Johannes
Carl Friedrich Gauss was born in southern Germany in the late 1700's. Though he is
not as famous as Newton, Gauss is still considered to be a great contributor to
world of mathematics. His works included developments in number theory, such as
advances in algebra, and statistics. He we also know for some of with work in the
world of physics, as many mathematicians where in his time. He died in the mid
eighteen hundreds.
While in his
early years Gauss was known to be a child prodigy. He seemed to have a natural
ability with mathematics, and often times astonished people. One day while still
in his youth Gauss went to school as he normally would. However, the professor
was especially grumpy that day. Many variations of the story say he was old,
and cranky, and would beat kids with his cane, making him seem more menacing
then he probably was. The grumpy professor walked into the one room classroom,
and since he did not feel like putting up with the children, he told them to
add all the numbers from one to a hundred. At this time there was not pens and
paper, but instead the children had blackboard slates. Now seconds after the
professor had challenged the boys, Gauss runs up and puts his slate face down
on the teacher’s desk. Thus submitting his answer. At this point in time Gauss
is not the oldest boy in the room, and many of the other older boys look down on him.
They were said to be thinking, poor little gauss, he has gone with a guess.
Well time goes on, and all the other boys start working out the problem, using
addition, multiplication, subtractions, and division, anything to try to get
done first. Once the boys finished the problem they did like Gauss had
previously done, putting their slate face down on the teachers desk, stacking
them up. Once the time was up, the professor walked up to the desk and flipped
all of the slates over. Now all the slates were in order from first to last,
with the answers facing upward. Sitting on top of the pile was Gauss' answer.
Gauss' answer was the correct answer, which is 5050.
So how did Gauss
do it? Gauss, then explained, that the problem needs to be looked at in a
different way. The most brute force way would be to add one plus two plus
three, all the way to one hundred. Instead of doing it like that Gauss looked
at in a different way. If all the number one through hundred are laid out one a
number line, then Gauss' method can be seen. If you take one and add it to one
hundred the answer is one hundred and one. The answer is the same if you add
two and ninety nine. The same would apply to all the pair of numbers that can
be created, all the way down to fifty and fifty-one. All Gauss had to do was
take one hundred and one, and multiple by the number of pairs. In this case
that would be fifty, so the answer would be fifty times one hundred and one.
Fifty times one hundred and one is 5050.
Going back to the problem at the start of post. Using the same process, start by writing out all the numbers from one to a million. Now notice that if you add to one to 999,998 it equals 999,999. This process can be done again and again. This process is the same pairing process gauss used. So then i would take the sum of all the digits of 999,999 and multiple that by the number of pairs. If zero is included that would make 500,000 pars. So 54 times a half a million would be twenty seven million. Not forgetting the one in one million would make the solution to the original problem 27,000,001.
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