Scientific Method Steps http://www.youtube.com/watch?v=bNc9vWLDSCA create oiyv >¥se theory modify the/ \o make a theory / prediction Observation! [ Prediction | design an perform the\ / experiment experiment\ /to test the prediction Pre-listening. 1. The speaker mentions five steps you go through in the scientific method. Try to order these steps Experiment Research Conclusion Question Hypothesis 2. These are the verbs that combine with the steps - nouns. Match nouns and verbs, come up with do form draw design ask pose test conduct define run reject prove Listening. Listen to and watch the video and answer Qs. 1. In the speaker's opinion, what is important for doing science?......................................... 2. What is typical for the scientist regarding his attitude to research?.................................. 3. What kind of a question does the speaker pose?................................................................. 4. What is the difference between a scientific and non-scientific hypothesis?..................... 5. What sort of an experiment would the speaker do?............................................................ 6. What is drawing conclusions a tricky step?........................................................................... 7. What is the difference between a hypothesis and a theory?.............................................. 8. Which theory does the speaker mention which was rejected by scientists in the past? Listen to the beginning of the talk again and try to correct mistakes in these automatic subtitles. depending on your teacher your textbook you'll see many different ways of breaking down the scientific method but no matter how many were versions i've seen the all seem to share some way these five steps the first thing that whenever you're doing science is ask a question do some research come up with a hypothesis tested with a hypothesis it started with an experiment and then finally draw conclusion now some people have this idea that in order to do sign do you have to have been wearing a lab coat and have a p_h_d_ that's not it at all and its heart decides on his views beatrix instructions can questions about the world for example of this could range anywhere from isn't important indeed as how much your cancer too as simple as what's the fastest way for me to get from san jose to san francisco now that i've asked a question it's time for me to find out what the people who know more thinking up and this is where some people wonder running into trouble because a scientist or somebody who's thinking like a scientist when they do their research that always skeptical about but they're reading Section 2 Development 4. Read this: Axioms Some mathematical laws are accepted without proof. These fundamental laws are known as axioms. In algebra, for example, there are the following axioms. 1) x+y=y+x 2) xy = yx 3) x+(y+z) = (x+y)+z 4) x(yz) = (xy)z 5) x(y+z) = xy + xz 6) x+0 = x 7) lx = x 8) For every real number x, there is a real number y such that x + y = 0. 9) For every non-zero real number x, there is a real number y such thatxy=l. ?S~ I " f j •2tis axiqn^iticjjthatifx+y = Mheny+x = a. 1 Write similar sentences about the other axioms. 5. Read this: • Axiom lisillustratedbythefollowingexample:2 + 3 = 5; 3 + 2 = 5. Write similar sentences using these examples: a) 1x22 = 22 b) 5 is a non-zero real number; 1/5 is a real number c) 5x7 = 35;7x5=3S d) 3x(ll + 19) = 3 x 30 = 90;(3xll) + (3 x 19) = 33 + 57 = 90 e) 3+0 = 3 f) 3+(4+5) = 3+9= 12;(3+4) + 5 = 7 + 5 = 12 g) 5 is a real number: — 5 is a real number h) 2x(3x4) = 2xl2 = 24;(2x3)x4 = 6x4 = 24 Section 3 Reading 6. Read this: Theorems From the axioms given in the last section we can prove all the laws of algebra. Laws which are not axioms are called theorems. There are various methods of proving theorems. Some examples are given here. * 1 Proof by deduction Theorem (I) To prove: If a + b = a + c, then b = c. Proof By axiom (8), there is a number y such that y + a = 0. Adding y to both sides gives y + (a + b) = y + (a + c). By axiom (3), we have (y +a) + b = (y + a) + c. Buty+a = 0, thereforeO + b = 0+c. By axiom(6),0 + b = b andO+c = c. Therefore b = c. 2 and 3 Proofs by elimination and contradiction Thereom (2) To prove: Given a and b, there is one and only one x such that a + x = b. Proof-There are three possibilities: more than one x, less than one x, exactly one x. If we eliminate two of these possibilities, then the third must be true. We can divide the proof into three parts, a) To prove that there is not less than one x such that a + x = b. By axiom (8), there is a number y such that a +y = 0. fC Let x = y + b. Thena + x = a + (y + b). By axiom (3) a + x = (a+y) + b. Buta + y = 0, therefore a + x = 0+b. By axiom (6), a + x = b. Hence there is not less than one x such that a + x = b. b) To prove there is not more than one x such that a + x = b. We can prove this by using proof by contradiction, i.e. we assume the opposite of what we are trying to prove and show that this leads to a contradiction. Assume that there are several different x, such that x, ?*x2/x3, etc. and a+Xj = b, a + x2 = b, a + x3 = b, etc. i.e. a+x, = a + x2 = a + x3,etc. By theorem (1), we have x, = x2 = x3, etc., but this contradicts our assumption that x, ^ x2 ^ x3, etc. Therefore there is not more than one x such that a + x = b. c) Two possibilities have been eliminated, therefore the only remaining possibility is that there is one and only one x such that a + x = b. Say whether the following statements are true or false. Correct the false statements. a) Theorems are fundamental laws. b) There are not less than three ways of proving theorems. c) All of the symbols a, b, x, y used in the above proofs refer to real numbers. d) All of the symbols a, b, x, y used in the above proofs refer to non-negative numbers. e) The three kinds of proof shown are mutually exclusive. 7. Look and read: Here are the proofs of two more theorems. Complete them by inserting the correct words and say what kinds of proof are used. a) Theorem (3): If ab = ac and a # 0, then b = c. By......there is a number y......ya = 1. __both sides by y,......y(ab) = y{ac). By......(ya)b = (ya)c. Butya = 1,___lb = lc. By.......lb = band lc = c. _b = c. b) Theorem (4): Given a ^ 0 and b, there is one and only one x such that ax = b. There are three possibilities,......one x.......one x,___one i) By......there is a number y......ya = 1 ^ OJ^fjLX ■jjuw ^WAxu ~%uUJU . r •--r—r— X = yb. --_ax = a(yb). By......,wehaveax = (ay)b. -ay = 1,--ax = lb. By......ax = b. --there is...... ") -axj = b.axj = b,ax3=betc. andxj^x2 ^x3etc. --ax, = axj = ax3 etc. By.......wehavex, = x2 * x3etc. But this -our-that xt * x2 # x3 etc. ■--there is...... m) -the only-_ possibility is that there is. Proof: Beginnings (Trzeciak Jerzy,Writing Mathematical Papers in English. European Mathematical Society, 1995) We prove (show, recall, observe) that...........Let us first outline (give the main ideas of) the proof. We claim that.... Our proof starts with the observation that....... Suppose....... Assuming..........The proof falls naturally into three parts. To deduce (3) from (2), take..............We have divided the proof into a sequence of lemmas. To see that........ Proof: Arguments By definition... .By assumption... By a similar argument.... Define .... Theorem 4 now shows that.... Since/is compact......Therefore Lf= 0 by Theorem 6. From this we conclude...........We now prove that........From what has already been proven......According to...From the above it follows that.......We conclude from (5) that ....................Changing the formula we obtain.... Proof: Conclusions This proves the theorem. This completes the proof. This establishes the formula. This is the desired conclusion. Q.E.D. This contradicts our assumption that.... The same proof works for.... Note that we have actually proven that.....The proof is strongly dependent on the assumption that.............Since......we reached a contradiction.....The details are left to the reader. For more details we refer the reader to .... Proof that e is irrational From Wikipedia, the free encyclopedia In mathematics, the series representation of Euler's number e n=0 can be used to prove that e is irrational. Of the many representations of e, this is the Taylor series for the exponential function ey evaluated at>> = 1. Summary of the proof This is Joseph Fourier's proof by contradiction. Initially e is assumed to be a rational number of the form alb. We then analyze a blown-up difference x of the series representing e and its strictly smaller 6th partial sum, which approximates the limiting value e. By choosing the magnifying factor to be the factorial of b, the fraction alb and the 6 th partial sum are turned into integers, hence x must be a positive integer. However, the fast convergence of the series representation implies that the magnified approximation error x is still strictly smaller than 1. From this contradiction we deduce that e is irrational. Now go through the proof and fill in the missing expressions using examples from proof writing. Proof Towards a contradiction, 1)..........e is a rational number. Then 2)..............a and b such that e = alb where clearly b > 1. 3)........the number *=6!(e-£i) To see that if e is rational, then x is an integer, 4).........ee = a/b into this definition to 5). ~«(?-£.i)-*-i>'-E5- v n=0 7 n=0 The first term is an integer, and every fraction in the sum is actually an integer because n < b for each term. 6)..........x is an integer. 7)..............0 ° v n=0 7i=0 / n=6+l because all the terms with n < b cancel and the remaining ones are strictly positive. We now prove that x < 1. For 9)........with n > b + 1 we have the upper estimate bl 1 ^ 1 nl (&+l)(&+2)...(6+(n-6)) ~ (6+1)«"*' This inequality is strict for every n > b + 2. 10).........the index of summation to k = n -b and using the formula for the infinite geometric series, we obtain °° b\ °° 1 1 1 / 1 \ 1 x=„£,^< „£, =5 =^ =*< Since there is no integer strictly between 0 and 1, we have 11)............, and so e must be irrational. 12).............