The long answer is actually the same as the short answer. This is almost word-for-word the explanation that I now use in class if a student asks -- otherwise I just tell them to follow the directions :. Bu now you know how to solve this. When we squared, we lost the sign information. The equation no longer indicates what it came from.
Notice that the step after squaring is the same for both examples. So, when you square both sides of an equation, you can get extraneous answers because you are losing the negative sign.
That is, you don't know which one of the two square roots of the right hand side was there before you squared it. Others have addressed why new solutions are introduced so I won't repeat that. But I do want to constructively comment on a more fundamental source for this issue.
By the way, I strongly discourage the terminology "can do to an equation". It reeks of students just trying to figure out what they have to do to make the teacher happy. A teacher especially should be very careful to avoid that phrase and use more appropriate phrases:. If I was instructing students in algebra, I'd require them to write arrows between their steps from early on.
The fundamental problem is that when novice students and even some non-novices ones manipulate equations, they get the false impression that anything implied by their original expression will also imply their original expression. To take an intentionally absurd example absurd because we actually have the solution written down to start with , they reason like this:. And this is indeed a mistake people sometimes make when solving inequalities.
One-way implications do not necessarily lead to solutions. The problem stems from being taught various mechanisms for manipulating in equalities, but losing track of what these manipulations actually mean in terms of correct deductions about the values involved. If the operation applied to both sides is an injection then of course it's reversible and so we have two-way implication, and in that case the deduction would be valid. In short: you can square both sides, of course. Sign up to join this community.
The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Why can't you square both sides of an equation?
Ask Question. Asked 7 years, 11 months ago. Active 2 years ago. Viewed 94k times. Damian Yerrick 1 1 silver badge 8 8 bronze badges. Jeff Jeff 3, 3 3 gold badges 27 27 silver badges 40 40 bronze badges. Show 1 more comment. Active Oldest Votes. Bruno Joyal Bruno Joyal A slightly higher-level way to phrase the issue is that the usual square root is not a "function" at all.
Strictly speaking, it's not a function at all because it doesn't pass the 'vertical line test' i. Remarkably, if you count all the tiny triangles in each design—both green and white—the numbers are square numbers! Build a stair-step arrangement of Cuisenaire rods, say W, R, G.
Then build the very next stair-step: W, R, G, P. Put the two consecutive triangles together, and they make a square:.
This square is the same size as 16 white rods arranged in a square. When placed together, these make a square whose area is 64, again the square of the length in white rods of the longest rod. Stair steps that go up and then back down again, like this, also contain a square number of tiles.
Here are two examples. Color is used here to help you see what is being described. For each new square, add two rods that match the sides of the previous square, and a new W to fill the corner.
And how does "correcting" it improve mathematics? Just so you know, I used the phrase "happens to be" somewhat ironically. It's easy to prove that it is true for all numbers except zero, starting from the definition. Why do you take this narrow corollary to be more important than the broader definition? But the important thing is this: We can change definitions, but only because doing so produces interesting or useful new mathematics. All you appear to be doing is to remove some mathematics; no one is going to go along with you.
Saying "I believe" does not make it true. It is not sufficient to define how many pieces zero splits itself into, evidenced by contradictions in other operations. It is not sufficient to define the square root of zero, evidenced by contradictions in other operations. Whether these arguments contribute to or otherwise the utility of square root in the greater landscape of mathematics ultimately won't affect what square root is or means.
We can study it to learn about it, but it's not up to us to decide what we get out of it. The only contradictory equations I can come up with such as the aforementioned division and logarithm examples would still fail if given zero without a square root in the mix.
Finding examples where zero itself isn't a troublemaker or where zero is a requirement could shed some light on this. Last edited: Jan 13, What you're saying is that if any operation results in zero, you're going to disallow it, because there are some things you can't do with zero. Frankly, this is nonsense. As they say, "hard cases make bad law"; likewise, "special cases make bad definitions".
Let the definition be what it is; if you get a number you can't do something with, just don't do that! It sounds like you need to see evidence that we need to be able to take the square root of zero! If the answer isn't zero, then we can't find the distance from a point to itself!
But clearly it's zero Subhotosh Khan Super Moderator Staff member. Joined Jun 18, Messages 25, I remember, that laws of exponentiation does not apply to 0 or 1 from high school. Cannot prove it but I have learned to take it as truth. Subhotosh Khan said:. Again I needed that emoji showing "tongue planted in cheek". I gave up because all the Google results I clicked on were aimed at beginners.
But it is useful to know I hope I got this right, and complete. BTW I did not consider a,b,or c as complex numbers. HallsofIvy Elite Member. Joined Jan 27, Messages 7,
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