[L1] NUMBER SYSTEM (6).pdf
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Surabhi Gangwar has over 6 years of teaching experience and has mentored over 10,000 students, including 5,000 students for NTSE and Olympiad exams. She provides lessons on topics related to number systems, including natural numbers, whole numbers, integers, fractions, irrational numbers, real numbers, and rational numbers. She explains their properties and provides examples of representing them on a number line.
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[L1] NUMBER SYSTEM (6).pdf
- 1. Surabhi Gangwar B. Tech, M.Tech 6+ years Teaching experience 10,000+ students mentored 5,000+ students mentored for NTSE and Olympiad NUMBER SYSTEM Foundation
- 4. Fractions Irrational Numbers Negative integers Zero Natural Numbers Whole Numbers Real Numbers Rational Numbers Integers
- 5. Natural Numbers Set of all non-fractional numbers from 1 to n. Denoted by N. Natural Numbers
- 6. Whole Numbers If zero is adjoint to natural number then the collection is called whole numbers. Whole numbers are Denoted by W. Whole Numbers
- 7. All natural numbers, negatives of natural numbers and 0, together form the set Z or I of all integers. Integers Integers -3, -2, - 1, 0, 1, 2, 3
- 8. Fractions A numerical quantity that is not a whole number (e.g. 1/2, 0.5).
- 9. 1. Number which can be written as a ratio of two integers. 2. Rational numbers are generally denoted by Q. Rational Numbers Rational Numbers
- 10. Is zero a Rational number? Example
- 11. 1. Irrational means not rational. 2. Irrational number can not be represent in the ratio of integers, With no common factors (co primes). Irrational Numbers Irrationa l Numbers
- 12. Prove this √2 is an irrational number. Example
- 13. Real Numbers A number that can be found on the number line. Real numbers denoted by R. Real Numbers
- 14. DECIMAL EXPANSION TERMINATING (remainder becomes zero) NON TERMINATING (remainder never becomes zero) NON RECURRING/NON REPEATING (Remainder stops repeating ) RECURRING/REPEATING (Remainder repeates after certain stage)
- 15. Conversion of decimal number in p/q form Step-1 : Obtain the rational number. 1. When number is of terminating nature. Step-2 : Determine the number of digits in its decimal part Step-3 : Remove decimal point from the numerator. Write 1 in the denominator and put as many zeros on the right side of 1 as the number of digits in the decimal part of the given rational number Step-4 : Find a common divisor of the numerator and denominator and express the rational number to lowest terms by dividing its numerator and denominator by the common divisor.
- 16. Express this in p/q form. -10.275 Example
- 17. Step-1 : Obtain the repeating decimal and put it equal to x (say) 2. When number is of non terminating pure repeating/recurring nature Step-2 :Determine the number of digits having bar on their heads. Step-3 : If the repeating decimal has 1 place repetition, multiply by 10; a two place repetition, multiply by 100; a three place repetition, multiply by 1000 and so on.
- 18. Step-4 : Subtract the number in step 2 from the number obtained in step 3 Step-5 : Divide both sides of the equation by the coefficient of x. Step-6 : Write the rational number in its simplest
- 20. 3. When number is of non terminating and mixed repeating / recurring nature. Step-1 : Obtain the mixed recurring decimal and write it equal to x (say) Step-2 : Determine the number of digits after the decimal point which do not have bar on them. Let there be n digits without bar just after the decimal point Step-3 : Multiply both sides of x by 10 n so that only the repeating decimal is on the right side of the decimal point. Step-4 : Use the method of converting pure recurring decimal to the form p/q and obtain the value of x
- 21. Express this in p/q form. (1) 15.71212121212……. Example
- 22. 1. Let x = p/q be a rational number, such that the prime factorisation of q is of the form 2n × 5m where n, m are non negative integers. Then x has a decimal expansion which terminates. 1. Let x = p/q be a rational number such that the prime factorisation of q is not of the form 2m × 5n, where m, n are non negative integers. Then x has a decimal expansion which is non-terminating and recurring. Points To Remember
- 23. 1. All the terminating & non terminating recurring numbers are rational. Points To Remember 2. All the non terminating numbers are irrational. 3. Sum and difference of an rational and irrational number is irrational. 4. Negative of an irrational number is irrational 5. Sum, difference, product and quotient of two irrational need not to be irrational.
- 24. Find the six rational numbers between 3 and 4 and solve the problem solve this problem by two methods. Example
- 25. Find 3 irrational numbers between 3 and 5. Example
- 26. Insert 2 irrational number between and √2 and √3. Example
- 27. Insert 2 irrational number between and 0.12 and 0.13. Example
- 28. Representation of Rational and Irrational numbers on number Line:
- 29. Example Represent 3.765 on the number line.
- 30. Represent √2 and √3 on the number line. Example
- 31. Represent √4.3 on the number line. Example
- 33. Represent 4.2626262….. on the number line. Example
- 34. Prove that 2+√3 is an irrational number. Example