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25.11.2020, adminWhether you work in law, 10th class algebra textbook pdf, politics, economics, or mathematics, you will use proof to establish the validity of your propositions. In systems of logic and mathematics, a proof is a finite sequence of well-formed formulas. Our ncert solutions for class 10th english a tiger in the zoo full of geometric proof has its roots in ancient history. In B. In these books, Euclid made such effective use of proof in geometry that the books ncert solutions for class 10th english a tiger in the zoo full the standard in mathematics for over years.
In the seventeenth century, a French mathematician, Ren Descartes, solved geometric problems by applying the newly developed skills in algebra. This new mathematics was called analytic geometry. It did not displace the geometry of Euclid, but became an additional tool for mathematicians to use.
In this chapter, you will be introduced to the concept of proof and will learn how to construct a proof that presents a convincing mathematical argument. Solutions where some of the integers are zero are possible but not interesting, and are known as trivial solutions. Fermat wrote the statement in the margins of his Latin edition of a book called Ncert solutions for class 10th english a tiger in the zoo full, written by the Greek mathematician Diophantus in the third century A.
Fermat claimed, I have discovered a truly marvellous proof of this, which, however, this margin is too small to contain. The result has come to be known as Fermats Last Theorem because it was the last of his conjectures to remain unre- solved after his papers were published. It became famous among mathematicians because for hundreds of years many great mathematicians attempted to prove it and achieved only partial results. Investigate Fermats Last Theorem is closely related to the Pythagorean Theorem, and it is known that x 2 y 2 z 2 has integer solutions.
For example, 3 2 4 2 5 2. The numbers 3, 4, 5 are called a Pythagorean triple. There are an infinite number of Pythagorean triples. Try it. Then prove that it works for any a and b. Similar to Fermats Last Theorem is Leonhard Eulers conjecture that there are no non-trivial solutions to x 4 y 4 z 4 w 4. This question remained unresolved for over years until, inNaom Elkies found that 2 4 15 4 18 4 20 4 Since x 2 y 2 z 2 has infinitely many solutions and x 4 y 4 z 4 w 4 has at least one solution, it is hard to believe that x n y n z n has no solution.
Does x 3 y 3 z 3 have a non-trivial solution? Why is there no need to consider negative integer solutions to x n y n z n? That is, if we knew there were no solutions among the positive integers, how could we be sure there were no solutions among the negative integers? When Fermats Last Theorem was finally proven, its proof made headlines in newspapers around the world. Do you think the attention was justified?
The concept of proof lies at the very heart of mathematics. When we construct a proof, we use careful and convincing reasoning to demonstrate the truth of a mathematical statement. In this chapter, we will learn how proofs are constructed and how convincing mathematical arguments can be presented. Early mathematicians in Egypt proved their theories by considering a number of specific cases. For example, if we want to show that an isosceles triangle has two equal angles, we can construct a triangle such as the one shown and fold vertex B over onto vertex C.
In this example, B C; but that is only for this triangle. What if BC is lengthened or shortened? Even if we construct hundreds of triangles, can we conclude that B C for every isosceles triangle imaginable?
Consider the following example. One day in class Sunil was multiplying some numbers and made the following observation: 1 2 1 11 2 2 12 2 1 11 2 He concluded that he had found a very simple number pattern for 10th class algebra textbook pdf a num- ber consisting only of 1s.
The class immediately jumped in to verify these calcu- lations and was astonished when Jennifer said, This pattern breaks. The class checked and found that she was right. How many 1s did Jennifer use? From this example, we can see that some patterns that appear to be true for a few terms are not necessarily true when extended.
The Greek mathematicians 10th class algebra textbook pdf first endeavoured to establish proofs applying to all situations took a giant step forward in the development of mathematics. We follow their lead in establishing the concept of proof. In the example just considered, the 10th class algebra textbook pdf breaks down quickly.
Other examples, however, are much less obvious. Consider the statement, The expression 1 n 2where n is a positive integer, never generates a perfect square. Is this statement true for all values of n? Does this expression ever generate a perfect square? We start by trying small values of n. It turns out that the expression is not a perfect square for integers from 1 through to 30 It is a perfect square for the next integer, which illustrates that we must be careful about drawing conclusions based on calculations.
It takes only one case where the conclusion is incorrect a counter example to prove that a statement is wrong. We can use calculations or collected data to draw general conclusions. InJohn Snow, a medical doctor in London, England, was trying to establish the source of a cholera epidemic that killed large numbers of people.
By examining the location of infection and analyzing the data collected, he concluded that the source of the epidemic was contaminated water. The water was obtained from the Thames River, downstream from sewage outlets. By shutting off the contaminated water, the epidemic was controlled. This type of reasoning, in which we draw general conclusions from collected evidence or data, is called inductive reason- ing. Inductive reasoning rarely leads to statements of absolute certainty.
We will consider a very powerful form of proof called inductive proof later in this book. After we collect and analyze data, the best we can normally say is that there is evidence either to support or deny the hypothesis posed. Our conclusion depends on the quality of the data we collect and the tests we use to test our hypothesis.
In mathematics, there is no dependence upon collected data, although collected evidence can lead us to statements we can prove. Mathematics depends on being able to draw conclusions based on rules of logic and a minimal number of assumptions that we agree are true at the outset.
Frequently, we also rely on definitions and other ideas that have already been proven to be true. In other words, we develop a chain of unshakeable facts in which the proof of any state- ment can be used in proving subsequent statements. In writing a proof, it is important to explain our reasoning and to make sure that assumptions and definitions are clearly indicated to the person who is reading the proof. When a proof is completed ncert solutions for class 10th english a tiger in the zoo full there is agreement that a particular statement can be useful, the statement is called a theorem.
A theorem is a proven statement that can be added to our problem-solving arsenal for use in proving subsequent statements. Theorems can be used to help prove other ideas and to draw conclusions about specific situations.
Theorems are derived using deductive reasoning. Deductive reasoning allows us to prove a statement to be true. Inductive reasoning can give us a hypothesis, which might then be proved using deductive reasoning. As an example of inductive reasoning, note that if we write triples of consecutive integers, say 11, 12, 13exactly one of the three is divisible by 3.
If a number of such triples are written say two or three by everyone in the class ncert solutions for class 10th english a tiger in the zoo full, we can observe that every triple has exactly one number that is ncert solutions for class 10th english a tiger in the zoo full multiple of 3.
This provides strong evidence for us to conclude inductively that every such triple contains exactly one multiple of 3, but it is not proof.
We will consider deductive proof in the other sections of this chapter. Part A In each of the following exercises, you are given a mathematical statement. Using inductive reasoning that is, 10th class algebra textbook pdf specific casesdetermine whether or not the claim made is likely to be true. For those that appear to be true, try to develop a deductive proof to support the claim. All integers ending in 5 create a number that when squared ends in Test for the first ten positive integers ending in 5.
The expression f n n 3 5n 2 5n 6, where n is a positive integer, gives a composite number for all ncert solutions for class 10th english a tiger in the zoo full of n. Test for n 1, 2, 3, 4, 5, 6. In ncert solutions for class 10th english a tiger in the zoo full set of four positive integers such that the second, third, and fourth are each greater by 5 than the one preceding, there is always one divisible by 4.
One such set is 1, 6, 11, Test using 5, 6, 7, 8, 11, 14, and 17 as first numbers in the set. Part B 4. The expression n 2 n 5 generates a prime number for every positive integer value of n. The expression n 2 n 11 generates a prime number for every positive integer value of n. The expression n 2 n 41 generates a prime number for every positive integer value of n.
Inductive reasoning is a method of reasoning in which specific examples lead to a general conclusion. Straight lines are drawn in a plane such that no two are parallel and no three meet in a common point.
It is claimed that the nth line creates n new regions. For example, the first line divides the plane into two regions, creating one region in addition to the original one. Test this claim for n 1, 2, 3, 4, 5. If 9 is subtracted from the square of an even integer n greater than 2, the result is a number that is composite has factors other than 1. Test this claim for n 4, 6, 8, 10, If 9 is subtracted from the square of an odd integer n greater than 3, the result is a number that is divisible by 8.
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