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3. Carl Friedrich Gauss “Mathematics is the queen of the sciences and arithmetic is the queen of mathematics” 1777 - 1855

4. Story Carl Friedrich Gauss was born in Brunswick, Germany on 30 th 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 well- authenticated 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. 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.

5. 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. Sum (1  40) = ? 1 + 2 + 3 + 4 +... + 38 + 39 + 40 1 + 2 + 3 + 4 +... + 98 + 99 + 100 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. Sum (1  100) = ? 820

6. 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

7. 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 Using the same method as before, what multiplication sum do we have to do to work it out? Sum (1  20) = ? There are 10 pairs each totalling 21 so 10 x 21 = 210 Sum (1  20) = 210

8. 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 +... + 38 + 39 + 40 Using the same method as before, what multiplication sum do we have to do to work it out? Sum (1  40) = ? There are 20 pairs each totalling 41 so 20 x 41 = 820 Sum (1  40) = 820 Now consider your original task to add up all the numbers from 1 to 40.

9. 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 +... + 98 + 99 + 100 Sum (1  100) = ? There are 50 pairs each totalling 101 so 50 x 101 = 5050 Sum (1  100) = 5050 Now consider your Carl’s task to add up all the numbers from 1 to 100. This is a very easy calculation to perform mentally since: 50 x 100 = 5000 + 50 x 1 = 50 5050

10. 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  30) 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. Sum (1  12) Sum (1  50) Sum (1  60) Sum (1  78) Sum (1  90) 15 x 31 = 465 6 x 13 = 78 25 x 51 = 1275 30 x 61 = 1830 39 x 79 = 3081 45 x 91 = 4095 a b c d ef

11. 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  10) 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. Sum (1  100) Sum (1  1000) 5 x 11 = 55 50 x 101 = 5050 500 x 1001 = 500500 Sum (1  10,000) 50005000 Sum (1  100,000) 5000050000 Sum (1  1,000,000) 500000500000 a d b e c f

12. General 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) = ? There are ½n pairs each totalling n+1 so ½n x (n+1)= Because we are repeating the same procedure each time we should be able to derive a formula that will speed things up even more. n(n+1) 2 Sum (1  n) = n(n+1) 2 If n is any positive integer, what is the number before n? What is the number before n - 1? 1 + 2 + 3 +... + + + n ? ? n-1 n-2 Can you work out the formula?

13. 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 Sum (1  15) = 15 x 16 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. 8 = 8 x 15 = 120 Sum (1  24) = 24 x 25 2 12 = 12 x 25 = 300 You should be able to do the first two in your head! There is always going to be an even number on top.

14. 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 87 x 88 Sum (1  87) = 2 44 = 87 x 44 = 3828 Sum (1  100) = 100 x 101 2 50 = 50 x 101 = 5050 Sum (1  93) = 93 x 94 2 47 = 93 x 47 = 4371

15. How did Carl add up all the numbers so quickly? Sum (1  n) = n(n+1) 2 Sum (1  14) Sum (1  72) Sum (1  279) ab cd e f Sum (1  28) Sum (1  36) Sum (1  143) 105 406 666 2628 10 296 39 060 Use the formula method to work out the questions below

16. Triangular Sum (1  n) = t n = n(n+1) 2 We can use our formula to calculate the nth triangular number (t n ). Can you see why by considering the case of the 10 th triangular number (t 10 ) as shown?

17. Sum (1  n) = t n = n(n+1) 2 10 x 11 2 t 10 = = 55 n(n+1) 2 t n = So in general :

18. t n = n(n+1) 2 t 16 a b c d e f 136 903 2926 5050 20 100 Use the formula to work out the triangular numbers below. t 42 t 76 t 81 t 100 t 200 3321

19. Sum (1  30) 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. Sum (1  12) Sum (1  50) Sum (1  60) Sum (1  78) Sum (1  90) a b c d ef Worksheet 1

20. Sum (1  10) 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. Sum (1  100) Sum (1  1000) Sum (1  10,000) Sum (1  100,000) Sum (1  1,000,000) a d b e c f Worksheet 2

21. Sum (1  n) = n(n+1) 2 Sum (1  14) Sum (1  72) Sum (1  279) ab cd e f Sum (1  28) Sum (1  36) Sum (1  143) Use the formula method to work out the questions below. Worksheet 3

22. t n = n(n+1) 2 t 16 a b c d e f Use the formula to work out the triangular numbers below. t 42 t 76 t 81 t 100 t 200 Worksheet 4

23. Story Carl Friedrich Gauss was born in Brunswick, Germany on 30 th 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 well- authenticated 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. 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.