Transcript Gauss` Story - beingamathematician

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Carl Friedrich Gauss
1777 - 1855
“Mathematics is the queen of the
sciences and arithmetic is the
queen of mathematics”
Carl Friedrich Gauss was born in Brunswick, Germany on 30th April 1777 into a
poor uneducated family. His father was a hard working Brunswick labourer,
stubborn in his views who tried to stop his son from receiving an appropriate
education. Carl’s mother took a different view and being un-educated herself she
encouraged Carl to study.
As a very young child, Carl taught himself to read, write and to do arithmetic. He
excelled in all he did and was a remarkable infant prodigy, who according to a wellauthenticated story, corrected a mistake in his father’s arithmetic at the age of
three.
Carl didn’t start elementary school until the age of eight. His class teacher was a
renowned task master and on Carl’s first day at school he ordered the class to add
up the first 100 numbers, with instructions that each should place their slate on a
table as soon as the task was finished. Almost immediately Carl placed his slate on
the table saying., “there it is”. The teacher looked at him scornfully as the other
pupils continued to work diligently. After about an hour when the teacher had
inspected everyone's results, to his astonishment Carl was the only one to arrive
at the correct answer. Eventually, Carl’s mathematical powers so overwhelmed his
school masters that by the age of ten they freely admitted that there was nothing
more that they could teach the boy.
Story
Carl went on to become one of the greatest mathematicians that the world has ever
known, as well as making major contributions in Physics and Astronomy.
Throughout her life, Carl’s mother took great pride in all his achievements until her
death at the age of ninety-four.
How did Carl add up all the numbers so quickly?
Using pencil and paper only, add up all the numbers from 1 to 40. We
will see how long it takes and how many people get the right answer.
1 + 2 + 3 + 4 + . . . + 38 + 39 + 40
Sum (1  40) = ?
820
How did Carl add up all the numbers from 1 to 100 instantly. He
obviously could not have done it in the usual sequential way.
1 + 2 + 3 + 4 + . . . + 98 + 99 + 100
Sum (1  100) = ?
How did Carl add up all the numbers so quickly?
In maths it is often a good idea to examine a simpler situation first
before trying to tackle a more difficult problem. Consider trying to
add up the numbers from 1 to 10 in a non-sequential way.
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
Can you see a possible method that might help speed things up.
Clue 1:
Clue 2:
Clue 3:
We simply add them together in pairs from either end. There
are 5 pairs that each total 11 so we simply work out 5 x 11 = 55.
Sum (1  10) = 55
How did Carl add up all the numbers so quickly?
In maths it is often a good idea to examine a simpler situation first
before trying to tackle a more difficult problem.
1 + 2 + 3 +
. . .
+ 18 + 19 + 20
Sum (1  20) = ?
Using the same method as before, what multiplication sum do we
have to do to work it out?
There are 10 pairs each totalling 21 so 10 x 21 = 210
Sum (1  20) = 210
How did Carl add up all the numbers so quickly?
In maths it is often a good idea to examine a simpler situation first
before trying to tackle a more difficult problem.
Now consider your original task to add up all the numbers from 1 to 40.
1 + 2 + 3 +
. . .
+ 38 + 39 + 40
Sum (1  40) = ?
Using the same method as before, what multiplication sum do we
have to do to work it out?
There are 20 pairs each totalling 41 so 20 x 41 = 820
Sum (1  40) = 820
How did Carl add up all the numbers so quickly?
In maths it is often a good idea to examine a simpler situation first
before trying to tackle a more difficult problem.
Now consider your Carl’s task to add up all the numbers from 1 to 100.
1 + 2 + 3 +
. . .
+ 98 + 99 + 100
Sum (1  100) = ?
There are 50 pairs each totalling 101 so 50 x 101 = 5050
Sum (1  100) = 5050
This is a very easy
calculation to perform
mentally since:
50 x 100 = 5000
+
50 x 1 = 50
5050
How did Carl add up all the numbers so quickly?
In maths it is often a good idea to examine a simpler situation first
before trying to tackle a more difficult problem.
Work out the following either in your head or using pencil and paper.
None of them should take longer than a minute at the outside.
a
c
Sum (1  30)
15 x 31 = 465
Sum (1  50)
b
d
25 x 51 = 1275
e
Sum (1  78)
39 x 79 = 3081
f
Sum (1  12)
6 x 13 = 78
Sum (1  60)
30 x 61 = 1830
Sum (1  90)
45 x 91 = 4095
How did Carl add up all the numbers so quickly?
In maths it is often a good idea to examine a simpler situation first
before trying to tackle a more difficult problem.
These are bigger sums but they are easier than the previous ones once you
spot the pattern. Separators have been deliberately omitted from the answers.
d
a
Sum (1  10)
b
5 x 11 = 55
Sum (1  100)
50 x 101 = 5050
c
Sum (1  1000)
Sum (1  10,000)
50005000
e Sum (1  100,000)
5000050000
f
Sum (1  1,000,000)
500 x 1001 = 500500
500000500000
How did Carl add up all the numbers so quickly?
In maths it is often a good idea to examine a simpler situation first
before trying to tackle a more difficult problem.
Because we are repeating the same procedure each time we should be able to
derive a formula that will speed things up even more.
1 + 2 + 3 +
. . .
+ n-2
? + n
? + n-1
Sum (1  n) = ?
If n is any positive integer, what is the number before n?
What is the number before n - 1?
Can you work out the formula?
There are ½n pairs each totalling n+1 so ½n x (n+1)= n(n+1)
2
Sum (1  n) = n(n+1)
General
2
Formula
How did Carl add up all the numbers so quickly?
In maths it is often a good idea to examine a simpler situation first
before trying to tackle a more difficult problem.
Sum (1  n) = n(n+1)
2
This method is easier than thinking about adding in pairs (particularly if n is
odd) although it is exactly the same thing really. We will use the formula to
work out some problems (without a calculator) and look for further short cuts.
You should be able to do the first two in your head!
8
There is always
going to be an even
number on top.
Sum (1  15) = 15 x 16 = 8 x 15 = 120
2
12
Sum (1  24) = 24 x 25 = 12 x 25 = 300
2
How did Carl add up all the numbers so quickly?
In maths it is often a good idea to examine a simpler situation first
before trying to tackle a more difficult problem.
Sum (1  n) = n(n+1)
2
44
Sum (1  87) = 87 x 88 = 87 x 44 = 3828
2
50
Sum (1  100) = 100 x 101 = 50 x 101 = 5050
2
Sum (1  93) =
47
93 x 94
= 93 x 47 = 4371
2
How did Carl add up all the numbers so quickly?
Sum (1  n) = n(n+1)
2
Use the formula method to work out the questions below
a
Sum (1  14)
b
105
c
e
Sum (1  36)
666
Sum (1  143)
10 296
Sum (1  28)
406
d
Sum (1  72)
2628
f
Sum (1  279)
39 060
n(n+1)
Sum (1
 n) = tn=
Triangular
2
We can use our formula
to calculate the nth
triangular number (tn).
Can you see why by
considering the case
of the 10th triangular
number (t10) as shown?
n(n+1)
Sum (1  n) = tn=
2
t10
10 x 11
=
= 55
2
So in general :
n(n+1)
tn =
2
tn = n(n+1)
2
Use the formula to work out the triangular numbers below.
a
t16
b
136
c
t76
903
d
2926
e
t100
5050
t42
t81
3321
f
t200
20 100
Worksheet 1
Work out the following either in your head or using pencil and paper.
None of them should take longer than a minute at the outside.
a
Sum (1  30)
b
c
Sum (1  50)
d
e
Sum (1  78)
f
Sum (1  12)
Sum (1  60)
Sum (1  90)
Worksheet 2
These are bigger sums but they are easier than the previous ones once you
spot the pattern. Separators have been deliberately omitted from the answers.
a
Sum (1  10)
b
Sum (1  100)
c
Sum (1  1000)
d
Sum (1  10,000)
e
Sum (1  100,000)
f
Sum (1  1,000,000)
Sum (1  n) = n(n+1)
2
Use the formula method to work out the questions below.
a
Sum (1  14)
b
Sum (1  28)
c
Sum (1  36)
d
Sum (1  72)
e
Sum (1  143)
f
Sum (1  279)
Worksheet 3
tn = n(n+1)
2
Use the formula to work out the triangular numbers below.
a
t16
b
t42
c
t76
d
t81
e
t100
f
t200
Worksheet 4
Carl Friedrich Gauss was born in Brunswick, Germany on 30th April 1777 into a
poor uneducated family. His father was a hard working Brunswick labourer,
stubborn in his views who tried to stop his son from receiving an appropriate
education. Carl’s mother took a different view and being un-educated herself she
encouraged Carl to study.
As a very young child, Carl taught himself to read, write and to do arithmetic. He
excelled in all he did and was a remarkable infant prodigy, who according to a wellauthenticated story, corrected a mistake in his father’s arithmetic at the age of
three.
Carl didn’t start elementary school until the age of eight. His class teacher was a
renowned task master and on Carl’s first day at school he ordered the class to add
up the first 100 numbers, with instructions that each should place their slate on a
table as soon as the task was finished. Almost immediately Carl placed his slate on
the table saying., “there it is”. The teacher looked at him scornfully as the other
pupils continued to work diligently. After about an hour when the teacher had
inspected everyone's results, to his astonishment Carl was the only one to arrive
at the correct answer. Eventually, Carl’s mathematical powers so overwhelmed his
school masters that by the age of ten they freely admitted that there was nothing
more that they could teach the boy.
Story
Carl went on to become one of the greatest mathematicians that the world has ever
known, as well as making major contributions in Physics and Astronomy.
Throughout her life, Carl’s mother took great pride in all his achievements until her
death at the age of ninety-four.