Well actually it’s the other way around. The reason imaginary numbers were invented was to solve a problem we’d been crying over for centuries.
Then, as in most cases, solving one problem opens the door to millions of other problems like why in the fuck does the universe use these imaginary numbers we made up to solve cube roots?
Why is i a core part of the unit circle with like ei*pi ? “Oh that’s because i is just perpendicular to the real number line” ?! Say that sentence again, how the fuck did we go from throwing sharp sticks to utterly deranged sentences like that? More importantly why do utterly deranged sentences like that accurately describe our universe and what is the next ludicrous math concept we’re going to discover is integral to the function of the universe?
how the fuck did we go from throwing sharp sticks to utterly deranged sentences like that? More importantly why do utterly deranged sentences like that accurately describe our universe and what is the next ludicrous math concept we’re going to discover is integral to the function of the universe?
You need 3Blue1Brown in your life.
Imaginary is a poor name for them because it implies they dont exist. Complex numbers[/component] is more appropriate
Imaginary numbers are any number multiplied by the square root of negative 1, a complex number has both a real and imaginary part. 10+i is complex, 10*i is imaginary.
the name seems to be an unfortunate choice that stems from their historical usage as “a means to an end”. i.e, they were first used as part of a method to find some solutions to cubic equations. this method would require algebraic manipulations of complex numbers, but the ultimate goal was to discover a real root. the complex roots would be discarded once a real root was found (if it existed).
the wikipedia article attributes the name to Descartes:
… sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine.
which i think helps to highlight how skeptical the people at that time were about the existence of the “imaginary” numbers.
source: memories of my first complex analysis class, and https://en.wikipedia.org/wiki/Complex_number#History
i’d strongly recommend reading the history section of that wikipedia page to anyone interested in the topic, it has some pretty fun history
The Italian Bombelli in 1572 seemed to toy with both concepts but called imaginary numbers “quantità silvestri” (silvestri meaning ‘wild’) and complex numbers “numeri complessi”. Interesting the imaginary is a quantity and the complex is a number, but maybe old Italian didn’t have that distinction.
I suppose Descartes would agree with you, he first coined the term “imaginary” because he didn’t think they’d serve much purpose. Euler made use of them and continued using the term. Complex number is a complex - a number with a real and an imaginary component.
Imaginary numbers are as real as negative numbers.
Fine, then you figure out what the square root of a negative number actually is!
https://www.wolframalpha.com/input?i=sqrt(-25)
No idea why it doesn’t just say 5i lol.
Calculators also say that dividing by 0 is an error, but logic says that the answer is infinite. (If i recall, it’s more correctly ‘undefined’, but I’m years out of math classes now.)
That is, as you divide a number by a smaller and smaller number, the product increases. 1/.1=10, 1/.01=100, 1/.001=1000, etc. As the denominator approaches 0, the product approaches infinity. But you can’t quantify infinity per se, which results in an undefined error.
If someone that’s a mathematician wants to explain this correctly, I’m all ears.
Serious question because I am math-challenged.
What things are we able to quantify by finding the square root of a negative number aside from square roots of negative numbers?
Electrical engineers use them for calculating AC-circuits. In a DC circuit, you only have to worry about how much volt and amperes are in each part of the circuit. In an AC circuit, you also have to worry about the phase, cause the voltage goes up and down. The phase means where in that up and down you are.
The complex number is interpreted as a point on a 2-dimensional plane; the complex plane. You have the “normal” number as 1 axis, and orthogonal to that the imaginary axis. The angle of the vector to that point gives the phase.
They can be generally used for such “wavy” (ie periodical) processes. But I think this particular field of electrical engineering is the main application.
Equations. When we model things with equations, sometimes they don’t have a ‘solution’ at a particular place, unless we use the formal math rules of ‘imaginary’ numbers like i. Someone else in the comments mentioned electric conductance/resistance in circuits as an example.
Hate to break it to you, but all numbers are imaginary.
I’m not sure I agree with this… it’s like if no one is around to hear a tree fall, does it make a sound?
All numbers are imaginary
They are a logistical concept, invented with the purpose of counting and calculating quantities.
Strictly speaking “Math” doesn’t exist in nature.
Circles are not round because of Pi. A triangle’s sides are not consistent because of the Pythagorean theorem. A thrown ball doesn’t travel in a parabola because of Algebra.
Math is a tool CREATED to understand natural phenomena. Though its logistical power is so strong that it can be stretched to understand almost everything that can be measured.
If a tree falls in the woods it vibrates the air at an audible frequence. Your ears absorb the vibrations and send a signal to your brain that we understand as sound. But the tree never makes a sound. The tree exists and interacts with the environment. Your brain interprets some of those interactions as sound.
You can think of the numbers as the sound. You can understand them clearly, but they’re just an interpretation of a natural phenomena.
But the tree never makes a sound.
That depends on how you define ‘sound’. If it’s only perception and interpretation that creates sound, then sure, a tree falling with nothing to hear or perceive it makes no sound. But if you label sound as the vibration created independent of the perception of the phenomena, then sound happens regardless of whether it’s perceived or not. Since we label some sounds as imperceptible, or outside of human hearing ranges, my interpretation would be that the phenomena is the sound, rather than the perception of it.
I always thought the tree falling in the forest thing was an inpenetrable koan from the depth of ancient philosophy, but it’s actually a pretty simple tool to highlight the difference between sound as a physical thing and sound as perception, two related but different concepts for which we only use one word, hence the confusion.
https://m.youtube.com/watch?v=ymGt7I4Yn3k
Its a bit long but I think its the best explanation of the Russel paradox. Essentially, numbers really aren’t what we think they are.
Just thought it might be interesting to someone at least.
