An injective function is the basic function we always think of, I don't understand what you mean. The square root of x does happen to be an injective function by it's definition.
It depends on whether one considers the principal square root function or the multivalued relation. Choosing between R and C is irrelevant (for what anyway? domain? codomain?). You can map from reals to reals with a real image and still have both the positives and negatives. It just won't be a function. R vs C is irrelevant.
Maybe it's a cultural thing, when I took math I was required to express the results to a square root always as +/- (unless imaginary number) and if the root was part of a bigger operation record both results (when the value is negative and when positive)
Not really. In this case, it's not really clear whether or not we're talking about the just the principal value or not. Unlike written out mathematical notation, words can be ambiguous on this matter. So we clarify,
You start with 4, take the (principal) square root, and you will only have 2. Likewise, if you start with the (principal) square root of 4, then you will still only have 2.
On the contrary,
When you do the action of square rooting, you must consider the positive and negative cases. Since you do not know whether the square root was principal or not, you must consider both cases. The positive case is the one shown in the content - the principal square root. The negative case will give you -2.
If you start with simply "square root of 4", you have skipped the action of square rooting, and jump right to evaluating the positive case (the principal square root).
As displayed by the graph of a square root function (shown on the left), there are no negative y values, and thus f(x) can never be negative [ sqrt(2) can never equal - 2 ]. This is because the square root operation IS A function, and thus their can only be 1 y value for every x value (but not necessarily the other way around).
You are stating one of the most common misconceptions people have once they are taught to solve quadratic equations, observe.
1. x^2 = 4
2. x = sqrt (4)
3. x = ± 2
You make the assumption that the square root of 4 is equal to +/- 2, however, by actual mathematical definition, line 2 holds a mistake. This is because x DOES NOT equal sqrt(4). The correct solution is:
1. x^2 = 4
2. x = ± sqrt(4)
3. x = ± 2
Just as you cannot take the square root of a negative number, the square root of a positive number can never be negative. The radical symbol by definition is only the principal square root, i.e., always positive.
Wait wait, you're totally wrong. If the square root of a positive number can never be negative, why is -2 squared 4? That one you have up is only the positive, not negative roots.
I am. If you take the root of 4, its both -2 and 2. But if you already have the square root of four, its just two. I do understand the definition of a square root, with is a number that when multiplied by itself gives you the original. -2 times -2 is 4.
But when you take the square root, you are taking the positive and negative root, and you SHOULD technically be using the notation "±sqrt (a)", otherwise what you are writing is simply incorrect by means of notation. You may be doing the right thing in your head, but you're not writing down what you should be.
Type the square root of x into any graphing software and I assure you it will either give you that exact graph (might also include the imaginary part as well, a reflection along the y-axis)
The only bad part I see is "the square root of a positive number can never be negative". However, given the context one can easily tell that he's still referring to the principal square root.
Besides this, how is he incorrect? (unless this is what you meant)
the man on the pedestal is claiming to be something greater, better, or more important than those around him. The man being led away is speaking out and trying to show them that he's not anything more than they are, because the square root of 4 is 2.
Reminds me of the film 10,000 BC. "He is not a god!" Whilst prior everyone trembled before him. Yeah, it wasn't a very good movie. I thought it was okay.
The code in the program is only made to recognize a certain number of significant figures. The algorithm they use to calculate square roots must therefore produce very slightly incorrect results.
I now wonder what it would be like if a presidential candidate ignored political platforms and just promised the voters game sequels... "America, we have waited too long, elect me president, and WE WILL have Half-life 3!"
Length is a scalar, thinking in negative terms confuses calculations, so convention says to work with the positive root. But many times in mathematics, and physics, it is important to consider the negative root also.
I'm confused, where in your back up did it state that the function itself could not be negative? It said it COULD be, but we disregard it for the sake of simplicity.
It said it could be DEFINED as such, but it is not currently because that wouldn't be as simple.
A function by definition cannot produce multiple outcomes so the range is restricted.
f(x)=sqrt(x) is (in this case) a function that takes on the positive value of the sqrt of x. This bechause what you said, for a given x, f(x) can only have one value, so we forget about the negative one. It's still a solution to f(x) tho.
Every positive real number has two square roots, one positive and one negative. For example, the two square roots of 25 are 5 and −5. The positive square root is also known as the principal square root, and is denoted with a radical sign.
No.
The radical sign is itself a function (the f(x)= means it's a function too). If you want to include the negative half, you have to write it like so.
plus/minus sign indicates two solutions. The square root of a number has two real solutions. What you wrote has therefore 2*2=4 solutions, it's just that (in the case of x=4) the solutions ++2 and --2 are identical, just as -+2 and +-2. therefore, in the case you wrote, the plus/minus sign adds no information, it merely reminds us that square roots do indeed have two real solutions - one positive and one negative.
I know that the plus or minus is used to demonstrate that's it's both solutions, but when we first learned about square roots in middle school or whenever, we were taught that they always had two solutions without the plus or minus. However, once into harder math courses, the plus or minus sign was used for clarification. I thought that technically it still had two solutions, but you just used the principal value.
The symbol over the 4 means "square root". If you "square" a number, it means you multiply it by itself. An example is, 5 "squared" equals 25. The "square root" is when you take the number, such as 25, and find the number it is the square of, which in my example, is 5, so the square root of 25 is 5.
The joke is that the square root of 4 is 2, but only one person understood that, so everyone else thought he was crazy. In a more metaphorical sense, it is saying that people worship other people, even when those people are no different than themselves, but shun anyone who tells the truth of the matter.
I personally see it as religion in a nutshell, especially in the earlier days. Sqrt 4 saying he's better than the 2s, and people being punished for saying that he's just another 2.
Didn't say I agree, just showing the rational behind it.