Negative Numbers

Posted Monday, November 16, 2009 3:02 PM by WesleySonofCornelius

In math, you were always told, "well, you were told you cannot ____, but really you can."Mr. Owl

Lies, all lies!

Recall that in 1st or 2nd grade you were told you cannot subtract a larger number from a smaller number (for example 3 - 5 = doesn't exist).

Well, then you enter 5th or 6th grade and learn that you can.  The number is called negative.  But never take the square root of a negative number (for example sqrt(9) = + or - 3 but sqrt(-9) = doesn't exist).

However, you then enter 9th or 10th grade, and guess what?  You really can take the square root of a negative number.  You call this number imaginary, and give it the letter i.  For example sqrt(-9) = + or - 3i.  However, the square root function is okay.  BUT never, ever, ever take the logarithm of a negative number (for example log(100) = 2 but log(-1) = doesn't exist).

Or does the log(-1) exist?  Hmmm?!  Might need to go ask Mr. Owl.

Comments

# re: Negative Numbers

Tuesday, November 17, 2009 3:00 PM by :o)}

Mr. Owl says:

The logarithm function can be extended to the complex logarithm, which applies to negative and complex numbers and yields a complex number. The value is not unique though, since for example e2πi = e0 = 1 which implies that both 2πi and 0 are equally valid logarithms to base e of 1.

When z is a complex number, say z = x + iy where x and y are real, the logarithm of z is found by putting z in polar form that is, z = reiθ = r(cos θ + i sin θ), where  r = |z| = sqrt(x2+y2) and θ = arg(z) is any angle such that x = r cos θ and y = r sin θ. The function arg is a multivalued function.

If the base of the logarithm is chosen as Euler's number e, that is, using loge (denoted by ln and called the natural logarithm), the complex logarithm is:

log(z)=ln|z| + iarg(z)

which is, just like arg, also a multi-valued function. The principal value of the logarithm, Log (denoted by a capital first letter), is a single-valued function.

Therefore, the principal value of the logarithm of a negative number r is:

 Log(r) = ln|r| + i(pi).

Any dummy should have known that!

:o)}

# re: Negative Numbers

Thursday, November 19, 2009 3:19 PM by WesleySonofCornelius

The interesting thing is that

log(-1) = pi*log(e)*i

Why do we get pi and and Euler's constant, e?  This mysterious world of mathematics!

# re: Negative Numbers

Thursday, November 19, 2009 9:24 PM by leon

Well, I don't know the details but I do admire the symmetry and the depth beyond our comprehension.

The Great Mathmetician must have a chuckle over all the problems we say cannot be done that sit before our very eyes each day.