How did imaginary numbers come into existence? What was the first problem that required use of imaginary number?
In: 760
I think the best way to answer this question is by this fantastic video by Veritasium : [How Imaginary Numbers Were Invented](https://www.youtube.com/watch?v=cUzklzVXJwo&t=7s&pp=ygURaW1hZ2luYXJ5IG51bWJlcnM%3D)
It goes through the entire history and necessity for such a tool in Mathematics. Really fascinating
Not ELI5, but complex numbers are a two-dimensional field consisting of pairs of real numbers with a specific addition and multiplication, and additive identity (0,0) and multiplicative identity (1,0). Using C, the real numbers sit inside them as a special case.
While it is an unordered field it does have most of the other properties of the real numbers such as being “complete”.
You can do calculus on them. And when you do, the exponential function is easily related to the sin and cosine functions in the complex numbers. And the roots of polynomials in C are simple.
Complex numbers allow many of the ‘holes’ in real-number math to be filled in nicely. Just like integers (including 0 and negative numbers) fill in theoretical ‘holes’ if you are only working with natural numbers.
Once you go from N to Z to Q to R to C, I believe most analytical math becomes as elegant as possible. (Not counting out vector spaces and such, or trans-finite stuff.)
[Here](https://www.youtube.com/watch?v=cUzklzVXJwo) Veritasium does formidable job explaining exactly this
The “imaginary” in “imaginary numbers” denotes their perpendicular orientation to real numbers in the complex plane, not their validity or reality.
You mean like onee and thrwo?
I think imaginary numbers are a bit of a misnomer. It’s really just applied algebra, in a sense. Setting a letter equal to something, and working with it.
To solve a quadratic (something that looks like 3x^2 + 2x + 9 = 0) you can use the quadratic formula. The [quadratic formula](https://www.google.com/search?client=firefox-b-d&q=quadratic+formula) is basically just a way of [completing the square](https://www.mathsisfun.com/algebra/completing-square.html) but using ax^2 + bx + c = 0 as the quadratic.
The problem with this though is if you look at the quadratic formula a cannot be 0 (assumption #1). If it were 0 we would be dividing by 2*0 and that’s a no-no. So a > 0. But we can also see b^2 – 4ac is under the square root. In order for it to make sense this must also be greater than or equal to 0 because its impossible to multiply any two numbers together to get a negative square number (assumption #2).
So b^2 -4ac >= 0 or b >= sqrt(4ac). 4ac is under the square root here too we know a > 0 so it must also be true that c >= 0 in order to prevent negative numbers under the root. So the quadratic formula only works for values of a > 0, b >= sqrt(4ac) and c >= 0. But we have a problem… we know that for (a = 1, b = 0, c = -9) or x^2 – 9 = 0 has a solution. Look, add 9 to both sides x^2 = 9, square root both sides x= +/-3. So c must be able to be negative, even though we have just shown it must be c >= 0
So one of our assumptions must have been wrong. It can’t be 1, which leaves 2… the only wrong assumption here is that the value under a square root must be a positive number. It therfor is possible to multiply two of the same numbers together and get a negative square. (here 3i * 3i = -9, or in otherwords sqrt(-9) = 3i)
“Zero: Biography of a Dangerous Idea” is an awesome history on my favorite “imaginary number”. 🙂
This is gonna be long like a history lesson but i’ll explain it the best way i was taught.
In 820AD Mohammad bin Musa Al-Khwarismi made the quadratic formula Ax^2 + Bx + C where a b c are numbers and X is a variable and it proceeded to be used until today to describe many things.
This formula was obtained by Simon Stevin in 1594. In 1637 René Descartes published La Géométrie containing the quadratic formula in the form we know today but it’s the same thing just different syntax.
When newton made calculus we learned how to derive differential equations soon after to represent many phenomena like population growth or mechanics.
Eventually we had something happening that was in the of y”+y’+y= N a lot where y is a variable, y’ is the rate of change of that variable otherwise known as derivative, y” is the rate of change of the rate of change or second derivative and N is some number. For example if a car is moving distance Y, the velocity would be Y’ and acceleration would be Y” or if you’re familiar with an object falling if gravity is acceleration Y” = A, velocity would be Y’= V= A*t + V0 with t being time and V0 being initial velocity and vertical motion down is Y= At^2+V0t+ Y0 with Y0 being the initial distance.
Because we established many ways to solve the quadratic equations since it’s over a thousand years old it was easier for mathematics to convert the differential equation y”+y’+y= 0 to m^2 +m+ 1 = 0. We do this by letting Y = e^mx which means Y’= me^mx,Y”= m^2 e^mx and replacing in the original and dividing by e^mx this is called an axillary form.
So now we turned the hard differential into something we can easily solve but there is a catch quadratic solved by delta sometimes gives us K(-1)^1/2
were K is some number but the (-1)^1/2 doesn’t exist so we just called it “i” for an imaginary thing we don’t know much about. The reason this thing pops up is because the quadratic equation solved by the delta formula that’s ( -b + sqrt(b^2 – 4ac) /2a & ( -b – sqrt(b^2 – 4ac) /2a can give an imaginary value if -4ac is larger than b^2.
In math if we can’t identify something we just say it’s wrong like dividing by zero we just say doesn’t exist but for this we call it imaginary. Why? well because we found a solution…
See take a plate being heated and you wanna measure how it’s being heated we found it’s differential equation
AY”+ BY’+ CY = 0
if say A was 1, B was 0 and C was 4 you get Y” + 4Y= 0 , changing to axillary you get m^2 + 4 = 0 and solving the delta you get two values for m or more easily just take 4 to the other side you just get m = -2i, +2i.
So measured the heat transfer and got C1(cos2x) + C2(sin2x) basically +i and – i became cosine and sine and is the fundamental thing we take as mechanical engineers in differential equations because we work on it more in boundary value problem and it gets more complicated with heat transfer and thermodynamics.
To the average person they won’t ever use it but to engineers working with any turbine blades we care about heat on a plate because the turbine blades are curved pointy plates that we want to cover in ceramic to protect it from the heat because it operates at a temperature high enough to damage the blade but by coating it we can have high pressure turbines what we use to generate power from basically every nuclear and fossil fuel power plant. If the blade was coated too much it’s heavier and will turn slower so we get less power and loss of energy so we need to accurately coat it and that’s done by our friend mister imaginary number that not so imaginary after all.
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