Ask a Mathematician!

[Leoreth]

I have a feeling I won't understand the answer anyway, but how does the concept of square-rooting negative one make sense, when any number squared is positive? In other words, how CAN the square root of negative one exist when it breaks other mathematical rules?

The "any number squared is positive" rule is just something high school teachers like to tell their students because they know they'll never reach the point where this statement becomes false in high school, and those that will study math later on will get by anyway.

It's like telling children in elementary schools that you can't divide 3 by 2 without rest because they don't know rational numbers.

Now "sense" is of course a completely different thing. For mathematicians, sense is secondary, if they can define a set of numbers that allow x² = -1 etc. to have a solution and everything's consistent, they're satisfied. If it has a practical purpose is another issue, but as it turned out in physics, in certain fields there's actually a meaning behind imaginary numbers.

 

[Atticus]

Imaginary numbers are a bit like negative integers: When you were kid and first heard of them, you thought that they're stupid, but when you get used to them, they're perfectly ok. You can also use term "complex number", which is prehaps a little better.

The history behind them is that in middle ages or renaissance some dudes used them like variables in their calculations, but discarded them in the answers. So when they were calculating, they just named this nasty thing i, and later on it could vanish and only real numbers remain. Then they checked if their solution was right, and it was. So you could think it as a nice technique which is ill found however.

When people had been doing this for sometime, some of hem started to use it like a regular number, and if it can be consistently used, why not call it a number? Why would be it any worse than irrational numbers. (Which have similarly funny name, and which too were at least problematic until 19th century).

Then there's the geometric representation of complex (imaginary) numbers: You can think real numbers as a real line, but you can think of complex numbers as a plane. The real numbers is one axis of that plane and imaginary numbers other. I won't go into detail, but wikipedia probably has something about it, or someone else might want to.

Many other areas of maths have evolved the same way, calculus for example, or people integrating "delta function" (which is not a function in the usual sense at ll, and thus can not be integrated). The job of mathematicians is to try and make them well found.

 

[Mise]

What I've learnt about imaginary numbers:

Everybody conceptualises imaginary numbers differently.

 

[Snorrius]

Complex and quoternions was found to be useful in a number of applications but what about octonions, sedenions and higher numbers?

 

[Leoreth]

I've once read an article which was about utilizing octonions to simplify something with multidimensional string theory. Half of it flew over my head when I read it and I've forgotten the rest by now, though

 

[SS-18 ICBM]

Well, what base a number is in doesn't actually affect mathematics that much, especially since a number in base 10 is still just as valid a number as a number in base 12! However, based purely on convenience, it would have been a lot nicer for civilization if we had developed using a base 12 number system, yes. Much more convenient to have that many divisors. (This is actually one of the reasons why the Babylonian's base 60 is so useful: it has divisors 2, 3, 4, 5, AND 6!) Alas, God/Nature/Evolution gave us ten fingers, and humans love to use their fingers for things.

Do you think it's worth changing now, though? Or would it be too much cost for not enough return?

 

[gangleri2001]

What's your definition of natural number? Do you call the number 0 a natural number or not?

And another question. What's in you opinion the most comfortable algorithm to multiply manually? I know of pretty comfortable algorithms to add, substract and divide manually but multiplication is a hell of a operation if you want to do it manually.

 

[GhostWriter16]

The "any number squared is positive" rule is just something high school teachers like to tell their students because they know they'll never reach the point where this statement becomes false in high school, and those that will study math later on will get by anyway.

It's like telling children in elementary schools that you can't divide 3 by 2 without rest because they don't know rational numbers.

Now "sense" is of course a completely different thing. For mathematicians, sense is secondary, if they can define a set of numbers that allow x² = -1 etc. to have a solution and everything's consistent, they're satisfied. If it has a practical purpose is another issue, but as it turned out in physics, in certain fields there's actually a meaning behind imaginary numbers.

Fair enough. I guess it doesn't matter much to me then since I don't want to study math!

 

[IdiotsOpposite]

What's your definition of natural number? Do you call the number 0 a natural number or not?

My definition of a natural number is the standard one, i.e. the set of integers greater than zero. Since zero isn't greater than zero, it's not a natural number.

And another question. What's in you opinion the most comfortable algorithm to multiply manually? I know of pretty comfortable algorithms to add, substract and divide manually but multiplication is a hell of a operation if you want to do it manually.

Hm... well, for relatively small numbers, i.e. numbers less than 100, I find it pretty easily to separate them into tens and ones and then use (a+b)(c+d)=ac+bc+ad+bd. For example, if I were to multiply 37 and 73, I would do something like this:

(30+7)(70+3) = 2100+490+90+21

Then just add the four together. Of course, this gets really difficult for larger numbers. If you're multiplying two three-digit numbers or bigger, I'd probably pull out a calculator at that point.

 

[ParadigmShifter]

Huh? Just do long multiplication... It's not hard.

I'm with you on the natural numbers being 1, 2, ... though. Dutchfire disagrees!

 

[IdiotsOpposite]

Huh? Just do long multiplication... It's not hard.

I'm with you on the natural numbers being 1, 2, ... though. Dutchfire disagrees!

Just because it's not hard doesn't make it not tedious.

 

[ParadigmShifter]

Well, do successive doubling and addition then! Much more exciting. Convert one of the numbers into binary and work it through.

EXAMPLE: 1536 * 11

11 is 8 + 2 + 1 in binary.

1536 * 1 = 1536
1536 * 2 = 3072
1536 * 4 = 6144
1536 * 8 = 12288

so 1536 * 11 = 1536 + 3072 + 12288 = 4608 + 12288 = 16896.

And checked with a calc - right first time

EDIT: Although 15360 + 1536 is probably easier

 

[Mise]

I use my phone.

EDIT: And yeah, timesing by 11 doesn't need any special methods

 

[ParadigmShifter]

Well it's completely general, I just used 11 as an example. Was going to use 7 but that's 1 + 2 + 4 so doesn't involve any redundant calculation.

 

[gangleri2001]

Well, do successive doubling and addition then! Much more exciting. Convert one of the numbers into binary and work it through.

EXAMPLE: 1536 * 11

11 is 8 + 2 + 1 in binary.

1536 * 1 = 1536
1536 * 2 = 3072
1536 * 4 = 6144
1536 * 8 = 12288

so 1536 * 11 = 1536 + 3072 + 12288 = 4608 + 12288 = 16896.

And checked with a calc - right first time

EDIT: Although 15360 + 1536 is probably easier

You missed the point. I was asking for a comfortable algorithm to multiply manually. This algorithm is not comfortable when both factors are pretty large numbers. Do you understand? In other words, what algorithm would you use to solve the multiplication such as 27711.84 * 204929.25 manually?

 

[Atticus]

In other words, what algorithm would you use to solve the multiplication such as 27711.84 * 204929.25 manually?

Write one line under the other like this:
027711.84
204929.25
========
And then start multiplying the upper row with the leftmost digit in the lower row: 4*5=20, write 0 down, and add 2 to the next multiplication, 5*8+2. Write 2 again down and add 4 to the next multiplication 5*1 etc.

The first digit you write down this way goes under the leftmost 5, and you fill from right to left.

Then you start multiplying the upper row with the penultimate digit of the lower row, 4, and start filling from right to left.

At the end you add all the rows you write down, and move the decimal point 2+2=4 left from the end of the sum, and there's the result.

They taught us this when we were 7 or 8 or something like that, and I don't suppose there is any method more comfortable that would be worth learning.

 

[ParadigmShifter]

Yeah, long multiplication, I already said that was easy. Apparently it's tedious as well

 

[gangleri2001]

So no way to get rid of the long multiplicaiton?

PS: You suck at algorithmics CFC.

 

[Atticus]

I don't think it's tedious. Just some little calculating. If it's too bothersome, then use a calculator.

 

[ParadigmShifter]

Using a calculator is the easy way to get rid of long multiplication.

Most short cut algorithms depend on nice properties of one of the numbers.

EDIT: If there was an easier way, don't you think they'd teach you it in school?