This generation has a certain wonderment, confusion, and disbelief regarding imaginary numbers. I have found it very helpful to realize that in previous centuries, the same feelings prevailed regarding negative numbers. No one really understood or believed that you could have less than zero of a thing. (Zero was once in this category too, but let’s not get too far afield.)
Yet, by redefining a number less than zero as a debt, it suddenly becomes a bit more understandable to the layman, and incredibly more useful to the accountant (as Andy_L wrote above).
Similarly, someone once tried to explain imaginary and complex numbers by analogy to a 2-dimensional grid, where one axis is for numbers and the other axis is for “something else”. If you consider “3i + 2” as another way of describing the point “(3,2)”, then it becomes a little more clear. (I still don’t really get that, to be honest, but I’m not giving up yet.)
One thing to note – “never repeat” just means that pi doesn’t turn into a series of infinitely repeating sequences , the way 1/7 = 0.142857142857142857, for instance, with that “142857” being repeated over and over again into infinity. Certainly individual numbers repeat over and over (“1” appears twice in the first four digits), and so do sequences of 2 numbers (“79” appears three times in the first 100 digits), and so on. I believe it’s been shown that any arbitrary sequence of numbers appears in pi, eventually. That would include sections containing, say, “2323232323232323”, but that still doesn’t violate the claim that the digits in pi “never repeat”, even though we would colloquially say that the pattern was repeating.
In fact, nearly all of them are non-repeating. And by nearly all, I mean for all intents and purposes, all. If you choose a random point on the number line, the chance that it is not irrational is as close to 0 as to be 0.
And they form complex numbers. For most people, regular numbers are complex enough, so hearing that these other numbers are labeled as complex makes many just check out.
For that matter, the term “irrational numbers” might make people think they’re illogical or don’t make sense. But rational numbers are just numbers that are ratios (of integers), and irrational numbers aren’t.
When you ask what do we need all those digits for, what do you mean? Whether or not we need them for some task seems irrelevant as to whether or not they exist.
I know someone that asks what the purpose of mosquitos is. Like it can’t exist without serving a (human centric) purpose.
First of all complex numbers are complex as in two houses sharing a wall and you insert a door in the wall you get a complex. They are not complex as complicated things as the are actually very simple.
Like it’s already written here a complex plane is an extension of a real line where multiplying with i means turn 90 degrees counterclockwise. It is nothing more and as such it’s very simple.
The name imaginary number comes from the idea that solving a third degree polynomial equation that has three different real roots one must imagine the steps how to calculate them algebraicly as one cannot do that within real numbers.
Today we say that imaginary numbers lie 90 degrees removed from the real line. Nothing more imaginable than the real line itself which it’s points with no dimension is as imaginary as anything imaginary numeber line is.
If you want it there’s a whole part of district mathematics where you never meet as much as a rational number. Everything can be expressed in integers. The theory of differences is as large area as is calculus and can fullfill the same need when discreet equations are about. And they are: computers anyone.
There is a curious question about whether “all those numbers” even exist. A real number is just a thing in and of itself. We can do our usual thing of working out ways of classifying them in different ways, and one of the ways we can classify them is to see which of them can be constructed as a ratio of two integers. One could view that as just an arbitrary box and label we decide on. Then someone decides that there is another box and label, those numbers that can be constructed with a ratio of two integers, where the second integer used is divisible by ten and only ten. What a funny idea! Makes no difference to the numbers themselves what box we decide to put them in. They are still just numbers.
Doing mathematics that involves real numbers we usually never worry about such things either. Mathematics tends to worry about symbolic manipulation of expressions and application of logic to abstract relationships between numbers and expressions. Some of those symbols represent numbers in the abstract, sometimes we need to place constraints on the numbers, but very rarely would we ever impose a constraint involving which of those odd little boxes we categorise the numbers into.
Eventually we might actually need to do some arithmetic to evaluate our mathematics to yield a concrete result. Maybe we are building a bridge, or working out our tax. So we substitute some concrete numbers into our mathematics and calculate a result. It is only at this point that we might discover that mundane issues about the representation of our numbers matters. Our formula might contain nasty looking numbers like e or \pi as well as perfectly sensible numbers like \small{1\over2}. In reality all we care about is that nothing in or final evaluation of the answer results in nasty ill-conditioned behaviour and that we evaluate the result to enough precision for the task at hand.
The number or decimal places involved in representing the number are immaterial. Nothing affects the actual mathematics, not the real world application of the mathematics.
From a philosophical point of view it is important, as understanding the nature of the various categories of numbers and the manner of constructing them (or not) forms a part of the philosophical underpinnings of mathematics, with all its successes and failures. The discovery of irrationals exposed a flaw in an early philosophy of mathematics, which was clearly important. But if all you care about is whether your bridge falls down or not, you just don’t care, as it is meaningless to you. Not just unimportant.
Not quite. It’s true that Cardano’s cubic solution implied their existence but Cardano didn’t really understand that.
It was Descartes who later coined the “imaginary” as he deliberately wanted to be derisive about a concept that, to him, was silly. Unfortunately, it stuck.
Descartes was a clever man, but some of his ideas have done an awful lot of harm. What you say, for instance, or his claim that animals have no feelings, only reflexes, so you may torture them without consequences.
Yep. The positional notation (with a zero digit) was a great advance for applied mathematics–especially the people doing the arithmetic. But it has done a disservice to people understanding the concept of what a number is. Many people think the representation is the number.
Questions about repeating or non-repeating or this or that property of the decimal notation of a number are curiosities. I do find them interesting, but it’s nothing fundamental about the numbers themselves.
I’ll also note that multi-component entities like vectors, matrices, tensors, etc are all numbers as well. It’s commonly understood that complex numbers are isomorphic to a subset of matrices. And just as we don’t need an isomorphism from natural numbers to real numbers to call real numbers “numbers”, we don’t need an isomorphism from natural numbers to matrices to call them numbers.
That is not what it “is”. I mean, it’s not wrong, but it requires proof and kind of misses the main point, which is: you start with the real numbers, and adjoin a new element i satisfying i^2=-1 (it is not a real number because the square of any real number cannot be negative). Seems not too interesting, but now you can prove that ANY equation of the form
This doesn’t actually work (probably isn’t well-defined). All repeating numbers repeat with a certain period (1 digit, 17 digits, 1328348734 digits). That period can be expressed as a positive integer. So the number “closest to never repeating” is one with “the longest” period, but there’s no largest positive integer, so there’s no longest period, and thus no number that’s “closest to not repeating”.
But let’s assume for a moment that there is a largest integer, n, and you have a repeating decimal with period n. What do you get when you change a single digit in a repeating decimal with an n-digit period? A repeating decimal with an n-digit period.
For example, here’s a repeating decimal with a 2-digit period: 0.1515151515…
What happens if we change one of those digits? Here’s another repeating decimal with a 2-digit period:
It’s a suggested “game” for someone how has a hard time understanding how irrational numbers never repeat. It’s mathematically badly defined because it’s intended to be.
As the rest of your post points out it is also wrong though.
It is virtually certain that if you look far enough into the decimal expansion of \pi you will find 314159265359 repeating a googleplex of time but then the next digit is a 4. And this behavior will also be repeated a googleplex of times. But there is no repeating forever.
