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A. Math’s Project

A. Math’s Project. Let's Have "SUM" Fun!. Introduction. “ Sum”(addition) is the first thing we learn in mathematics, i.e. the most basic and important element that you must not ignore!! Combining with the word “Fun”, that means some funny mathematics that you must not miss!!!!

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A. Math’s Project

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  1. A. Math’s Project Let's Have "SUM" Fun!

  2. Introduction • “Sum”(addition) is the first thing we learn in mathematics, i.e. the most basic and important element that you must not ignore!! • Combining with the word “Fun”, that means some funny mathematics that you must not miss!!!! • So, let’s go on immediately to see how fun mathematics is!!!

  3. Content • ★ ‘Sum’ Fun about Numbers. • ☆ ‘Sum’ Fun about Mysteries Maths. • ★ ‘Sum’ Fun about Mathematics in Arts. • ☆ ‘Sum’ Fun about Mathematics • in Daily Life. • ★ ‘Sum’ Fun about Math’s jokes. • ☆ ‘Sum’ Fun about IQ question.

  4. ★‘Sum’ Fun about numbers. • Why we have to doing ×÷ before ＋－? • Why subtracting a negative no. = adding a same but positive no. • Is 0.9999….. = 1? ┐ Strange • What’s the strange use of 9. |- Fractions • Moving in a circle incessantly. │ • What’s special about 1/2? ┘

  5. ‘Sum’ Fun about Number (Cont.) • Something about numbers. • What is 71562435×11? • Don’t be cheated by the numbers! • What is the Chinese • unit of some numbers?

  6. Why we have to do ×÷ before＋－? • Since primary school, our teachers always told us that if, for example, 12+11×3- 4/8, we have to do multiplication or division before addition or subtraction. But why? • Maybe someone will say that it will be easier to calculate as there is one answer only and so people set this rule. But can it be set in an opposite way?

  7. We must know that as arithmetic is used in our daily life. Therefore, we must know that this rule must be reasonable and convenient to us. • In fact the reason behind this question is very simple. As there’s more situation for us to multiply or divide something before plus or minus something.So, we set the rule like this.

  8. Let’s give a simple example to show how common it is: • E.g. $3/orange, $3.5/apple, $4/pear. Now I want to buy 9 oranges, 5 apples and 6 pears. How much I have to pay? • $3×9 + $3.5×5 + $4×6 = $68.5 • In this case, multiplying all the numbers first is reasonable and convenient. This rule can also reflect the need in daily life.

  9. Why subtracting a negative no. = adding a same but positive no.? • When we learn positive and negative numbers, somebody will wonder why 5-(-2)=5+2? Is that mean we have $5 and while we don’t need to pay a $2 bill, we hill have $7 in our pocket? • But we have to know when we calculate in daily life, a bigger no. cannot minus a smaller one. For instance, there’s 4 oranges, can we eat 6? i.e. 4-6= -2 in mathematics?

  10. Of course, everybody know that it is impossible. But actually sometimes it really happen in our daily life. • E.g.1) One object move 4m towards the right and 6m towards the left. • E.g.2) The lift move 5m upwards and 9m downwards. • E.g.3) One person only have $100, but due to some reasons, he need to take out $120.

  11. Like these examples, in mathematics, we can express like this: • E.g.1) 4-6 = -2. E.g.2) 5-9= -4. E.g.3) 100-120 = -20 • Here, positive numbers means right hand side, upwards and savings while negative numbers means left hand side, downwards and expenditures. • Although Mathematics exists in daily life, but sometimes we cannot use it, like 1/2man,1/3 animal or $√-2etc.

  12. If you know this concept, it is not difficult for you to know about why 5-(-2) = 7. If you need to pay the bill, the $5 in your pocket is not all yours. Only while you don’t need to pay the $2, you can actually own the $5. Therefore, subtracting a negative number is actually equal to adding a same but positive number.

  13. Strange FractionIs 0.999…… = 1 ??? • Below is the relationship between a fraction with 9 as denominator and decimals: • 1/9 = 1÷9 = 0.111... 5/9 = 5÷9 =0.555... • 2/9 = 2÷9 = 0.222... 6/9 = 6÷9 = 0.666... • 3/9 = 3÷9 = 0.333... 7/9 = 7÷9 = 0.777... • 4/9 = 4÷9 = 0.444... 8/9 = 8÷9 = 0.888...

  14. We can notice that all are recurring decimals and their numerators is the recurrent no. Therefore, 8/9 = 0.8, then 9/9 = 0.9. But 9/9 = 1. So 1 = 9/9 = 0.9. It can be calculated: • 0.9999….. • 9 ) 9. 0 • 8 1 • 9 0 • 8 1 • 9 0 • 8 1 • 9 0 • 8 1 • 9

  15. Not just 9/9, also 7/7, 8/8, 6/6 (numerator = denominator) = 0.999.... • 0.9999….0.9999…. • 7 )7.0 8 ) 8.0 • 6 37 2 • 7 0 8 0 • 6 37 2 • 7 0 8 0 • 6 37 2 • 7 0 8 0 • 6 37 2 • 78

  16. Strange use of 9 • Except n/9 ( 0＜n≦9 ) can be easily changed into recurring decimals without calculators. Also, denominator is 99,999,9999….etc. has this characteristic. For example, 13/99 = 13÷ 99 = 0.13 • 7/99 = 7÷ 99 = 0.07 • 115/99 =115÷ 99 = 0.115

  17. Move in a circle incessantly • When we talk about recurring decimals, we must not miss the fraction with 7 as denominator. • Let’s see the following numbers: • 1/7 = 0.142857 4/7 = 0.571428 • 2/7 = 0.285714 5/7 = 0.714285 • 3/7 = 0.428571 6/7 = 0.857142

  18. Because of this strange characteristic, when we encounter some decimals in the calculator, we can easily change it back to fractions. • For example, 0.63 = 63/99 = 7/11 • 0.135 = 135/999 = 5/37

  19. All these recurring decimals are made up of 6 numbers. That is 1,4,2,8,5,7. Maybe you haven’t seen the knack yet. Let’s put these numbers into a circle and easier for us to observe. • 1 • 7 5/7 1/7 3/7 4 • 5 4/7 6/7 2/7 2 • 8

  20. With this circle, we can find the decimals from 1/7-6/7. E.g. 3/7 You read from ‘4’ clockwisely. All the 6 numbers are the recurring no. i.e. 0.428571. Others are the same. If you don’t believe, try it and check with the calculator! • In the kingdom of Mathematics, fraction is acting a clown. It is changeable.

  21. E.g. 1/10 can represent million dollars of the property of a millionaire or $1 of a beggar. Chinese people use fraction to create idioms or phrases. • E.g.九死一生(extremely dangerous) • 九牛一毛 (very little). • Fraction can be changed into percentage. So we want to talk a little about it:

  22. In fact in our daily life, we will say ‘九成九’. But what’s mean by that? It’s related to percentage. Percentage takes 100 as a whole, 100% = 1 (all). Nowadays, we always use percentage to show the rate of birth, death, gain, loss, inflation, interest, discount etc. • But in China, we use 10 as and there’s some idioms related to 10.

  23. E.g.十全十美 (perfect),十拿九穩 (very confidence),十中無一 (very rare),為山九仞,功虧一簣 (almost finish but fail because of a minor mistake),人生不如意事十常八九 . From these idioms, we know that our ancestors use 10 as a whole. ‘十成’ means 10/10 = 100％. Therefore ‘九成’means 9/10, , ‘九成九’means 9.9/10 = 99/100 (almost equal to 1). So when we say九成九, it means it’s near a must or a fact.

  24. Something about 1/2 • ½ is a very “unstable” number, for example, in a meeting, if no. of people supporting a plan is ½ of all ,that means ½ people of all will against the plan (assume all of them have give their opinions). Then the meeting will not have any conclusion.

  25. In a voting, if 2 people gain the same number of votes, i.e. each have ½ of all votes, then we will not be able to find out the better one from the 2 people.

  26. But on the other hand, ½ can also be a stable and equilibrium number. If we draw a line, let the end on the left be 0 and the end on the right be 1, then ½ will be at the middle. The distance between 0 and ½ and the distance between ½ and 1 will be exactly the same. It is just like the seesaw we play in the playground. There is only a point on the seesaw can make it equilibrium  the point on the centre, that is ½ .

  27. ½ can be a straight number also. • Let’s see in the following case. • Edward has a long string, he decided to cut away a half everyday. His brother, Edmond, then thinks, “ ½ + ½ =1, after 2 days, is that means all the string will have been cut off?”

  28. In this case, of course everyone of us will know that what Edmond said is not true. On the first day, ½ of the string will be cut off; one the second day, not ½, but ½ × ½ = ¼ string will be cut off . Then, how long do we need to cut off all the string? • The answer is !…… because: • 1/2 + 1/4 + 1/8 + 1/16 + 1/64 +…… 1 • Therefore maybe after 10 years, there will still be a piece of string left !!!

  29. Something about numbers • Let’s see how can we use the number 1 to 9 : • 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 × 9 =100 • 1 × 2 + 34 + 56 + 7 - 8 + 9 = 100 • 1 + 2 + 34 - 5 + 67 - 8 + 9 = 100 • 123 - 4 - 5 - 6 - 7 + 8 - 9 = 100 • 123 - 45 - 67 + 89 = 100 • 98 - 76 + 54 +3 + 21 = 100

  30. We can also group them into 3 simple equations: • 7 + 1 =8 , 9 - 6 = 3 , 5 × 4 = 20 • We can arrange them into 4 square of other rational numbers: • 9 , 16 , 784 , 3025 • ( 3 sq. , 4 sq.. , 28 sq.. , 55 sq. )

  31. We can also arrange them into 3 square of other rational numbers: • 361 , 529 , 784 • ( 19 sq. , 23 sq. , 28 sq. ) • Make into 2 prime numbers (by using 1-9 two times): • 23456789 , 1234567891

  32. We can arrange them into a square of a large rational number: • 923187456 ( 30384 sq. ) • 1398543276 ( 11826 sq. ) • Make into : • 3948012567 (= 3.14156  )

  33. Group as some equations (*): • 290 × 38 = 76 × 145 • 138 × 42 = 5796 • 297 × 18 = 5346 • Group as some equations (): • 35 / 70 = 48 / 96 = 1 / 2 • See, the Numbers 1- 9 is how amazing !!!

  34. What is 71562435×11? • Without using the calculator or a pen, can you say out the answer of 71562435×11 in a vice versa way? Haha! We can even do it within one minute!! The answer is 587681787!! • You also can do it just memorizing the following rule:

  35. Let’s use a simple example to explain this. • 692×11 • First, copy the most right hand side number i.e. ‘2’. So, the digit of the answer is 2. • 692×11 • = 2 • Second, from left to right, add its right hand side’s number. i.e. ‘9+2’. So the tenth digit of the answer is 1 and give ‘1’ to the HUNDRED.

  36. Do this process until the most left hand side’s integer. Then we have to add 6 to 9 = 5 and give’1’ to the THOUSAND . The HUNDRED number is 6. • 692×11 • = 612 • Finally, copy the most left hand side’s integer as the most left hand side’s integer of the answer. i.e. ‘6+1’. • 632×11 • = 7612

  37. Let’s check the answer with the calculator. Aha. It’s correct! • Now it’s easy for you to calculate 71562435×11. You may found that it’s even easier to read it out in a vice versa way!!

  38. Don’t be cheated by the numbers! • A merchant have said that, If you want to make the disadvantages of a product into advantages, the best method is to compare it with another good product, and make the disadvantages into advantages.”

  39. It is just like an advertisement. It said that, “According to the survey and some teachers in the universities,said the ‘ABC Mathematics Textbook’ is the best textbook since the information inside the book is 25% more than the other mathematics textbook on the textbook market.”

  40. But if we think in detail,(first let’s ignore the ‘survey’ and ‘teachers in the universities’ since we don’t know whether it is true or not) what means by “25% more”? What do we use to compare with the ‘ABC Mathematics Textbook’?

  41. If we use the textbook we used in primary school, then the 25% more is useless. Also, 25% more is only the quantity more, how about the quality? Maybe the quality this book is the worst on the textbook market!

  42. Therefore, we must be very careful if we see some information or details. We need to know what the product it compare with. Just like a company, the manager can said that the profit they made have increased this year (compare with 10 years ago), have no change in profit (compare with 5 years ago), or even have decreased (compare with last year). It’s just depend on what we compared.

  43. Sometimes, a concept or a definition can be also very important. For example, in 1949, a person in Russia said that there are 14,000,000 people didn’t have job in USA. He said that this problem was very serious and should not be allowed. But after reading the report of USA, we find that the number of people who have no job were only 4,000,000. Why there are such big difference?

  44. It was because the definition of “No job” were different in 2 places. In USA, the meaning of “No job” is people didn’t have a job and is finding now. But in Russia, they think that people doing work less than 8 hours a day were consider as no job or part time job. Therefore their are such a big difference.

  45. In our daily life, sometimes we are also cheated by the static too. For example, number of accidents in aeroplanes increase compare to 50 years ago and number of people died because of cancer increase compare with 25 years ago. Then can we conclude that using aeroplanes to go to other countries is very dangerous? Can we conclude that nowadays, people are easier to have cancer?

  46. Of course we cannot. The number of accidents increase is because the number of aeroplanes increase almost 100 times, and number of people also increase a lot then 50 years ago. At the same time, although number of people dead of cancer increase a lot, but actually, the quality of medicine 25 years ago is not as good as now. Even some people dead in cancer, the doctors at that time may not be able to find out the reason of death.

  47. Therefore at the ‘REASON’ column, most of the time written was “Unknown’. Compare to nowadays, the medical have improve a lot, we can find out the reason of death more easily. Also, the population nowadays have increase a lot. So we cannot conclude something immediately after we have the information. We must think carefully before we conclude.

  48. Sometimes, using percentage to present something can also mislead the others. In the USA John Hopkin University, there was a report which made many people puzzled and felt unbelievable: “There were 33.3% girl students get married with the professors at university!” It’s really hard to believe. But actually, it is the true, it was because there were totally only 3 girls studying in the university and one of the girl get married with the professor!

  49. These are the traps that we always didn’t recognize. Therefore we must be very careful when we handle the static and information! Otherwise, we will be cheated by the number easily!

  50. What is the Chinese unit of some numbers? • 無量大數 1 0000 0000 0000 0000 0000 0000 0000 0000 • 0000 0000 0000 0000 0000 0000 0000 0000 • 0000 0000 0000 0000 0000 0000 • 不可思議 1 0000 0000 0000 0000 0000 0000 0000 0000 • 0000 0000 0000 0000 0000 0000 0000 0000 • 0000 0000 0000 0000 • 那由他 1 0000 0000 0000 0000 0000 0000 0000 0000 • 0000 0000 0000 0000 0000 0000 0000 0000 • 0000 0000

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