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#1 Introductions » Hi from Brazil! » 2024-04-30 14:13:17
- G4B_r09
- Replies: 3
Hello! My name is Gabriel, and I am a freshman (9th grader) from Brazil. I have used Math is Fun as a very useful learning tool for a couple of years, but never acknowledged the existence of this forum. I participate in science olympiads, mostly math-based. My dream is becoming an engineer, and for short-term it is getting into the IMO, which I consider one of the most prestigious Math olympiads worldwide.
Good to be here!
(I might include a bible verse in my signature or elsewhere because I'm Christian, hope that doesn't bother anybody.) 
#2 Re: Help Me ! » Principal Square Root of A Number » 2024-04-30 14:05:39
Heya!
A number has two square roots: one negative and one positive. The principal square root is the positive root, and so it is nonnegative. Basically y is the nonnegative number which multiplied by itself gives x as a result. The principal square root of 4 is 2 (2 * 2 = 4), the principal square root of 9 is 3 (3 * 3 = 9), and so on. All nonnegative real numbers have a square and a square root. The negative real numbers only have a square, they don't have a square root. An important thing to note is that the two roots of a number (the positive and the negative) are always opposite, they have the same numerical value but are in opposite sides on the number line. This can also happen in quadratic functions and equations, but not always, since these involve numerical expressions and more complicated square roots.
Hope I could help ya! 
#3 Re: Help Me ! » Square Root {Negative Number} » 2024-04-30 13:46:50
Hey there!
Grab a graphic calculator or access one that is on-line, like Desmos. When you input the function y = x², which takes every real x value and equates it to its square, you'll notice that the y value is never negative, and y always tends upwards, so there's no way it'll curve down to bypass the x axis and become negative. And so it is the case that no number multiplied by itself (squared) results in a negative number.
Why does this happen? Well, if we multiply a positive number by itself, we get a positive number back (e.g. 3 * 3 = 9). If we multiply a negative number by itself, we also get a positive number (e.g. (-3) * (-3) = 9), and when we multiply 0 by itself, well, we get 0. In none of these cases we get a negative number after squaring, so logically for any real number x, its square will be positive (or 0, if x = 0)
It's not that the square root of a negative number doesn't exist - well, at least in doesn't in the set we're used to working with, the real numbers set, which includes all rational and irrational numbers. There is a set that includes the set of real numbers, but transcends the real plane, with values that are outside of that plane (which we call the Cartesian plane). That set is the Complex set, and its main determinant is a little factor called i, which is equal to the square root of -1. We can get some interesting properties from this number; like i^2 = -1, i^3 = -1 and i^4 = 1. At least in my country, the complex set is explored more around the second half of High School.
The square root of a, if a is negative and less than -1, is not exactly undefined; it is nonexistent when we consider only the real numbers set, but it does exist when we consider the complex set.
Hope I could make it a bit more clear for ya.
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