55: Useless

Explain xkcd: It's 'cause you're dumb.
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Even the identity matrix doesn't work normally
Title text: Even the identity matrix doesn't work normally


Randall is attempting to apply mathematics to the concept of love to no avail. Specifically, he is attempting his "normal approach", which is a term used in mathematics for the method one typically uses to solve a certain type of problem. However, as love is not a well-defined mathematical entity, his normal approach is useless. Simply put: he's saying he has found no way of describing love using only the tools of mathematics.

From the top, moving left to right, he tries the square root of love, the cosine of love, and the derivative of love with respect to x. He then attempts to left-multiply love by a 2x2 identity matrix, and finally he defines a function of love as a Fourier transform. These are all "normal approaches" to solving certain math problems.

The message of the comic is that for someone who uses math to solve all their problems, defining love is impossible. It also indicates that love is not always a rational (or irrational) phenomenon.

This comic has been made into a t-shirt in the xkcd store, with a Laplace transform in place of the bottom integral.

Basic explanations of the functions

Note: The Wikipedia links will provide far more detailed explanations of the mathematics.
  • The square root of x is the number that, when multiplied by itself, equals x.
  • Cosine is a trigonometric function that, when given the measure of an angle in a right triangle as an input, outputs the ratio of the lengths of two sides of that triangle (for cosine, it is the non-hypotenuse side adjacent to the angle and the hypotenuse).
  • A derivative of a function is the rate of change of that function at a given value of x. It is a primary focus of calculus. A basic example is where "velocity" is the rate of change of displacement at a given time, the derivative of velocity is "acceleration", which is the rate of change of velocity at a given time.
  • Identity matrices are matrices that consist of only zeros and ones, with zeros everywhere except along the main diagonal. Multiplying a matrix by the equal-sized identity matrix will result in the same output in the same way that multiplying a non-matrix by 1 does not change the original term. The title text suggests that multiplying love by the identity matrix does not return the same "love" value.
  • A Fourier transform converts a function from one (sophisticated) function into an endless continuous series of (more simple) functions, where each next part is bringing the equation closer to the real result. This means that you can stop your calculations after a few iterations and you are very close to the real result, and it also can be used to deconstruct signals.


[Different mathematical equations, all with a heart on left side, and all ending up with question marks.]
√♥ = ?
cos ♥ = ?
d/dx ♥ = ?
[1 0]♥ = ?
[0 1]
F{♥} = 1/√2π ∫-∞f(t)eit♥dt = ?
[Caption below:]
My normal approach is useless here.


  • This was the 52nd and last comic originally posted to LiveJournal.
    • The previous was 53: Hobby.
    • The next comic 56: The Cure was not also posted on LiveJournal
      • This thus ends this repeating trivia section that has been present for all those 52 LiveJournal comics.
      • Also after this there has been no issue with comics released at two times, since they were now released only on xkcd.
  • This comic kept its original title: "Useless"
    • It is part of the last six comics on LiveJournal which all had a title without the word "Drawing" in it.
    • Five of these had exactly the same title on both sites.
    • Only 11 comics have the same title on both sites.
    • Apart from the thirteen first comics posted to LiveJournal, there were only three other comics without the word "Drawing" in the title before these last six.
  • There was no original Randall quote for this comic.
  • This comic was the last of the last 11 comics posted on LiveJournal.
    • These 11 comics were posted both on LiveJournal and xkcd after the xkcd site opened on the 1st of January 2006.
    • The first six comics were posted on both sites on the same day.
    • And then this first happened again with this last comic.
  • The version used on the t-shirt and in xkcd: volume 0 is slightly different. The derivative is with respect to time (t) instead of x, and the function at the bottom is a Laplace transform.
  • This comic was adapted to a wedding cake featured on an installment of "Sunday Sweets", a regular feature on popular blog Cake Wrecks.

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There seems to me to be a philosophical monologue going on here: What is the root of love? What is the angle of love? What is the derivative of love? What is the identity of love?

Unfortunately, I don't know much about the Fourier transform, so I'm at a loss for describing it in layman's terms. Anyone wanna lend a hand? 16:59, 13 February 2013 (UTC)MagnusVortex

I think the better reading is:
  • What is the root of love? (i.e. Where does love come from?)
  • What are the signs of love? (Sine is a periodic function, and laypeople would confuse sin(heart) with sinning against love)
  • How do you derive love?
  • How do you identify love? (i.e. How do you know when you've fallen in love? How do you know when someone truly loves you?)
The last one is a bit harder to interpret. Possible interpretations include:
  • What is the frequency/wavelength of love?
  • How often do you fall in love?
  • How do you transform love?
  • What is the spectrum of love? (gay, straight, bi, asexual, &c.)
  • How do you find love analytically?
Randall Munroe would likely consider the possibility of multiple interpretations of the last one to be a feature, not a bug.
I worry that most xkcd readers would not realize that Munroe is posing specific questions.
—DrDnar 22:52, 29 June 2016 (UTC)

I believe the last one is:
What is the frequency of love?

-JD 18:02, 25 February 2013 (UTC)

For those of you who have used Mathematica, if you replace the heart with "Indeterminate", you'll find yourself in a similar situation: essentially all functions of Indeterminate yield Indeterminate. It can be frustrating. --Quicksilver (talk) 20:11, 25 August 2013 (UTC)

It is clear that the author has yet to study non-linear dynamics as this approach has already been covered in

Strogatz, S. H. (1988) Love affairs and differential equations. Math. Magazine 61,35.
Strogatz, S. H. (1994) Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering. (Perseus Books, Reading, Massachusetts)

One is forced to conclude that love is chaotic.

-Boyd 07:59, 10 October 2013 (UTC)

I feel like there's a pun within the alt-text: "Even the Identity matrix doesn't work normally" but I'm not sure I "get" the pun. It could simply be a reference to the fact that love tends to change who you are, in marriage the idea of two becoming one, or it could be referencing the Identity (0) directly. Even the Identity formula doesn't output zero. 00:35, 24 September 2017 (UTC) Sam

These observations should have been enough of a warning to those of us who don't have "other approches" figures out to just stay away... it wasn't in my case. I regret that.--TheTimeBandit (talk) 21:55, 30 October 2017 (UTC)

Ok, I know this comic was posted 16 years ago and someone who was born the day it was published would be in high school today. (Feel old?) However, it looks like no one has mentioned it til now, that this comic is almost a direct rebuttal to the previous comic, 54: Science. --mezimm 20:34, 1 March 2022 (UTC)