Square roots of negative numbers.
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Porii Sames
Active Member
Yeah...
In Algebra, the square root of a negative is, well, impossible. So, we use i to signify the square root of negative 1, since it can come up often in math. So, 3i would be 3 times the square root of negative 1, even if the number itself doesn't exist.
In Algebra, the square root of a negative is, well, impossible. So, we use i to signify the square root of negative 1, since it can come up often in math. So, 3i would be 3 times the square root of negative 1, even if the number itself doesn't exist.
z-man
New Member
In short, it doesn't work. This is one of those impossibilities, but mathematicians usually setup the "what if" scenario using 'i'. If you could root a negative, this is what would happen. i = sqrt(-1), i^2 = -1, i^3=-sqrt(-1), and i^4=1. Just something you'll need to learn. It is very useful in theoretical mathematics.
Cool note: Did you know that e^(pi*i)=-1?
Cool note: Did you know that e^(pi*i)=-1?
The main reason for the imaginary number, as people have said, is that it gives "solutions" to quadratic equations (equations of the form ax^2+bx+c) that would otherwise be impossible. Since a real number squared is always positive, there is no real number x that satisfies, for example, the equation x^2 = -4.
However, this is where i comes in: it is formally defined as the square root of -1.
The imaginary number is used in complex analysis, which is a tricky subject.
In regards to why e^(i*pi) = -1, it is because e^(i*x) = cos(x) + i*sin(x). This is called Euler's formula.
However, this is where i comes in: it is formally defined as the square root of -1.
The imaginary number is used in complex analysis, which is a tricky subject.
In regards to why e^(i*pi) = -1, it is because e^(i*x) = cos(x) + i*sin(x). This is called Euler's formula.
Kayle
Active Member
I'll try...
With the square root of a positive number, like, rt(4) [the square root of 4], well, that's just an operation you happen to know already. The square root of 4 is 2.
However, rt(-4) is weird and we don't know exactly how to evaluate it.
Some clever guy (euler? I'm not up on math history) realized that rt(-4) = rt(-1) x rt(4). He could then evaluate rt(4) to be 2, so he got (2 x rt(-1)).
Then he just decided to let i stand for the square root of -1. So, rt(-4) = 2i.
People call it the imaginary number because no one knows exactly how it's supposed to work (but it does).
amirite
prodigal_fanboy
New Member
Objection. Imaginary numbers were first a thought exercise that had useful implications for vector and matrix algebra and differential equations, but since the formulation of the Schrodinger equation in 1926, they're very much a real part of the world around you.
The wave functions of hydrogenic (ions with a single electron) atoms actually have the square root of negative one sitting right in there, without which it wouldn't be solvable in anything like its present form.
I'll grant you it's something that just must be learned, and it is definitely useful in theoretical mathematics, but to say imaginary numbers don't really exist isn't quite true.
NoPoke
Active Member
imaginary numbers have been useful way before 1926!.
The "imaginary" name isn't very helpful and I believe it was initially used as a mocking description to imply that the concept would be useless. Boy were the critics wrong. Adding imaginary numbers to the reals unlocks and simplifies vast swathes of mathematics. It is actually this last bit that is important and not what root(-1) might be that matters.
Imaginary numbers are just a really useful addition to mathematics, they are every bit as "real" as the normal numbers. They are used because they simplify mathematics.
I can't get the formula to appear nicely here but e^(i pi) = -1 has to be one of the most beautiful expressions in mathematics.
root(-1) is important because it makes mathematics easier.
The "imaginary" name isn't very helpful and I believe it was initially used as a mocking description to imply that the concept would be useless. Boy were the critics wrong. Adding imaginary numbers to the reals unlocks and simplifies vast swathes of mathematics. It is actually this last bit that is important and not what root(-1) might be that matters.
Imaginary numbers are just a really useful addition to mathematics, they are every bit as "real" as the normal numbers. They are used because they simplify mathematics.
I can't get the formula to appear nicely here but e^(i pi) = -1 has to be one of the most beautiful expressions in mathematics.
root(-1) is important because it makes mathematics easier.
Even in university I cant wrap my head around this lol. I mean, I know how to calculate with them but I still dont understand the point. My father studies maths and always goes like "well sqrt(2) is just an idea as well right" but at leats you can get close to sqrt 2, but i is just, I dont get it xD
I mean, its not like a description that is close enough to something real, its just completly made up and I dont see how it can ever work lol. Anyone has any example where you can see WHY this works? It just seems so completly random, what if I define x/0 = 32? or idk, x/0 = f. Seems just as random.
im not hating on the concept but I cant wrap my head around WHY this works lol
I mean, its not like a description that is close enough to something real, its just completly made up and I dont see how it can ever work lol. Anyone has any example where you can see WHY this works? It just seems so completly random, what if I define x/0 = 32? or idk, x/0 = f. Seems just as random.
im not hating on the concept but I cant wrap my head around WHY this works lol
prodigal_fanboy
New Member
Don't worry, I said they were useful before 1926... but in 1926 the first (to my knowledge) concrete predictions of real-world behavior involving them was formulated.
And Yoshi, "i" is just something you multiply by itself to get -1. Numbers aren't "real", in the sense that all our mathematics system is based upon fundamental assumptions. I don't know enough about the theory to condense it for you, but ask your father about mathematical axioms.
Kayle
Active Member
NoPoke
Active Member
@yoshi, your dad is right. He has made the leap from using numbers to count things to using them to represent ideas. Students are first exposed to the potential for this when negative numbers are introduced and then when rational numbers like a 1/2 are included. I'm reasonably certain that the majority of mathematics teachers in the UK at junior school don't even realise that there is a fundamental shift from counting things to the abstraction of describing the location of points on a line taking place. They talk about the "number line" without getting that it is a major leap.
The next leap takes place when you want to describe points on a plane. A single number isn't enough and a coordinate system is just slipped in without the fanfare it deserves. There are multiple ways to set up the coordinate system for a plane and relationships between them, this is usually not even mentioned
Mathematics is taught in little boxes and the chance to have pupils think on a grander scale is missed. By the time students are 17 or older it is now quite hard to not think in little boxes. Counting has become automatic, ingrained, second nature.
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Is a plane geometry potentially useful? If yes how do we make use of it? This is the same as asking how do you perform calculations upon points on a plane . Answering these questions will inevitably lead to the invention/discovery of root(-1) as a valuable idea. That it can be used to glue together different areas of mathematical thinking into something both simpler and more grand is what makes it beautiful.
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JMO but much of the problem with the understanding of mathematics is that it is taught in compartments often by teachers with very little training in mathematics. I was shocked to find out that there wasn't a single teacher in any of the schools that my sons attended up to the age of 11 that had any formal qualification in mathematics, other than the exams they took when they were pupils themselves, sciences were similarly absent. Given the latter it isn't at all surprising that very big shifts in mathematical thinking are often unexplored. unexplored if only because of the needs of the curriculum and its boxes of skills that pupils have to learn without any requirement to understand. That those really big shifts are often very subtle doesn't help either.
counting --> points on a line --> points in space.
I think I was in my very late teens when I started to truely let go of counting and to treat numbers as abstractions. I'd been exposed to imaginary numbers one way and another for around 6 years at that point. Now I look back and can't really think why I held on to counting for so long.
The next leap takes place when you want to describe points on a plane. A single number isn't enough and a coordinate system is just slipped in without the fanfare it deserves. There are multiple ways to set up the coordinate system for a plane and relationships between them, this is usually not even mentioned
Mathematics is taught in little boxes and the chance to have pupils think on a grander scale is missed. By the time students are 17 or older it is now quite hard to not think in little boxes. Counting has become automatic, ingrained, second nature.
===
Is a plane geometry potentially useful? If yes how do we make use of it? This is the same as asking how do you perform calculations upon points on a plane . Answering these questions will inevitably lead to the invention/discovery of root(-1) as a valuable idea. That it can be used to glue together different areas of mathematical thinking into something both simpler and more grand is what makes it beautiful.
----
JMO but much of the problem with the understanding of mathematics is that it is taught in compartments often by teachers with very little training in mathematics. I was shocked to find out that there wasn't a single teacher in any of the schools that my sons attended up to the age of 11 that had any formal qualification in mathematics, other than the exams they took when they were pupils themselves, sciences were similarly absent. Given the latter it isn't at all surprising that very big shifts in mathematical thinking are often unexplored. unexplored if only because of the needs of the curriculum and its boxes of skills that pupils have to learn without any requirement to understand. That those really big shifts are often very subtle doesn't help either.
counting --> points on a line --> points in space.
I think I was in my very late teens when I started to truely let go of counting and to treat numbers as abstractions. I'd been exposed to imaginary numbers one way and another for around 6 years at that point. Now I look back and can't really think why I held on to counting for so long.
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It actually appears a lot beyond calculus. It is frequently used in engineering and linear algebra and differential equations.
