How did imaginary numbers come into existence? What was the first problem that required use of imaginary number?

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How did imaginary numbers come into existence? What was the first problem that required use of imaginary number?

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Anonymous 0 Comments

The same problems that they are used for now: square roots of negative numbers.

There are many problems in physics where you do have to ha square root of a negative number as part of the process of finding out the answer even if that itself will never ultimately be the answer.

Anonymous 0 Comments

This is a fascinating subject, and it involves a story of intrigue, duplicity, death and betrayal in medieval Europe. Imaginary numbers appeared in efforts to solve cubic equations hundreds of years ago (equations with cubic terms like x^3). Nearly all mathematicians who encountered problems that seemed to require using imaginary numbers dismissed those solutions as nonsensical. A literal handful however, followed the math to where it led, and developed solutions that required the use of imaginary numbers. Over time, mathematicians and physicists discovered (uncovered?) more and more real world applications where the use of imaginary numbers was the best (and often only) way to complete complex calculations. The universe seems to incorporate imaginary numbers into its operations. This (https://www.youtube.com/watch?v=cUzklzVXJwo) does an excellent job telling the story of how imaginary numbers entered the mathematical lexicon.

Anonymous 0 Comments

Solving cubics.

The guy credited with initially developing imaginary numbers was Gerolamo Cardano, a 16th century Italian mathematician (and doctor, chemist, astronomer, scientist). He was one of the big developers of algebra and a pioneer of negative numbers. He also did a lot of work on cubic and quartic equations.

Working with negative numbers, and with cubics, he found he needed a way to deal with negative square roots, so acknowledged the existence of imaginary numbers but didn’t really do anything with them or fully understand them, largely dismissing them as useless.

About 30 years after Cardano’s *Ars Magna*, another Italian mathematician Rafael Bombelli published a book just called *L’Algebra*. This was the first book to use some kind of index notation for powers, and also developed some key rules for what we now call complex numbers. He talked about “plus of minus” (what we would call *i*) and “minus of minus” (what we would call *-i*) and set out the rules for addition and multiplication of them in the same way he did for negative numbers.

René Descartes coined the term “imaginary” to refer to these numbers, and other people like Abraham de Moivre and Euler did a bunch of work with them as well.

It is worth emphasising that complex numbers aren’t some radical modern thing; they were developed alongside negative numbers, and were already being used before much of modern algebra was developed (including x^(2) notation).

Anonymous 0 Comments

Historically it came about when people were solving cubic equations, but I prefer the below introductory “lesson”:

Suppose you want to solve a regular, first-degree equation in one variable. For example:

2x + 3 = 7

This is easy to see that you can subtract three, then divide by 2. So x = 2.

In general, this type of equation can always be solved in this way. So equations of the type:

ax + b = c (think of a, b and c as ANY numbers you want)

Yields a simple solution, x = (c – b) / a

So that’s the “first-degree equation”. Now lets advance to the second degree. Equations of this type look like:

ax^2 + bx + c = 0 (now there’s an x^2 term, and for simplicity, I moved the “constant” from the right hand side over to the left, so now it’s incorporated into the value of c).

As it happens, there’s a great solution to this equation as well, and it’s the quadratic formula you’re probably familiar with:

x = [-b +/- sqrt(b^2 – 4ac)] / 2a

A little bit of proof goes into this formula, but it definitely works out nicely and always yields two roots (since squares of negatives are also positive).

However, you can now see a potential problem. Consider the quadratic:

x^2 + 1 = 0

You can apply quadratic formula, but you don’t even really need to because you can still solve it a simpler way, by subtracting 1 from each side and then taking the square root. When you do so, the solution seems to be the positive and negative square root of -1.

Now, here’s where we find out if you’re a mathematician or not. When confronted with this conundrum, you *could* simply say “no number when squared could ever be -1, so thus this equation *has no solutions*”. In fact if you graphed that quadratic on an xy plane, you’d see that it has no x-intercepts, which is essentially the same thing as saying the equation has no solutions.

But some enterprising mathematical minds decided instead to ask the question “but, what if we said it *does* have a solution?” and thus the imaginary number is born.

So the imaginary numbers came about because people wanted to not be restricted by equations like that. In other words, we prefer to live in a world where algebra has all of it’s well-formed equations have solutions. But this requires a set of numbers beyond simply the real numbers, and must include imaginary numbers.

Then of course, in the years to come, many other uses for imaginary (and complex) numbers became apparent. There are a number of interesting applications in physics, electricity/magnetism, quantum physics, etc. and the complex numbers allow us to model certain situations in ways that make the mathematics very easy to work with. So this particular development may have begun as algebrists trying to “force” solutions to equations to exist, but has since developed into a whole new approach for problem-solving.

Anonymous 0 Comments

Rotation. Multiplying with imaginary unit (i) rotates by 90 degrees in 2D complex plane which is very useful because universe is mostly and utterly sinusoidal.

Anonymous 0 Comments

Several commenters have answered when our *understanding* of imaginary numbers were developed. However, the specific phrasing here – when did they come into *existence* – lets us touch on an interesting point in mathematics:

It is currently debated in the philosophy of mathematics if mathematical truths are invented or discovered. That is to say, it’s not clear to us if mathematics are a property of the universe, in which case it is discovered as a branch of science, or if they are a logical construct where mathematics are developed from philosophy ex nihilo.

By that first interpretation, for instance, we would expect that imaginary numbers came into existence with the Big Bang, and were left *undiscovered* until attempts to solve the cubic. While in the second, they didn’t exist until we thought about them.

Anonymous 0 Comments

An imaginary number is just the square root of a negative number. When people were first doing square roots they would they would have realized they existed but just ignored them. It’s worth mentioning that applying problems to the real world you ignore any imaginary parts of the answer.

Anonymous 0 Comments

Well the idea came quite naturally. Imagine youre looking to solve a quadratic equation or a cubic equation. You’ll find that a cubic equation should have 2 solutions. But sometimes you’ll only find one and the other is not solvable because you would have to take the root of a negative number. But there should be a solution says your intution. So you imagine what if there is a number i with i^2=-1. Then you could solve those negative roots. Now build a new set of numbers that includes all real numbers and those created with your newly imagined I. And it should behave like real numbers when there is no i involved. And voila you get complex numbers.

Anonymous 0 Comments

I would like to bring an understanding that has blown my mind recently. For numbered things, you can imagine negative numbers. For real positive numbers, you can define a positive surface (like square meters). For real positive surfaces, you can imagine negative surfaces. Necessarily these negative squared meters are the result of something that squares to negative numbers.

I believe that simple idea/understanding may lead to understand advanced math better.

Anonymous 0 Comments

I’ll just add on to others. These numbers are called imaginary only because they have no physical representation. An imaginary number exists as a concept.

The concept of, say, a dragon has no actual impact on reality without human interaction. Imaginary numbers would affect reality regardless of the presence of humankind. They are only imaginary because we define things as real/imaginary in relation to *our* ability to interact with them.