Why use 'i' in calculations?

Question

I know that it stands far an imaginary number, namely the square root of a negative 1, but why is this number needed in calculations if it is imaginary?

Answer 1

It's called Imaginary because it's impossible to calculate its value, since there is no way to mulitply a number by itself, and get a negative answer. There's no such thing as a "half-negative" or whatever. However, in doing quadratic equations, you come across situations a lot where one of the solutions is Imaginary. Mathematics acknowledges their importance, because when you go on beyond just Algebra, and start getting into Physics, Pre-Calculus & Calculus, you will sometimes get an Imaginary value along the way, but when you finish working the problem out to its conclusion, there are no more Imaginary values; they've resolved themselves into Real numbers, or been eliminated. If we didn't allow for Imaginary values, we wouldn't have been able to solve these problems completely.

Answer 2

'Imaginary' is just a name.

A complex number, a number with a 'Real' part and an 'Imaginary' part, is just a number that doesn't lie along a single number line. In other words, it has more than one dimension.

Numbers with one 'real' and one 'imaginary' part can define a two-dimensional vector. The 'real' part just describes the portion that lies on the x-axis, while the imaginary part describes the part that lies on the y-axis. If the number has no 'real' part, then the number lies entirely on the y-axis.

Numbers with a 'real' part of 0 and three 'imaginary' parts define a three-dimensional vector. (William Hamilton, the inventor of 'quaternions', couldn't come up with a set of mathematical rules that would work for a number with one 'real' part and two 'imaginary' parts.)

They could have chosen any name. The choice of 'imaginary' was because 'real' numbers were the name chosen to describe all numbers lying along a single number line (one-dimensional numbers). In fact, choosing 'imaginary' for the name could be said to display a lack of 'imagination' by mathematicians. (hee hee).

Answer 3

First of all.........
the majority of mathematics has been developed out of a need to make calculations needed to design things and predict behavior of things. I often feel rather sad that mathematics is taught without first defining and explaining a requirement for the particular calculation. This always gives rise to the sort of "why" question that you have asked.
Many years ago, I met a mathematician who had been recruited to teach math to engineers. He said that finding "real world" problems for the engineers to practice their maths had opened his eyes to a new world. He had previously taught maths in isolation.

On to your question.......
Numbers are called imaginary because it is not possible to write them down just by using ordinary numerals. A number made up of ordinary numbers and imaginary numbers are called complex numbers and many quantities in engineering are expressed using complex numbers.

I do not want to frighten you but using something called Euler's identity we can express cos(x) as (e^(jx) + e^(-jx))/2
and sin(x) as (e^(jx) - e^(-jx))/2j
Using these you can work out all those nasty trig identities that people try to make you remember.

Euler can also be used to obtain the very simple identity
e^(j.pi) = -1
This is rather nice because it allows us to link e, pi and j

So, j (if you are an engineer) or i (if you are a mathematician) can really make life a lot easier.

If we have a result that gives us a comple number of the form (a +jb), we obviously cannot buy a bit of wood that long. But, recogniaing that a complex number represents a vector quanity whe can calculate a scalar value and lay the wood down in the correct orientation.

Answer 4

The choice of the word 'imaginary' is actually a kinda poor one. The square root of -1 is, quite simply, a different kind of number with some fairly special properties that make it useful in problem solving (especially in vector problems)

It's like you can't express the square root of 2 with a fraction because it's an 'irrational' number (which means it can't be written as a 'rational' quotient of two integers)

And you can't represent the concept of 'two and one half' using just the integers. You have to develop the rational numbers before you can start talking about fractional parts of something.

As mans calculational needs became more and more sophisticated, we had to 'invent' these 'new' kinds of numbers to keep up

BTW, if you think of a+ib (a complex number) as an 'ordered pair' of numbers (a,b) then I'm sure you'll be ecstatic to learn that there are 'ordered quadruplets' of numbers (a,b,c,d) that are refered to as 'hyper-complex' numbers (or, sometimes, 'quaternions')

Spooky, isn't it?


Doug

Answer 5

Another writer already pointed out that, even though numbers related to "i" are called "imaginary," they still have real applications in physics and electrical engineering.

What it comes down to, really, is that the term "imaginary" is a misnomer. Quantities that we refer to in math circles as "imaginary" really do exist -- just not in a way that we can count them. There's a great explanation of the concept at the link below. Hope it helps!

Answer 6

I've asked that same question many times. I really wish I could give you a better answer as the quality of the questions deserves one.

The best I can do is to tell you that the concept of the imaginary number actually solves a number of problems in physics (and in other fields of study). As silly as it may sound, the results are undisputable.

There is another question presently open about "is math perfect?" This is one example of the imperfections of our understanding of math.

Answer 7

After being given documents on the fee and path of the asteroid and the fee and path of the Earth, i replaced into waiting to calculate precisely whilst the asteroid could hit the Earth and blow us all into oblivion.

Answer 8

This is only a guess, but I'd say to represent the concept of a negative number more simply. For example, is it easier to write "sqrt(-1)" or is it easier to write "i"?

Answer 9

It speeds along certain calculations. In my Trig class I could have taken MUCH longer to finish certain equations without i.

Answer 10

hmmm good question, We did a whole thing on 'i' in my Algebra 2 class but I didnt see what it was good for. All I know is that it can become real if you use a power on it.

As I look back all the adition subtraction multiplaction and complex conjugates with did with 'i' seem useless cuz idk what its even for now.