November 11, 2008
The cube root of negative 27 equals ...
One day when I was taking an electronics class at American River College
the teacher introduced the idea of logarithms and stated that this is
how calculators do powers. I then piped in that "that's why a calculator
cannot do the cube root of negative 27... "
Before I could finish my thought the teacher interrupted and said
"that's undefined!" When I objected that it wasn't he said "it was
yesterday!" at which the whole class erupted into laughter. He then
said "Besides, show me the cube root key on your calculator... "
Of course, the teacher and most of my classmates were wrong, and the
teacher was wrong about two things!
If you think about it, -3*-3=9 and 9*-3=-27 therefor -3 cubed = -27 So
the cube root of -27 is -3! Since any negative number sqaured is positive
you can't have a square root of a negative number, and the same would
go for any even root of a negative number. However, since a positive
times a negative is negative, you can take the cube root or any other odd
root of a negative number.
The other issue here is that the TI30
, the scientific calculator then used
by 75% of his students, can do any root through INV Yx, thus to take
the cube root of 27 you would type "27 INV yx 3 =" and you would get the
answer "3". If you typed "-27 INV yx 3 =" you would get an error even
though the answer is "-3" because it tries to take the logarithm of -27.
Try this on your windows scientific calculator (view scientific) and see
what it says!
You'd think that several of my 48 classmates would have realized this
but many of them mocked me and not one came to my defense.
That wasn't the last time that I was right while that teacher and the
entire class were wrong, but that's a story for another day.