Math - Educational
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mathematics class, 10th Chapter one topic, irrational numbers and irrational numbers. Hello, students hope you all are doing well. Let's discuss about irrational numbers now. Irrational numbers are those numbers, which cannot be written in the form p by Q, where P and Q are in teachers and Q is not equal to zero. For example, square root of two comma square root of three comma square out of 15 et cetera. The square root of all the numbers do not give an irrational number. For example, square root of two is an irrational number, but square root of four is equal to two, which is rational. Therefore, we can conclude that square roots of all primes numbers are irrational. Tear, um, important. If a prime number be the white a square, then B divides eight. Where a is a positive in danger. Let us consider a positive in teacher 12 factors of 12 hour 12 is equal to two into two into three. On squaring 12, we get 12. Whole power to is equal to 1. 44 factors of 1. 44 are 1. 44 is equal to to into too into two into two into three into three. If one of the factors, let's say two divides 1 44 then this factor divides 12. As it is one of its factors. Let's talk about one of the most important examples proving that square root of two is an irrational number. Let us assume that square root of two is rational. Square root of two can be written in a by B form, where A and B are in teachers and B is not equal 20. Therefore, A and B are called crimes. Square root of two is equal to a by B on squaring both sides. Two is equal to a square by be square to be square is equal to a square, therefore, to divide a squid as we proved in the above. Kerem, we know that when two divides a square, then to divide a also so we can write A is equal to to see where C is an integer. Putting a is equal to. To see in one e que en To be square is equal to a square. To be square is equal to to see whole square implies to be square is equal to four c square. We get the square is equal to two C square. Again, we can see if two divides be square, then to divide. Be also. Therefore, two is a common factor of A and B, but this contradicts the fact that a we have no common factor other than one. So it is concluded that square root of two is irrational exercise to be given to student just for practising. So students, you can solve more questions like this. For exercise you can do 0.1 five minus square root of three proof that this is irrational Point number two show that three square root of two is irrational. Let's move ahead to the next topic. Rational numbers Revisiting rational numbers and their decimal expansion. You have studied that irrational numbers have either a terminating decimal expansion or a non terminating decimal expansion. Terminating decimal expansion 6.25 is terminating because it dominates after finite number of steps. Non terminating decimal expansion. MHM repeating zero point three three three 333 three is non terminating, repeating because it repeats the number three again and again point B non REPEATING one point 0330 3033 is non terminating, non repeating because it neither terminates nor repeats the particular number in the process of division if any rational number is given, whose decimal expansion is terminating the prime factory isation of its denominator in the form of to the power of n Fight, the power of M comma in comma m are positive in teacher Why is worse. So if any rational number is given whose denominator is of the form two to the power of n 20.5 to the power of M, where N kama m are positive in teacher, then the number has terminating decimal expansion. If denominator is not in the form two to the power of n fight, the power of M, then the number has non terminating decimal expansion. Examples. State whether 543 by 225 has terminating or non terminating decimal expansion. 543 by 225 can be written as 543 by 225 is equal to 181 by 75. Fact rising the denominator we get 543 by 225 is equal to 181 by three in two five whole Power two. Since the factors of the denominator three in two five whole power to which is not of the form two to the power of m point fight the power of n so it is non terminating reckoning decimal expansion decimal expansion of 51 by two to the power of three in to fight the power of four will terminate after how many decimal places making the denominator in the form of two to the power of endpoint Fight to the power of n where m is equal to end should be done multiplying both numerator and denominator by two. 51 by two to the power of three in to fight the power of four is equal to 51 by two to the power of three in to fight the power of four into two by two is equal to 102 by two to the power of four into five to the power of four is equal to 102 by to into five whole to the power of four is equal to 102 by 10 holds the power of four is equal to 102 by 10,000 is equal to 0.102. It will terminate after four places of decimals. Solve these types of questions on your own to improve your skills. Chapter one has been completed.