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Tuesday, October 2, 2012

The two meanings of 4

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What is wrong with the following expression?

4=16=4×4=4×4=4.


It seems easy. 4 doesn't make sense. Right? Well, yes and no.

If we just consider the usual real numbers, then square of any number is positive and 4 simply does not exists.

However, there is something called complex numbers.  For complex numbers, square can be negative.  In this case, there are two ways of interpreting 4.

1.  4=2i, where i is the imaginary unit.  In this case, the equality 4×4=4×4 does not hold.

2. 4={±2i}, i.e. it is a set.  We would also reinterpreting 16={±4}. In this case, we have
{±4}=16=4×4=4×4={±2}×{±2}={±4},

here we use element-wise multiplication of two sets.

In general, there are two different meanings of x.

1. x is the unique positive square root of x, if x>0; 0=0; x=xi if x<0;  any one of the square root of x if x is not a real number.  Under this meaning, we have ab=a×b only if at least one of a and b are nonnegative.

Under this meaning, the formula for the quadratic equation ax2+bx+c=0 is b±b24ac2a.  As we have extended the definition of , this formula now makes senses for any complex numbers a0, b and c.

2. x is the set consists of the two roots of x.  Indeed, we can generalize it to nx being the set of all n-th roots of x.  For this meaning, we have nab=na×nb for all complex numbers a and b.

The formula for the quadratic equation is now simply b+b24ac2a, which is indeed the set of the two solutions.

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