Not all large numbers are complex
The Sunday Nation
Nairobi,
20 April 2014
Here is an old children’s puzzle: You are travelling along a road
heading to a certain village and then you come to a junction with two
branches. You don’t know which route to take to your destination.
Now at this junction there lives identical twin brothers. Despite
looking the alike, they are different in character. One is a perpetual
liar while the other is always truthful.
You find one of the twins on the road but you don’t know whether he is
the liar or the truthful brother. What one question would you ask in
order to be sure you find the correct route to take?
Then solution to the puzzle is this: “If I asked your brother the way to
the village, which road would he show?” Then whatever the answer you
get, take the other route.
If it is the liar that you meet, he knows that the brother would show
you the right way so he will lie and point you to the wrong one. If it
is the truthful twin, he knows that the brother would mislead you and
therefore he will also point you to the wrong way since that’s where his
lying brother would show! This puzzle illustrates the concept of a
double negative operation.
A negative number is one that is less than zero. Adding a negative
number to a positive one is equivalent to subtracting that quantity.
Thus +7 plus -5 is the same as 7 – 5; which is equal to 2.
Adding a negative number to another negative one results in a more
negative answer. Thus -7 plus -5 is the same as -7 – 5 = -12. -12 is a
greater distance below zero than -7.
But things change when we multiply. A positive number multiplied by a
positive number gives a positive result. A negative times a positive
gives a negative answer. However, a negative times a negative is a
positive – this is a double negative operation.
For this reason, the square-root of a number has two values – one
positive and the other negative. For example, the square-root of +16 is
either +4 or -4. Both these numbers when squared will yield +16.
The difficult question is this: what is the square-root of a negative
number? We can attempt to work it out in a two step process. For
example, to get the square root of -25, we first separate out the
negative and positive portions; that is, -25 = -1 x +25.
Next we find the square-roots of these two parts separately. Now the
square-root of +25 is either +/-5. But what is the square root of -1? To
get unstuck, mathematicians “invented” an imaginary number called “i”.
When i is squared, the result is negative one (-1). Therefore, the
square-root of -25 is either +/-5i.
Any number that contains the imaginary quantity, i, is said to be a
complex number. So, Philip Ochieng was wrong when he wrote that “…the
word for 10 is the most malleable in forming words for complex numbers”
in his column (Mark my word)
in the Saturday
Nation last week (12 April 2014). The numbers he was referring to
were just large; not complex. Complex numbers can be small or large, but
they must contain the square-root of -1.
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