An equation is a statement that says ‘this equals that’. It’s hard to imagine, from that raw and basic idea, that something called an equation could be of any use. Whether ‘this equals that’ or not seems a matter of little consequence!
But, in fact, we know that to state what things are equal can have powerful consequences.
Newton’s law – that the net force on something causes it to accelerate – is a matter of things being equal. Force = mass multiplied by acceleration. This law of nature governs an extraordinary panoply of phenomena: the solar system, the entire NASA program of space exploration, the working of engines, the nature of energy and thus of pressure and of temperature and thus our understanding of weather. It underlies thinking in engineering, geology, chemistry, biology, and, of course, physics. All from ‘this = that’.
That mass is a form of energy, the famous E = mc2, is a statement about things being equal. It has had profound consequences.
In market shopping, the price per pound of potatoes multiplied by the number of pounds in the package equals the purchase price. This is a matter of things being equal. The computation is the key ingredient of household economics.
The invention of the equation is surely comparable, in impact, to the invention of the wheel. But, in what resides the power of such a simple construct – the saying of what equals what?
I imagine that the saying and writing of equations is the observable expression of the mechanism of understanding taking place unawares inside the brain. The brain constructs equations in order to function. Eventually, as the use of writing evolved, this internal process flowered into an external expression of itself.
Thus the matter of sameness – equality – must be at the basis of ‘understanding’. Perceiving ‘what equals what’ allows us to catalogue experience.
On the significance of the evident applicability of mathematics to the physical world, see Is Symmetry Identity? For the idea that the pursuit of understanding is a quest for sameness see Physics As Symmetry
The modern equals sign (=) dates from 1557 England. The man, famous for introducing it is Robert Recorde. Before that, the words ‘is equal to’ were used. So the idea of ‘equation’ far antecedes the equals sign.