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Name: Jill
Status: student	
Age:  N/A
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Date: 12/13/2005


Question:
I read one of the answers to the question about i, but I am not completely sure if I understand. What is i^2?! Using two different methods, both mathematically correct, render you with two different answers, 1 and -1.

You can follow the rules where i^2 can be broken down into (-1^(1/2))^2. And the exponents multiply to give you -1^1 which is -1.

But if you break it down into two separate radicals, you have (sqrt(-1))*(sqrt(-1)), which can then be simplified by putting the product under the same radical, meaning sqrt(-1*-1). Thus rendering you with the sqrt(1) which is just 1. And compared with the first method, you then have that -1=1. How is this so?



Replies:
Jill,

You have to be careful about which rules you apply.

There are no real numbers which, when you square them, result in a negative real number.

But you can leave the "realm" of real numbers and enter the world of complex numbers which have a real part and a complex part (in the form a + bi).

This is a different mathematical model which is useful for many "real world" applications. Electrical engineers use complex numbers as a way of describing vectors of electrical power (where the length of the vector represents the magnitude of a voltage and the angle represents its phase, for example). In this realm of complex numbers, "i" represents some quantity that, when squared, gives you negative 1. You can represent any negative number using this defined quantity (a quantity squared that gives you -2 is 2i, a quantity squared that gives you -pi is (pi)i, etc).

So, by definition, i^2 is -1 in the realm of complex numbers. If you rewrite i^2 as sqrt(-1)*sqrt(-1), then you may NOT apply the rule (which works only in the realm of real numbers) that this is equal to sqrt((-1)*(-1)) -- because in the realm of real numbers, there is NO number that you can square to give you a negative number (so sqrt (-1) is undefined).

This can be confusing to students because they want to apply rules that work in the realm of real numbers to complex numbers. But different rules apply. How do you multiply (a + bi) * (c + di)? How do you add (a +bi) + (c + di)? The rules for manipulating complex numbers are different than the rules for real numbers.

Todd Clark, Office of Science
US Department of Energy


You have to be careful when you "square" a number, because there are two solutions (+N) and (-N) that both have the same value of (N)^2 -- (+N)*(+N) and (-N)*(-N). So in the case of "i": (+ i)*(+ i) = (i)^2 = -1 and (-i)*(-i) = (-1* i)* (-1*i) = [(-1)*(-1)]*[i * i] = [+1]*[i*i] = (i)^2 = -1. The confusion arises (and you are by no means alone) because using the notation for complex numbers (a + i*b) obscures the fundamental definition of multiplication of complex numbers, which is: Complex numbers are ordered pairs (a,b) of real numbers "a" and "b", that is (a,b) is not equal to (b,a). The multiplication of two complex numbers (a,b) and (c,d) is defined as an ordered pair of real numbers: (a,b)*(c,d) = (ac-bd, ad+bc). That is how mathematicians define complex numbers. The rest of us use the shorthand: (a+ib)*(c+id) = (a*c-(i)^2b*d) + i*(b*c+ a*d) and multiply using binomial multiplication, as we would do for real numbers. Where you have to be careful is when "squaring" a complex number. Usually, when one speaks about "squaring" a complex number, what is actually meant is multiply the complex number and its complex conjugate. The complex conjugate of (a+ ib) is defined as (a - ib), that is, the "i"'s are turned into (-"i"'s). So (a+ ib)*(a - ib) = a^2 + b^2 which is a real number. This is not the same quantity as (a+ ib)*(a+ ib) = (a^2- b^2, 2*a*b) or in the "i" notation: (a^2 - b^2) + i(2*a*b).

Vince Calder


Jill,

When dealing with numbers that are not real and positive, the relation sqrt(a)*sqrt(b)=sqrt(ab) is not necessarily true. When you go beyond positive numbers, "root" functions become more involved. When you have complex numbers, you cannot assume the answer will be positive. The square root of 1 can be either +1 or -1. The fourth root of 1 can be +1, +i, -1, or -i. All are fourth roots of 1 when working with complex numbers.

When working in only real numbers, sqrt(-1)*sqrt(-1) doesn't have a value. The value of i-squared is -1. This quantity allows us to work with square roots of negative numbers. It also provides a mathematical system that can provide some useful shortcuts when dealing with vectors.

Dr. Mellendorf



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