Does -1 (minus one, negative one) have a square root?
I’ve heard some hype over T.V. trivia shows, and in one of the radio promos in the Phladelphia area, they play a sound byte of one of the questions:
“What is the square root
of -1?”
I’ve seen it represented as “i”, but is there a square root of -1 ???
Help! 
Yes. You heard right, it is called ‘i’.
i stands (i’s had all i can stands, and i can’t stands no more!) for “imaginary” for want of a better term. It’s called that not because it doesn’t really exist, but because all numbers that don’t involve even roots of negative numbers are called “real” numbers, and since it’s not “real” it must be “imaginary”. Mathematicians have a really weird sense of humor.
If you’re curious, i has a square root, too:
sqrt(2)
You have the correct answer in your OP: the square root of -1 is i, better known as an “imaginary number” (more specifically, the imaginary number. All others are the product of i and some other numbers). And, just like 2 and -2 are the square root of 4, both i and -i are the square root of -1.
Mathematically speaking, then, the square root of any negative number is the same as the square root of its positive counterpart times i; e.g., the square root of -9 is 3i.
There’s a pretty good site on imaginary numbers here.
Negative numbers have no square roots among the real numbers. We must extend the definition of number to allow square roots of negative numbers. We begin by supposing that the square root of negative one exists and we give it a name, i If a is a real number, we call a number of the form a**i* imaginary.
Complex numbers are numbers of the form a + b**i*, where a and b are real numbers.
Also if i[sup]2[/sup] = -1, then (-i)[sup]2[/sup] = -1. So, i and -i are both square roots of -1, just as 2 and -2 are both square roots of 4.
Extending the definition of number is common. The first numbers used are natural numbers or counting numbers. The negative integers can be defined as an extention. Rational and irrational numbers are also extentions of the concept of number. Complex numbers are just the next step in this progression. Which system of numbers you use depends upon your application. If you are counting things, for example, you need no more that the natural numbers.
The answer to the question “Does -1 have a square root?” depends upon whether you consider only the reals or the complex numbers. If you are only considering the integers, the square root of 2 does not exist.
pldennison, Great site!
Thank you!
And if you want to know about the uses of complex numbers (that’s numbers with real & imaginary parts) there was a thread a while back, here.
Also, in response to what Joe_Cool says is imaginative nomenclature by mathematicians, it turns out that in some cases, real parts correspond to physical (real) effects, and imaginary parts don’t. So there is some rationale for calling them real and imaginary.
“Does -1 (minus one, negative one) have a square root?”
Only if you want it to, baby.
It’s true that certain mathematical systems have defined a term known as “i” such that i * i = -1. But it’s purely a matter of definition. You can set up a perfectly useful mathematical system where -1 has no square root at all.
Like language, math is our servant, not our master
I’m making a retraction of some of what I said in my previous post, in favor of the following :
Historically, it was the mathematicians and their imagination who came up with the term “imaginary” for these numbers.
In light of what we know today (applications & such) there is sufficient rationale to retain the label. It’s my opinion that it’s not merely coincidence, but I’m not going to get into it. Even if it did begin as a pure accident, “imaginary” currently makes sense for the square root of -1.
Also realize that ‘i’ is not the only designation. Among electrical engineers, sqrt(-1)=j. ‘j’ is always used, I assume since “i” is usually used to mean current.
Jman
I don’t have any references handy, but I can give you the general idea of some of the history of i.
I think around the 16th century or so there was a lot of effort put into solving cubic equations. Cardano and Tartaglia were a couple of the important people involved in this, it’s a really interesting story, worth looking up if you’re into that sort of thing, but I can’t remember exactly which of them did what offhand.
Anyway, specifically about i, one of them was using a formula for solving cubics and noticed that in some cases you had to take square roots of negative numbers as an intermediate step. The surprising thing at the time was that even though you couldn’t take square roots of negative numbers, the formula, even after those intermediate steps, still spit out the correct roots of the cubic polynomial! He (Tartaglia or Cardano, I can’t remember which) referred to these square roots as “fictitious numbers” (in Italian, but I don’t speak Italian). Nobody believed in the existence of such numbers, but it was certainly strange how they seemed to pop up in the calculations without screwing things up. He struggled to understand what was going on, but it wasn’t until much later when people like Euler really figured out the nature of the complex numbers. Still, the idea of them being “fictitious numbers” sticks with us, and we now, of course, know them as “imaginary numbers”.
Of course it’s still perfectly reasonable, because something that is not “real” has to be imaginary. I’m not trying to be a smartass. All real numbers can represent real, quantifiable “somethings” in our perception of the universe. Imaginary numbers cannot represent anything quantifiable (which is NOT to say they don’t have their uses).
The lowercase “i” usually means the imaginary number, but Jman’s right about the use of “j” in electricity. However, to represent current, we typically use the uppercase “I” so there would be no confusion.
The imaginary number is super important in electricty, but it always eventually resolves back to a real number when the problem is solved.
Well, in my experience (I’m a senior EE major), we usually use I for equations, but sometimes use the lowercase i when marking direction of current on a circuit diagram. Depends on who’s at the board really.
Anyway, complex numbers are just an everyday part of life…they help solve a lot of problems that can’t be solved using only reals.
Jman
Just a couple of comments…
There are some puritanical mathematicians who will say that although i^2 is defined as -1, it’s still meaningless to speak of the square root of -1. That is, it’s OK to say that something squared is negative but not OK to say that something negative has a square root. Semantics, not mathematics, but I can see their point occasionally.
Also, there are times when both the real and imaginary parts of a solution have physical meaning; for instance, in the complex solution for vibrations of an elastic material with damping. The imaginary part represents out-of-phase motion, the part that is lost as heat. Complex numbers can be thought of as just another kind of two-dimensional vector, and it’s familiar how both dimensions of the vector can have physical meaning.
A googel is 10[sup]100[/sup]. A googleplex is 10[sup]google[/sup]. They are real numbers, but I doubt you could argue that they “represent real, quntifiable ‘somethings’ in our perception of the universe”.
Which number system you use depends upon your application. It makes no sense, for example, to use negative numbers if you are measuring distance. If you are counting your chickens before they are hatched, you only need non-negative ingegers. The complex number system is just as “real” as the real number system.
Virtually yours,
DrMatrix - i am imaginary.
A slight correction:
The correct spelling is googol-googolplex.
For all you folks to whom the name rings a bell but don’t remember where you heard it, here is a tip: Back to the Future III. Specifically, the bar scene after Clara dumped Doc Brown and he is lamenting his bad fortune. While he is babbling gibberish to the local drunks he says in reference to Clara:
“She is one in a million, one in a googolplex…”
BTW, Dr.Matrix, very appropriate sig.
Googol/Googolplex first appeared in Mathematics and the Imagination by Edward Kasner and James R. Newman (from the 40’s, I believe). It had been coined by the young niece or nephew of one of the authors.
Often, lowercase i is used for AC, and uppercase I is used for DC current, especially if both are present in a circuit.
In some AC analyses, both real and imaginary components represent tangible quantities- in-phase and quadrature components, real and reactive power, etc. Complex numbers are often used two represent two-dimensional quantities, regardless of the “realness” of things.
Arjuna34
Thanks for the correction. I was working from memory.
Gosh, thanks.
Virtually yours,
DrMatrix - i am imaginary
Had a question on a math test last week.
Draw the graph of y=[sqrt(x)]^4
Remembering this thread, I simplified it to x^2, and drew the parabole. Got it wrong. But I figured since I got 97% on the test (that was my only “mistake”) and I also got a crap load of extra points for drawing 1/f(x) type questions, I decided it wasn’t worth it.
I hate my math teacher.
jbird3000: Had a question on a math test last week.
Draw the graph of y=[sqrt(x)]^4
jbird3000, unless your teacher specifically stated x>0, I think you have a case for a re-grade. The fact that a complex number appears in an intermediate step when x is negative is just a computational detail that can be ignored for the exponents that you were given.
Put more formally, the identity you used,
(x^a)^b = x^(ab),
is not true for arbitrary a and b (for instance, it’s not true for (a,b)=(0.5,0.5)), but you don’t need this to draw your parabola, you only need it to be true for the special case (a,b)=(0.5,4), which it is indeed. Therefore, there’s nothing wrong with writing
[sqrt(x)]^4=x^2.
As a former teacher, I’d normally discourage you from contesting your grade, but in this case, it looks like your teacher is trying to trick you but isn’t quite smart enough to do it correctly. In other words, your teacher is what is known to mathematicians as a “dickhead”.
I think he was trying to trick my “dimmer” classmates who are afraid of any power higher than 3 or lower than 1.
I’ve gotten tired of arguing with teachers over the years. I just let anything slide now…
Tony Hawkins in a renowned scientist…
The sun is a burning ball of gas…
An impair function is one that’s not pair…
The list goes on. I’d rather do my own thing than listen to him, because he tends to confuse me.