Reading equations forward and backward

There is no logical difference between writing A = B and writing B = A, but there is a psychological difference.

Equations are typically applied left to right. When you write A = B you imply that it may be useful to replace A with B. This is helpful to keep in mind when learning something new: the order in which an equation is written gives a hint as to how it may be applied. However, this way of thinking can also be a limitation. Clever applications often come from realizing that you can apply an equation in the opposite of the usual direction.

For example, Euler’s reflection formula says

Γ(z) Γ(1-z) = π / sin(πz).

Reading from left to right, this says that two unfamiliar/difficult things, values of the Gamma function, are related to a more familiar/simple thing, the sine function. It would be odd to look at this formula and say “Great! Now I can compute sines if I just know values of the Gamma function.” Instead, the usual reaction would be “Great! Now I can relate the value of Gamma at two different places by using sines.”

When we see Einstein’s equation

E = mc2

the first time, we think about creating energy from matter, such as the mass lost in nuclear fission. This applies the formula from left to right, relating what we want to know, an amount of energy, to what we do know, an amount of mass. But you could also read the equation from right to left, calculating the amount of energy, say in an accelerator, necessary to create a particle of a given mass.

Calculus textbooks typically have a list of equations, either inside the covers or in an appendix, that relate an integral on the left to a function or number on the right. This makes sense because calculus students compute integrals. But mathematicians often apply these equations in the opposite direction, replacing a number or function with an integral. To a calculus student this is madness: why replace a familiar thing with a scary thing? But integrals aren’t scary to mathematicians. Expressing a function as an integral is often progress. Properties of a function may be easier to see in integral form. Also, the integral may lend itself to some computational technique, such as reversing the order of integration in a double integral, or reversing the order to taking a limit and an integral.

Calculus textbooks also have lists of equations involving infinite sums, the summation always being on the left. Calculus students want to replace the scary thing, the infinite sum, with the familiar thing, the expression on the right. Generating functions turn this around, wanting to replace things with infinite sums. Again this would seem crazy to a calculus student, but it’s a powerful problem solving technique.

Differential equation students solve differential equations. They want to replace what they find scary, a differential equation, with something more familiar, a function that satisfies the differential equation. But mathematicians sometimes want to replace a function with a differential equation that it satisfies. This is common, for example, in studying special functions. Classical orthogonal polynomials satisfy 2nd order differential equations, and the differential equation takes a different form for different families of orthogonal polynomials. Why would you want to take something as tangible and familiar as a polynomial, something you might study as a sophomore in high school, and replace it with something as abstract and mysterious as a differential equation, something you might study as a sophomore in college? Because some properties, properties that you would not have cared about in high school, are more clearly seen via the differential equations.

5 thoughts on “Reading equations forward and backward

  1. My favorite brain-bending way to rearrange Einstein’s equation is as

    c = sqrt(E / m)

    That one hurts my head, especially from a dimensional analysis point of view.

  2. Nick Barrowman

    We often read equations causally: the left-hand side is seen as the effect and the right-hand side is seen as the cause. Judea Pearl has discussed this (see Slide 44 in a 1996 talk he gave on The Art and Science of Cause and Effect) . Causation is trickier than manipulating equations! I believe this is also related to issues of reversability of processes in thermodynamics.

  3. Note than people usually write things like “1/x = o(1)”, but “o(1) = 1/x” wouldn’t make any sense.
    As a student I was already using the big/little “o” notation for some time, when I realized it’s an abuse of notation, and you can’t write it backwards. The fact that it took me some time to realize this shows that there is, indeed, a psychological difference between writing A = B and writing B = A.

  4. Arguably the best example of this is the geometric series

    1 + x + x^2 + … = 1/(1-x)

    I was astonished when I first see it applied backwards when used to solve the Fibonacci series.

  5. I was looking for a way to make a book that works both ways as if you could read it backwards and forwards, A fractal of information that could be replicated forwards and backwards no matter how much you divided as to describe anything and everything. Calculus but for the layman I think this may have been a little more deep than I was expecting to find and I thank you for it! Unless I didn’t read it the right way which happens a lot.

    Erik Peterson

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