What Are Imaginary Numbers?Date: 7/24/96 at 16:52:29 From: Anonymous Subject: What are Imaginary Numbers? A fellow 6th grade teacher and I were lamenting the amount of math knowledge we have forgotten since our own high school and college educations. Her son is currently having trouble with imaginary numbers. I can't even remember what they are, their purpose, and how to use them. Can you give me a brief refresher summary? Thanks! Date: 7/25/96 at 14:29:48 From: Doctor Pete Subject: Re: What are Imaginary Numbers? Hello, Before I begin, let's define some notation. When used in an equation, * is multiplication, so 4*7 = 28. x^2 means "x squared," so ^ denotes exponentiation. Sqrt[ ] is the square root of the expression in brackets. Imaginary numbers were conceived in response to the question of whether or not we could think about the square root of negative numbers, or equivalently whether or not there existed a value that satisfied the equation x^2 + 1 = 0. If we decide that this equation _does_ have a solution, then we can give that solution a name: let's call it i. Then it's not too hard to show that i^2 = -1, and that -i also satisfies the equation x^2 + 1 = 0. The reason mathematicians chose "i" as this new number's name is that they still questioned its validity as a number, and questioned its right to co-mingle with the real numbers. After all, they had a lot to worry about - what physical significance does "i" have? (Several, in particular, in the physics of electric circuits.) Does its introduction lead to logical inconsistencies? (No - as a matter of fact, it opens up vast fields of study from abstract algebra to complex analysis.) But as time went on, people began to realize that "i" was a generally good idea - though the term "imaginary" stuck. How does one use "i" in calculation? Well, i follows most mathematical conventions, though care needs to be taken occasionally during multiplication and root extraction. For example, i+i = 2i, and (1+i)+(3-i) = 4. i^2 = i*i = -1, from the definition; note Sqrt[-4]*Sqrt[-1] is not equal to sqrt[(-4)(-1)] = 2, but rather Sqrt[-4]*Sqrt[-1] = i*Sqrt[4]*i = -2. This apparent contradiction arises simply because the rule where Sqrt[p]*Sqrt[q] = Sqrt[p*q] is only applicable when p and q are non-negative. Thus we have two sets of numbers: "real" and "imaginary." They are mutually exclusive (except perhaps for 0, which could be considered as both real and imaginary, though nearly always it is thought of as belonging to the former). The union of these sets forms the *complex* numbers; these are numbers of the form a+b*i, where a and b are reals. As can be verified, addition and multiplication are well-defined (non- ambiguous), there is an *additive identity* (zero), a *multiplicative identity* (one), there is always an *additive inverse* (i.e., for every complex number a+bi there exists a unique c+di such that (a+bi)+ (c+di) = 0), and a *multiplicative inverse* (i.e., for every a+bi there is a unique c+di such that (a+bi)(c+di) = 1). Finally, addition and multiplication are associative, commutative, and distributive. The most common purpose of the imaginary numbers is in the representation of roots of a polynomial equation in one variable. For example, what are the roots of x^2 + 2*x + 5 ? Using the quadratic formula, we find -2 + Sqrt[4 - 4*5] -2 - Sqrt[4 - 4*5] x = ------------------ , ------------------ 2 2 or x = {-1+2*i, -1-2*i}. An important theorem in algebra (which I stated for a particular case at the beginning of this letter) states that for a polynomial in one variable of degree n, there are exactly n roots, counting multiple roots. So a cubic equation has 3 roots, a quartic 4, and so on. There are other applications of complex numbers, some of them quite abstract - often they deal with the algebraic structure of these numbers, rather than the computational aspects. Hope this helps.... :) -Doctor Pete, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 7/25/96 at 19:17:28 From: Doctor Robert Subject: Re: imaginary numbers I think that the moniker "imaginary" is unfortunate, since in one sense, all numbers are imaginary in that they exist only in our minds. But, in mathematics, there is a distinction made between real and imaginary numbers. The real numbers are those that show up on the number line. Imaginary numbers arise because mathematicians could not find a solution to the equation x^2 + 1 = 0 in the set of real numbers. So, they decided to designate the square root of negative one by the small case letter i. If i = the square root of -1, then the square root of -4 is 2i. Now, complex numbers are those which have both a real part and an imaginary part. An example would be the number 2+3i. The young man studying imaginary numbers is probably learning to do basic operations with complex numbers like addition, subtraction, multiplication, and division. I hope that this helps. -Doctor Robert, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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