61–70 of 121 posts
Earlier quoted context omitted.
What should I read if I want to learn what a number is?
Learn set theory! Specifically ordinals (just one formal defintion of a number) This vsauce video is actually a really accurate (and entertaining!) introduction to set theory: https://youtu.be/s86-Z-CbaHA
Earlier quoted context omitted.
I think it’s clear you’ve never taught low level mathematics courses. There is a lot of hand waving and brain washing that happens. The vast majority of people don’t know what a number is in a precise, mathematical sense. At the level of the intended audience it would be wholly inappropriate talk about the definition of a number. My background on this topic is that I’ve taught intermediate algebra for over 20 years.
What should I read if I want to learn what a number is?
Earlier quoted context omitted.
I think it’s clear you’ve never taught low level mathematics courses. There is a lot of hand waving and brain washing that happens. The vast majority of people don’t know what a number is in a precise, mathematical sense. At the level of the intended audience it would be wholly inappropriate talk about the definition of a number. My background on this topic is that I’ve taught intermediate algebra for over 20 years.
What should I read if I want to learn what a number is?
We use the label 'number' to refer to a broad swathe of mathematical objects, objects that are different but also so similar they often appear interchangeable (for example counting numbers and fractions).
In the formal mathematical sense, a specific type of number is a group of objects which have been defined to have specific properties. I'm going to leave object and property as defined in the usual sense, I think most people have a good idea about what those are and not sure I can add anything to them.
There are no rules as to what properties you are allowed to give to objects, nor what group of objects you want to include, but generally if you are learning about some specific thing it's because people find them useful or interesting; the definitions we have for different types of numbers are the ones we have found useful or interesting.
Remember that the different types of number appear very similar. It is common to build a 'hierarchy' of differnt types of numbers, where we start with a simple type of number and then add new properties and objects when we find limitations we don't want.
The first type of number in this hierarchy are the natural numbers (sometimes called counting numbers). The most common properties defined for these today are called the Peano axioms [0]. There are quite a few of them, and the history of how we came to the formalisation is very interesting (a lot of it is about avoiding inconsistencies/contradictions) but the key ideas are:
- there is a natural number called 0
- every natural number has a successor, which is also a natural number - we can write S(n) is the successor of n
Most of the other axioms define what it means for two natural numbers to be equal (=).
Just having these objects isn't particularly useful, we typically want to do things like add, multiply, and compare numbers. To do that we include some operations: addition (+), multiplication (*), and total ordering(These are defined as, taking a, b, c as natural numbers:
- a + 0 = a
- a + S(b) = S(a + b) (this is recursive, so if we define 1 as 1:=S(0) then we have 1+1 = 1+S(0) = S(1+0) = S(1))
- a * 0 = 0
- a * S(b) = a + (a * b)
- a Importantly, using these definitions, we can say that the natural numbers are closed under addition and multiplication; whenever you add or multiply two natural numbers together you get another natural number.
To continue building the hierarchy we notice that there are operations we would like to do but are not possible for every natural number (please note I am skipping over the formalisations from hereon and talking about the motivation for different types of numbers).
We notice that if we can add two numbers together we should be able to subtract them again. If a + b = c, then c - b = a. However (for example) 0 - 1 is not a natural number. So we extend the natural numbers to the integers, such that the integers are closed under subtraction.
If we can multiply it makes sense to try and divide, but 2 / 3 is not an integer so we extend integers to the rationals (ratios of integers) which are closed under division. We add an object called 2/3 so that now when we can say 2 / 3 = 2/3.
The next step in the hierarchy is a bit more complex. We notice that we can define a subset of rational numbers that all meet a certain criteria, for example all rational numbers that are less than 2. We call 2 an upper bound of that subset. Notice that 2 is a rational number, and that there are no rational numbers smaller than 2 that are also an upper bound of our subset - 2 is the least upper bound. Define a new subset, where we say a rational number x is in the subset if x * x least upper bound for this set that is a rational number. We extend the rational numbers to include a least upper bound for every subset of rationals, and we call this the real numbers. The real numbers have a lot of nice properties, most notably they are complete under the normal ordering, which essentially means that there are no gaps.
The reals don't have everything though! We notice that we can create polynomial equations, like x * x - 1 = 0 and that sometimes these can be solved (in this case x = 1 or x = -1 solves the equation) and in other cases they can't. For example, there are no real numbers that are the solution to the equation x * x + 1 = 0. We can extend the real numbers to the complex numbers by adding a new object called i, which has the property i * i = -1. A complex number has the form a + b * i, where a and b are real numbers. The complex numbers are called algebraically closed, and there is a really nice result that shows that all polynomials have solutions in the complex numbers.
The hierarchy actually keeps going, but hopefully you can see that numbers are just objects with properties that behave in useful and interesting ways under different operations. The formal definitions we have today have been refined over a long period of time to avoid contradictions and other issues, but there is nothing stopping you from making up your own numbers with their own properties. If they are useful or interesting other people will probably use them too!
Earlier quoted context omitted.
At this level, associating numbers with potentially infinite decimals would not be too bad. The examples of π and √2 are shown as decimals, suggesting that students are already expected to think something like this. I think that this pair of definitions for real and irrational numbers are unacceptable - the examples of irrationals mean that they don't mislead anyone (unless you can find someone who thinks of complex…
> At this level, associating numbers with potentially infinite decimals would not be too bad. I find number with an infinite number of decimals quite an abstract concept. Another issue with this is that it confuses numbers with their representation.
If students are familiar with the idea that fractions can be represented as repeating decimals and are introduced to π as a non-repeating decimal, then I think the idea of the set of all possibly repeating decimals should make sense to them.
> Another issue with this is that it confuses numbers with their representation.
It's probably worth acknowledging that some numbers have multiple representations just to avoid arguments about whether 0.999.. = 1.
However, is it much worse to say that a real number is an equivalence class containing 1 or 2 infinite decimals than to say that a real number is a non-empty, downwards closed, proper subset of the rationals? I agree it's nicer to define the reals as a field with some extra properties, but I'm not convinced it's wrong to say that they are their representation.
Earlier quoted context omitted.
I think it’s clear you’ve never taught low level mathematics courses. There is a lot of hand waving and brain washing that happens. The vast majority of people don’t know what a number is in a precise, mathematical sense. At the level of the intended audience it would be wholly inappropriate talk about the definition of a number. My background on this topic is that I’ve taught intermediate algebra for over 20 years.
What should I read if I want to learn what a number is?
If you don't know what a rational number is, it's the equivalence class of every pair of integers that can be simplified to the same fraction. For example, (4,6) and (2,3) are both rational numbers, and in fact are the same rational number: two thirds.
If you don't know what an integer is, it's the natural numbers, but with negative numbers.
If you don't know what a natural number is, it's either zero, or a number that follows a natural number. For example one is the number that follows the natural number zero, and two is the number that follows the natural number that is the natural number that follows zero.
Earlier quoted context omitted.
Any given irrational number may not be possible to write down. a/b is a rational number. The set of real numbers is the union of rational and irrational number sets i.e. all possible rational and irrational numbers combined form the real number set.
> Any given irrational number may not be possible to write down. There's no "may" about it! By definition an irrational number has an infinitely long decimal expantion that does not repeat, so none of them can be written down (other than in symbolic representations like pi)
Earlier quoted context omitted.
> Any given irrational number may not be possible to write down. There's no "may" about it! By definition an irrational number has an infinitely long decimal expantion that does not repeat, so none of them can be written down (other than in symbolic representations like pi)
In fact, since there are only a countably infinite number of symbolic representations but an uncountable number of irrational numbers, nearly all of them cannot be written down at all!
Earlier quoted context omitted.
I think it’s clear you’ve never taught low level mathematics courses. There is a lot of hand waving and brain washing that happens. The vast majority of people don’t know what a number is in a precise, mathematical sense. At the level of the intended audience it would be wholly inappropriate talk about the definition of a number. My background on this topic is that I’ve taught intermediate algebra for over 20 years.
What should I read if I want to learn what a number is?
Earlier quoted context omitted.
In fact, since there are only a countably infinite number of symbolic representations but an uncountable number of irrational numbers, nearly all of them cannot be written down at all!
If you mean representations using finite sequences, sure. But the set of all infinite sequences (e.g., of digits) is uncountable.