As a first-time blogger, I'm going to quickly say "hello" to the internet, and then jump right in.
A few weeks ago, my friend (who is not a math student, but who is taking a philosophy of math course) asked me several questions. I decided to post the questions and most of my response. He told me it was enlightening, so I hope it is for you as well.
His questions included: "I know that our number system heavily relies on zero and the concept of a base of ten, but how can zero really be a number? Sure, it can subtract and add like other numbers, but we cannot divide with it, nor does it multiple like other numbers, where if we reverse the multiplication by division, we get the original number. So why does it qualify as a number if it can't do things that all (I'm presuming all) other numbers can? Do mathematicians just ignore these discrepancies or an am I missing something?"
And here was my response:
"Your question opens a GIANT can of worms, some of which could take a very long time to dicuss and explain. I'm up for an extended discussion, so I'll try to describe as much as I can as simply as I can:
To answer your question very succinctly--YES, zero is weird. But it is useful and important because it is weird. One is useful and important as well, because it is weird. Zero and one are the most important numbers because they do not act like the other numbers.
However, your claim that zero should not be counted as a number (that phrasing needs work), is not founded.
The definition of "number" is extremely abstract. I think what you mean is: "should zero be counted as an integer?" In which case, your claim is still unfounded. The true definition of an integer is half a semester's-worth into an introductory set theory class, and I don't remember it off the top of my head (even if I did, it wouldn't make sense at this point). However, I know it has nothing to do with being able to add, subtract, multiply, divide, or do anything else to the item in question. It relies only on a number's set theoretical form, which is complex and which I also don't remember exactly.
In any case, you are correct in that zero belongs in some groups of numbers and does not belong in others. Let's look at a few of these groups (which I should really call sets because groups are something entirely different):
The first set we learn about, and the most natural, is called (oddly enough) the natural numbers. It consists of all the positive integers: (1,2,3,4,....). As you can see, zero does not appear, and does not belong in this set.
This set is represented by a captial boldface N with the boldface part open. (Go to http://www-math.univ-poitiers.
The next set is called the whole numbers. It is the natural numbers and zero (0,1,2,3,4,....). Here, zero does belong. The whole numbers and the natural numbers are used for different things.
This set is represented by the lowercase Greek letter omega (the curly w).
The next set is the integers. It is the natural numbers, their negatives (opposites), and zero (...,-3,-2,-1,0,1,2,3,...).
This set is represented by the open boldface captial Z. (Why "Z"? I have no clue. Maybe because it's a sideways N...)
Ok, that's all very familiar. This probably is too: the next set is the rational numbers. This is all values that can be expressed as fractions (or ratios). It includes all the integers, 1/2, -2/3, 47/1098, -5.7, 3.89485, and many others. However, it does not include numbers like pi, e, or the square root of 2. Those are irrational (they cannot be expressed as fractions). In fact, all non-terminating, non-repeating decimals are irrational.
Rationals are represented by the open boldface capital Q (for quotient) and the irrationals don't have a letter all to themselves.
All of the sets so far are subsets of this next set: the real numbers. This is is simply all the rationals and all the irrationals together. Keep in mind that real is just a word we use to describe these numbers. Try to avoid associating the meaning of the English word "real" with these numbers. As one mathematician put it--"how real can any number be? I mean, when's the last time you stubbed your toe on a seven?"
The symbol for the reals is the capital open boldface R.
Ok now on to the non-familiar. The set of complex numbers is a set that contains all real numbers. It also contains all imaginary numbers. And it contains all combinations of real and imaginary numbers. Recall that i is the symbol for the square root of negative one. Imaginary numbers consist of a real number (besides zero) multiplied by i. So, i, 3i, -9i, 4.5i, pi*i, ei, (5/27)i are all imaginary numbers. Complex numbers consist of a real part, and an imaginary part (which are then added together). In mathematicians' terms, they are of the form a+bi where a and b are real numbers. Either or both of a and b can be zero. Therefore, 0 (0+0i), -3 (-3+0i), 5i (0+5i), 3+5i, pi-ei, and 4.08934759-43.90238098...i are complex numbers. Again, "imaginary" and "complex" are just words. Their English meanings have nothing to do with what they are about--imaginary numbers exist just as much as real numbers and are just as (if not more) useful. Think of it this way: there was a time when 0 was considered nonsense, and a time when negative numbers were considered nonsense. Now they are both vital to our society.
The symbol for the complex numbers is the capital open boldface C.
So why did I tell you all of this? Well, the members of each of these sets act differently when in the context of that set. 3 acts differently as a natural number from the way it acts as a whole number, and from the way it acts as an integer. For example, as a natural number, you cannot subtract 3 from 3. What would you get? 0 is not an option. As a whole number, you cannot subtract 5 from 3. -2 is not an option. As an integer, you cannot divide 3 by 5. 0.6 is not an option. So, when you hear that you cannot divide by 0, that's true, but it's not the only restriction that seems unreasonable. It just depends on your focus (which set you are examining or using). Side note: you still cannot divide by zero in the complex numbers. :( However, you may be able to define a set in which you can divide by zero--I don't know, I've never tried."
In this blog I hope to explore many areas of mathematics and life in general. I hope to post things that cover the very basics (even more basic than the above, possibly), and the very complex (possibly some challenging homework problems, or my research). I may also include stories of life as a graduate student, and information about computer science or LaTeX (if you think this is that stuff balloons are made out of, then you're probably not a mathematician, but you are welcome here anyway!), among other things. I also will aim to keep the atmosphere light, and even humorous where possible, because I think that's often why people are turned off by math: "It's so boring!" Well, I hope to show that math (and mathematicians) can be exciting, enlightening, and yes, even entertaining.
Happy mathing!
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