This is the last post in this series, where I state some generalizations of the Laws of Exponents for complex numbers.

In yesterday’s post, we saw that z^{w_1} z^{w_2} = z^{w_1 + w_2} as long as z \ne 0. This prevents something like 0^4 \cdot 0^{-3} = 0^1, since 0^{-3} is undefined.

Theorem. Let z \in \mathbb{C} \setminus \{ 0 \}, w \in \mathbb{C}, and n \in \mathbb{Z}. Then (z^w)^n = z^{wn}.

As we saw in a previous post, the conclusion could be incorrect outside of the above hypothesis, as \displaystyle \left[ (-1)^3 \right]^{1/2} \ne (-1)^{3/2}.

Theorem. Let u \in \mathbb{R} and z \in \mathbb{C}. Then (e^u)^z = e^{uz}.

Theorem. Let x, y > 0 be real numbers and z \in \mathbb{C}. Then x^z y^z = (xy)^z.

Again, the conclusion of the above theorem could be incorrect outside of these hypothesis, as (-2)^{1/2} (-3)^{1/2} \ne \left[ (-2) \cdot (-3) \right]^{1/2}.

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For completeness, here’s the movie that I use to engage my students when I begin this sequence of lectures.

 

 

Calculators and complex numbers (Part 23)

In this series of posts, I explore properties of complex numbers that explain some surprising answers to exponential and logarithmic problems using a calculator (see video at the bottom of this post). These posts form the basis for a sequence of lectures given to my future secondary teachers.

To begin, we recall that the trigonometric form of a complex number z = a+bi is

z = r(\cos \theta + i \sin \theta) = r e^{i \theta}

where r = |z| = \sqrt{a^2 + b^2} and \tan \theta = b/a, with \theta in the appropriate quadrant. As noted before, this is analogous to converting from rectangular coordinates to polar coordinates.

Theorem. If z = x + i y, where x and y are real numbers, then

e^z = e^x (\cos y + i \sin y)

Definition. Let z = r e^{i \theta} be a complex number so that -\pi < \theta \le \theta. Then we define

\log z = \ln r + i \theta.

Definition. Let z and w be complex numbers so that z \ne 0. Then we define

z^w = e^{w \log z}

Technical point: for the latter two definitions, these are the principal values of the functions. In complex analysis, these are usually considered multiply-defined functions. But I’m not going to worry about this technicality here and will only consider the principal values.

In the remaining posts in this series, I want to explore which properties of exponential functions remain true when complex numbers are used.

To begin, if w is a real rational number, then there is an alternative definition of z^w that matches De Moivre’s Theorem. Happily, the two definitions agree. Suppose that z = r e^{i \theta} with -\pi < \theta \le \pi. Then

z^w = e^{w \log z}

= e^{w [\ln r + i \theta]}

= e^{w \ln r + i w \theta}

= e^{w \ln r} e^{i w \theta}

= r^w (\cos w\theta + i \sin \theta)

Next, one of the Laws of Exponents remains true even for complex numbers:

z^{w_1} z^{w_2} = e^{w_1 \log z} e^{w_2 \log z}

= e^{w_1 \log z + w_2 \log z}

= e^{(w_1 + w_2) \log z}

= z^{w_1 + w_2}.

However, in previous posts, we’ve seen that the rules (x^y)^z = x^(yz) and x^z y^z = (xy)^z may not be true if nonpositive bases, let alone complex bases, are used.

We can also derive the usual rules z^0 = 1 and z^{-w} = \displaystyle \frac{1}{z^w}. First,

z^0 = e^{0 \log z} = e^0 = 1.

Next, we think like an MIT freshman and use the above Law of Exponents to observe that

z^w z^{-w} = z^{w-w} = z^0 = 1.

Dividing, we see that z^{-w} = \displaystyle \frac{1}{z^w}.

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For completeness, here’s the movie that I use to engage my students when I begin this sequence of lectures.

 

 

Calculators and complex numbers (Part 22)

In this series of posts, I explore properties of complex numbers that explain some surprising answers to exponential and logarithmic problems using a calculator (see video at the bottom of this post). These posts form the basis for a sequence of lectures given to my future secondary teachers.

To begin, we recall that the trigonometric form of a complex number z = a+bi is

z = r(\cos \theta + i \sin \theta) = r e^{i \theta}

where r = |z| = \sqrt{a^2 + b^2} and \tan \theta = b/a, with \theta in the appropriate quadrant. As noted before, this is analogous to converting from rectangular coordinates to polar coordinates.

Theorem. If z = x + i y, where x and y are real numbers, then

e^z = e^x (\cos y + i \sin y)

Definition. Let z = r e^{i \theta} be a complex number so that -\pi < \theta \le \theta. Then we define

\log z = \ln r + i \theta.

At long last, we are now in position to explain the last surprising results from the calculator video below.

Definition. Suppose that z and w are complex numbers so that z \ne 0. Then we define

z^w = e^{w \log z}

Naturally, this definition makes sense if z and w are real numbers.

For example, let’s consider the computation of i^i. For the base of i, we note that

\log i = \log e^{\pi i/ 2} = \displaystyle \frac{\pi i}{2}.

Therefore,

i^i = e^{i \log i} = e^{i \pi i/2} = e^{-\pi/2},

which is (surprisingly) a real number.

As a second example, let’s compute (-8)^i. To begin,

\log(-8) = \log \left( 8 e^{\pi i} \right) = \ln 8 + \pi i.

Therefore,

(-8)^i = e^{i \log(-8)}

= e^{i (\ln 8 + \pi i)}

= e^{-\pi + i \ln 8}

= e^{-\pi} (\cos [\ln 8] + i \sin [ \ln 8 ] )

= e^{-\pi} \cos (\ln 8) + i e^{-\pi} \sin (\ln 8)

In other words, a problem like this is a Precalculus teacher’s dream come true, as it contains e, \ln, \pi, \cos, \sin, and i in a single problem.

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For completeness, here’s the movie that I use to engage my students when I begin this sequence of lectures.

 

 

Calculators and complex numbers (Part 21)

In this series of posts, I explore properties of complex numbers that explain some surprising answers to exponential and logarithmic problems using a calculator (see video at the bottom of this post). These posts form the basis for a sequence of lectures given to my future secondary teachers.

To begin, we recall that the trigonometric form of a complex number z = a+bi is

z = r(\cos \theta + i \sin \theta) = r e^{i \theta}

where r = |z| = \sqrt{a^2 + b^2} and \tan \theta = b/a, with \theta in the appropriate quadrant. As noted before, this is analogous to converting from rectangular coordinates to polar coordinates.

Over the past few posts, we developed the following theorem for computing e^z in the case that z is a complex number.

Definition. Let z = r e^{i \theta} be a complex number so that -\pi < \theta \le \theta. Then we define

\log z = \ln r + i \theta.

Of course, this looks like what the definition ought to be if one formally applies the Laws of Logarithms to r e^{i \theta}. However, this complex logarithm doesn’t always work the way you’d think it work. For example,

\log \left(e^{2 \pi i} \right) = \log (\cos 2\pi + i \sin 2\pi) = \log 1 = \ln 1 = 0 \ne 2\pi i.

This is analogous to another situation when an inverse function is defined using a restricted domain, like

\sqrt{ (-3)^2 } = \sqrt{9} = 3 \ne -3

or

\sin^{-1} (\sin \pi) = \sin^{-1} 0 = 0 \ne \pi.

The Laws of Logarithms also may not work when nonpositive numbers are used. For example,

\log \left[ (-1) \cdot (-1) \right] = \log 1 = 0,

but

\log(-1) + \log(-1) = \log \left( e^{\pi i} \right) + \log \left( e^{\pi i} \right) = \pi i + \pi i = 2\pi i.

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For completeness, here’s the movie that I use to engage my students when I begin this sequence of lectures.