What exactly are square roots?

[Shadow Conception]

I've read the Wikipedia articles, I've used them to solve formulas, I've used them to figure out a leg of a right triangle and so on.

But why are they so useful? HOW did they come up with this? And why are they used so often in formulas? I mean, that definitely indicates that square roots have great significance in the mathematical world... but why?

With square roots, we were simply taught what to do with them. Root of 64 is 8, root of 121 is 11. Root of 2 would be an irrational number. But we've never been taught how to VISUALIZE it the same way we can visualize 2+2. Or is that even possible?

I can apply square roots in situations, but I don't understand why we use them. And looking over an AP Stats free-response exam (boredom), even THEY have formulas with square roots. I mean, it's STATISTICS, why the square roots?

Sorry if I sound stupid/narrow-minded on the subject, I'm not all too enlightened on the matter. I'm in geometry right now and I'll be taking Algebra II/Trig next year.

 

[bobsmith1492]

A number times itself = its square. Just do the reverse. A x A = B, so A is the square root of B.

Or you can think of it as a number to the 1/2 power which may or may not help you.

 

[firewolfsm]

To see why we use them in equations realize that it's just a 1/2 exponent. It's as useful as say, x^2. Exponents can be anything really, even equations themselves.

 

[Billb2]

You're stuck in the mindset that the root has to be a whole number. Any number has an nth root.

As to why they show up in math so often, that's because of the way god made the universe.

 

[BladeVenom]

I think the best way to visualize it is go by its name. Square root. Go from 2 dimensions to 1 dimension, like from a square to a line. The 2 dimensional square's roots which are 1 dimensional lines.

So if you have something that is nine square yards you think of it's roots, from 2 dimensions to one. So I visualize the square's roots which are lines 3 yards long.

That's the best visualization I can think of. I hope it makes sense.

Next would be cube roots, 3 dimensions.

 

[BrownTown]

I agree with BladeVenom, if you want to think of it in visual terms, the number is the area of a square, and its "square root" is the length of a side, just the same as the number could be the volume of a cube and its "cube root" is the length of one side.

Obviously though you see it all over the place, but any formula where something has a dependence on the square of something you can go the other way and its a square root. Just a simple example, power through a resistor is V^2/R, so the voltage is proportional to the square root of the power. In this case this arises because increasing the voltage increases power in 2 ways (higher voltage across the resistor plus more current through the resistor). Also, if you are talking about statistics, maybe you mean like the standard deviation using square roots? This isn't a law of nature, its just made be people, and the point is to give extra weight to outliers by squaring the distance they are from the middle. You can also have a similarly made up formula where they are not squared (variance).

 

[gsellis]

QED... A root is a base component. If you have the base component of a number that was squared, you have the Square Root. SqRt is important because of some dead Greek guy. Repeat after me, "The square on the hypotenuse is equal to the sum of the squares on the other two sides."

Applied in building, construction, tracking, etc. It you know two of the distances, you can calculate the third using a Square Root. Wish I remembered how to do it on a slide rule.

 

[Stiganator]

a square root isn't a thing it is a specific case, it is a parameter of the exponent operator (^). X^Y two parameters X and Y. ^ describes the action you perform, which is multiply X by X, Y number of time. If Y = 1/2, then X by X, 1/2 a time, you get the square root.

Many things in science are regressions, you'll get square roots, and cubic roots and all kinds of funky things based on how data can be fit most accurately. Linear and exponential are probably most common, but many things are non-linear.

 

[Cogman]

lol, math nerds! (so many answers to the question)

Math is like a giant rabbits hole, the deeper you go into it, the more useful everything you pass begins to look (To get the heck out while you still can lol)

For example, Algebra was different for me, and understanding why we would use a system like this was a stretch, (Ok, I did see a few practical applications) Then I took calculus and suddenly algebra became the most useful thing I could learn. Calculus actually has a lot of uses BTW that are quite apparent when you take it.

Square Roots are much the same way, initially understanding their uses is hard because we tend to just look at the function, after taking much more math, the uses of the square root extend indefinitely.

I can apply square roots in situations, but I don't understand why we use them. And looking over an AP Stats free-response exam (boredom), even THEY have formulas with square roots. I mean, it's STATISTICS, why the square roots?

he he, that is because statistics are based on calculus, where do you think all those neat functions they give you come from? Trust me, once you get to calculus the question wont be "Why do we have square roots" but "How to I get rid of these things" err I mean "How did I live without these!"

 

[gsellis]

Originally posted by: Cogman
he he, that is because statistics are based on calculus, where do you think all those neat functions they give you come from? Trust me, once you get to calculus the question wont be "Why do we have square roots" but "How to I get rid of these things" err I mean "How did I live without these!"

Oh be careful. Don't say statistics is 'math' in front of a math major. They get all out of sorts.

Key point is my example. A math major says that 2+2=4. A statistician says that it is improbable that 2+2 is something other than 4. It is a big difference

 

[Special K]

Roots only became hard to conceptualize for me once I considered imaginary roots. For example, 4^5 = 4*4*4*4*4, but what about 4^j (or 4^i for you mathematicians?) I know how to compute it, but what does it even mean to take a number raised to an imaginary power?

 

[BrownTown]

Originally posted by: Special K
Roots only became hard to conceptualize for me once I considered imaginary roots. For example, 4^5 = 4*4*4*4*4, but what about 4^j (or 4^i for you mathematicians?) I know how to compute it, but what does it even mean to take a number raised to an imaginary power?

they mean whatever you define them to mean, they are called "imaginary" for a reason because to my knowledge there isn't any physical usefulness for them. They are used in places obviously, but then meaning is defined in addition to just the root of -1, like in electrical engineering we use imaginary numbers all the time, but its simply redefining the variables of amplitude and phase to the coefficients of a complex number to make the easier to do ath on. Its not like a capacitor somehow has an imaginary resistance, you simply use the properties of the complex plane to make the calculations for the phase shift easier so you aren't doing all kind of differential equations up the ass.

 

[Peter]

All that "imaginary" business is just a different way of doing math on a 2D plane, rather than just back and forth a one-dimensional range of numbers.

 

[Cogman]

Originally posted by: BrownTown

Originally posted by: Special K
Roots only became hard to conceptualize for me once I considered imaginary roots. For example, 4^5 = 4*4*4*4*4, but what about 4^j (or 4^i for you mathematicians?) I know how to compute it, but what does it even mean to take a number raised to an imaginary power?

they mean whatever you define them to mean, they are called "imaginary" for a reason because to my knowledge there isn't any physical usefulness for them. They are used in places obviously, but then meaning is defined in addition to just the root of -1, like in electrical engineering we use imaginary numbers all the time, but its simply redefining the variables of amplitude and phase to the coefficients of a complex number to make the easier to do ath on. Its not like a capacitor somehow has an imaginary resistance, you simply use the properties of the complex plane to make the calculations for the phase shift easier so you aren't doing all kind of differential equations up the ass.

Hmm, I'll be interested to see the (2nd year studying Comp Engineering) as of yet, I haven't seen imaginary numbers used in any useful way. However, If they stop me from doing some of the crazy calculus I had to do in physics, then I'm all for it!

 

[hellokeith]

Originally posted by: Shadow Conception
But we've never been taught how to VISUALIZE it the same way we can visualize 2+2. Or is that even possible?

Yes it is possible and yes it is a huge failing of educations systems. Children from a very early age should be visually shown the length of mathematically significant numbers, such as SQRT(2), PI, e, golden ratio, etc.

Can you imagine how powerful would be a whole generation of people who know by heart and can instantly visualize these numbers? They'd be teaching calculus to 6th graders.

 

[CP5670]

Hmm, I'll be interested to see the (2nd year studying Comp Engineering) as of yet, I haven't seen imaginary numbers used in any useful way. However, If they stop me from doing some of the crazy calculus I had to do in physics, then I'm all for it!

It depends on what you consider to be useful. I can't think of any fundamental physical quantities that use them in an essential way (although that could arguably also be said about negative numbers ), but they are completely natural in many areas of math.

A simple example I can think of is for finding the region of convergence of a power series (you probably had to do that in calculus class at some point), which is controlled by properties of the function in the complex plane even if you're only working with real numbers.

 

[dorion]

Originally posted by: hellokeith

Originally posted by: Shadow Conception
But we've never been taught how to VISUALIZE it the same way we can visualize 2+2. Or is that even possible?

Yes it is possible and yes it is a huge failing of educations systems. Children from a very early age should be visually shown the length of mathematically significant numbers, such as SQRT(2), PI, e, golden ratio, etc.

Can you imagine how powerful would be a whole generation of people who know by heart and can instantly visualize these numbers? They'd be teaching calculus to 6th graders.

So we bring back the slide rule and teach logarithms right after you learn addition and multiplication. OR atleast pound it into the head of school children that logarithms reduce the complexity of arithmetic.

 

[DrPizza]

Square roots are the two vectors on the complex plane that are located 180 degrees apart, the direction angle of one of them is one half the direction angle of the original number, and whose lengths are equal to the sides of a square with the same area as the length of the original vector.

 

[BrownTown]

Originally posted by: Cogman
Hmm, I'll be interested to see the (2nd year studying Comp Engineering) as of yet, I haven't seen imaginary numbers used in any useful way. However, If they stop me from doing some of the crazy calculus I had to do in physics, then I'm all for it!

Well i'm not sure what the computer engineering degree is at your college, at mine its the exact same as the EE degree except you need 3 of the CS classes at some of your technical electives. If you are doing anything with electrical circuits then you are gonna be using imaginary numbers. I would say half my classes in college used imaginary numbers. Doing stuff like the Laplace transform, or the Fourier Transform, or using phasers for electrical circuits and stuff. The Laplace transform was what I was talking about to turn differential equations into more simple algebraic equations. You will definitely do those in differential equations if you haven't already.

Originally posted by: DrPizza
Square roots are the two vectors on the complex plane that are located 180 degrees apart, the direction angle of one of them is one half the direction angle of the original number, and whose lengths are equal to the sides of a square with the same area as the length of the original vector.

lol, are you trying to help people understand or make them more confused ?

 

[Born2bwire]

Originally posted by: BrownTown

Originally posted by: DrPizza
Square roots are the two vectors on the complex plane that are located 180 degrees apart, the direction angle of one of them is one half the direction angle of the original number, and whose lengths are equal to the sides of a square with the same area as the length of the original vector.

lol, are you trying to help people understand or make them more confused ?

Makes perfect sense to me.

 

[DrPizza]

Originally posted by: Born2bwire

Originally posted by: BrownTown

Originally posted by: DrPizza
Square roots are the two vectors on the complex plane that are located 180 degrees apart, the direction angle of one of them is one half the direction angle of the original number, and whose lengths are equal to the sides of a square with the same area as the length of the original vector.

lol, are you trying to help people understand or make them more confused ?

Makes perfect sense to me.

Eh, I actually gave my class a test on that today, so thought I'd post it. It was one of those tests where the seniors with any trace of senioritis failed... miserably, and the students who struggled all year aced the exam because it was so easy.

 

[BrownTown]

Oh well, seniors would all be into college by now anyways, so what do they care about failing

 

[Net]

a lot of reason. there are so many applications. just like why is multiplication so useful? because it saves time.

if i have to op amps and i want to optimize the overall gain then i would want each amps gain to be equal. just take the square root of the overall gain to find what each gain needs to be.

how about finding the absolute value? you square the number and take the square root. there is a ton of reason why you need your result to be positive all the time.

there are tons of applications.

 

[imported_Baloo]

Originally posted by: BrownTown

Originally posted by: Special K
Roots only became hard to conceptualize for me once I considered imaginary roots. For example, 4^5 = 4*4*4*4*4, but what about 4^j (or 4^i for you mathematicians?) I know how to compute it, but what does it even mean to take a number raised to an imaginary power?

they mean whatever you define them to mean, they are called "imaginary" for a reason because to my knowledge there isn't any physical usefulness for them. They are used in places obviously, but then meaning is defined in addition to just the root of -1, like in electrical engineering we use imaginary numbers all the time, but its simply redefining the variables of amplitude and phase to the coefficients of a complex number to make the easier to do ath on. Its not like a capacitor somehow has an imaginary resistance, you simply use the properties of the complex plane to make the calculations for the phase shift easier so you aren't doing all kind of differential equations up the ass.

In other words, they are used to simplify equations.