Confusion about a basic identity

[random_soldier1337]

So what I thought I knew years ago for some reason I am confused with right now. How shameful! Wonder which marble I lost? I must have had a brain fart.

 

Anyway, the identity goes (a - b)² = a² + b² - 2ab

 

Going from right to left, without knowing the left side, how would we know whether the negative term is a or b?

[timo]

You don't. It is a property of squaring a value. If you know that x²=4, you don't know whether x=+2 or x=-2. In a similar sense, the left side could be (a-b)² as well as (b-a)². And both terms of course also evaluate to the same value for any pair of a and b.

[random_soldier1337]

That's a bummer. I thought I had missed something. I would like to know what would you do if you had to simplify an expression. Further simplify or stop at this point on account of making an expression incorrect? Or does it depend on context like initial conditions for integrals or something?

Edited February 22, 2017 by random_soldier1337

[wtf]

[math](a - b)^2 = (b - a)^2[/math]. Your question is like noting that [math](-5)^2 = 25[/math], then asking if we are given [math]25[/math], how do we know if we "started" with [math]5[/math] or [math]-5[/math]. And the answer is that we don't. Squaring loses information. The squaring function maps two different values to the same value so you can't reliably go backwards.

 

ps -- There is a philosophical aspect to this point. If we view an equation as a statement that two different-looking expressions point to the same object, then from [math]a = b[/math] we may infer [math]b = a[/math]. Equality is a symmetric relation.

 

However if we regard an equation similarly to a formula in chemistry, a statement that one thing yield another thing via some process; then from [math]a = b[/math] we may not necessarily infer that [math]b = a[/math]. Not all transformations may be reversed.

 

This impacts our daily lives in the form of Internet security. Public key cryptography is based on the fact that [math]3 \times 5 = 15[/math] is a computationally easy problem; while [math]15 = 3 \times 5[/math] is not.

 

In your example we have a transformation (squaring) that's not reversible at all, because it loses information.

 

In the multiplication/factoring example we have a transformation that is in principle reversible, but one direction is computationally more expensive than the other.

 

When you say two things are equal, you have to be careful what you mean.

Edited February 22, 2017 by wtf