Wikipedia:Reference desk/Archives/Mathematics/2023 May 25

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May 25

What's the difference between a mathematical formula and a mathematical equation?

On StackOverflow, an user says that:

An equation is any expression with an equals sign, so your example is by definition an equation. Equations appear frequently in mathematics because mathematicians love to use equal signs.

A formula is a set of instructions for creating a desired result. Non-mathematical examples include such things as chemical formulas (two H and one O make H2O), or the formula for Coca-Cola (which is just a list of ingredients). You can argue that these examples are not equations, in the sense that hydrogen and oxygen are not "equal" to water, yet you can use them to make water.

If that's the case, then is there mathematical formulas that are not equations, or mathematical equations that are not formulas? If you make an equation for something, then by definition you would want the equation to give a desired result/do a desired action, so I'm not sure of the difference between the two terms here. CactiStaccingCrane (talk) 17:27, 25 May 2023 (UTC)

The difference is pragmatic. Purely formally, an equation consists of two expressions connected by an equals sign. One typically uses the term formula for an equation that gives the relation between an entity whose value is desired to be known, and other entities whose values it depends on, expressed as an equation. An example is the formula for the area $T$ of a triangle in terms of its height $h$ and the length $b$ of its base:

$T={\frac {1}{2}}bh.$

Another example is the formula for the solution of the linear equation $ax+b=0,$ where $a\neq 0$:

$x=-{\frac {b}{a}}.$

One can think of such formulas as representing conclusions of theorems. Equations also appear in building a mathematical model for a problem, such as, "In a cylindrical barrel with radius 2 m, the water is rising at a rate of 3 cm/s. What is the rate of increase of the volume of water?" (Note that the assumptions that the barrel is standing upright and is not leaky are not stated.) Letting $L$ stand for the water level in cm and $t$ for the elapsed time in seconds, we have $L=3t,$ and so on. One would not call these equations "formulas". Likewise for the quadratic equation $x^{2}+x=1$. These equations are not like the conclusion of some theorem.  --Lambiam 20:22, 25 May 2023 (UTC)

I should add that in informal use any slightly complex mathematical expression may be referred to as a "mathematical formula".^[1][2][3]  --Lambiam 07:07, 26 May 2023 (UTC)

I guess there are going to be a lot of different takes on this; both terms are part of the language used to describe mathematics and neither really has a formal definition. (There is a definition of 'equality' and the meaning of '=' itself in set theory, but that's not the same thing as an equation.) I would say an equation is a mathematical statement or condition that involves equality. An equation can be either true (e.g. 2+2=4) or false (e.g. 2+2=5). Frequently whether an equation is true depends on the value of some variable. For example 2x+10=4x+6 is an equation which is true or false depending on the value of x. Solving an equation is the process of finding values of the variable for which the equation is true, for example solving the previous equation produces x=2. A formula is an expression used to produce a desired result. For example the quadratic formula

${\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$

can be used to solve a quadratic equation ax²+bx+c=0. You can solve an equation which has an unknown variable, but there's no such thing as solving a formula. A formula will often be given as an equation, as in

$x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$

with the desired result on the left and the expression used to find that result on the right, but it's not strictly necessary. (Though there are probably a few math teachers who would deduct a point or two for not including "x=" in the quadratic formula on an exam answer.) But the difference is rather fuzzy since you can certainly treat a formula given as an equation like any other equation. For example A=πr² is the formula for the area of a circle in terms of it's radius. But you can solve for r and produce an equivalent formula $r={\sqrt {\frac {A}{\pi }}}$ for the radius terms of the area. Like I said at the start, these terms are part of the language of mathematics with no formal definition, so whether you consider something an equation, a formula, or neither can depend on context or what you're actually doing with it.

It might be worth mentioning that a statement involving <, >, ≤ or ≥ is often called an inequality. Like an equation, an inequality involving an unknown variable can be solved, though the process is different. --RDBury (talk) 07:39, 26 May 2023 (UTC)