1 / 31

Types of Numbers and Number Systems: Lecture 1

Learn about different types of numbers such as natural numbers, whole numbers, integers, even numbers, and odd numbers. Understand prime numbers, composite numbers, and how to determine if a number is prime or not. Discover tests of divisibility and arithmetic and geometric progressions.

clemonsk
Download Presentation

Types of Numbers and Number Systems: Lecture 1

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. PEA202 – Lecture #1 NUMBER SYSTEM

  2. DIGITS 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 NUMERAL A group of digits, denoting a number.

  3. 2 3 6 5 PEA504A Analytical Skills-II :: Vishal Ahuja

  4. TYPES OF NUMBERS NATURAL NUMBER 1 , 2 , 3 , 4 , 5 . . . WHOLE NUMBER All natural numbers including 0. INTEGERS All natural numbers, 0 & negative numbers {…, -3, -2, -1, 0, 1, 2, 3, …} • Positive Integers {1, 2, 3, …} • Negative Integers { -1, -2, -3, …} • Non-Positive & Non-Negative Integer is 0. PEA504A Analytical Skills-II :: Vishal Ahuja

  5. TYPES OF NUMBERS PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja

  6. EVEN & ODD NUMBERS EVEN NUMBER No’s divisible by 2. ODD NUMBER No’s not divisible by 2. PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja

  7. Facts about Even & Odd No’s Sum / Difference of two even numbers is an even number. Sum / Difference of two odd numbers is an even number. Sum / Difference of an even number and an odd numbers is an odd number. PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja

  8. Quick yak: Discuss Smallest prime no is 2 and 1 is not a prime no. 2 is the only even prime no. Rest all prime no are odd numbers. TYPES OF NUMBERS – Prime No A Prime Nois a natural no greater than 1 that has no positive divisors other than 1 and itself. Prime no’s upto 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. A natural number greater than 1 that is not a prime number is called a Composite No. Two no’s a & b are said to be Co-Primes, if their HCF is 1. PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja

  9. How to find whether a no is Prime or not Suppose we have to find whether 191 is prime or not. Now 14 > √191 Prime no less than 14 are 2,3,5,7,11,13. 191 is not divisible by any of these prime no. So 191 is a prime number. PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja

  10. Tests of Divisibility

  11. Tests of Divisibility Quick yak: For division with 7 & 13, use simple ways of division. Shortcuts are very complex.

  12. Division Algorithm Dividend = (Divisor * Quotient ) + Reminder P

  13. Basic Formulae (a + b)(a - b) = (a2 - b2) (a + b)2 = (a2 + b2 + 2ab) (a - b)2 = (a2 + b2 - 2ab) (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca) (a3 + b3) = (a + b)(a2 - ab + b2) (a3 - b3) = (a - b)(a2 + ab + b2) (a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ac) When a + b + c = 0, then a3 + b3 + c3 = 3abc. PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja P

  14. ARITHMATIC PROGRESSION An Arithmetic Progression (A.P.) is a sequence in which the difference between any two consecutive terms is constant. Let a = first term, d = common difference Then nth term PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja P

  15. Sum of an A.P. PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja P

  16. GEOMETRICAL PROGRESSION PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja P

  17. GEOMETRICAL PROGRESSION PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja P

  18. HOW TO FIND THE UNIT DIGIT OF A NUMBER Cyclicity P PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja

  19. Question Q 1.1 Which one of the following are prime number? A. 241 B. 337 C. 391 D. 571 PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja P

  20. Solution Q1.1 Clearly, 16 > √241 Prime numbers less than 16 are 2,3,5,7,11,13. 241 is not divisible by any one of them. So, 241 is a prime number. PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja P

  21. Question Q1.2 1397 x 1397 = ? A. 1951609 B. 1981709 C. 18362619 D. 2031719 E. None of these PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja P

  22. Solution Q1.2 Answer: Option A Explanation: 1397 x 1397 = (1397)2 = (1400 - 3)2 = (1400)2 + (3)2 - (2 x 1400 x 3) = 1960000 + 9 - 8400 = 1960009 - 8400 = 1951609. PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja P

  23. Question Q1.3 What least number must be added to 1056, so that the sum is completely divisible by 23 ? 2 B. 3 C. 18 D. 21 E. None of these PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja P

  24. Solution Q1.3 Answer: Option A PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja P

  25. Question Q1.4 Find the unit digit of 295 PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja P

  26. Solution Q1.4 The cyclicity of 2 is 4. 21 = 2 22 = 4 23 = 8 24 = 16 Divide 95 by 4. Remainder is 3. So, the unit digit is 8. PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja P

  27. Practice P1.1 The largest 4 digit number exactly divisible by 88 is: A. 9944 B. 9768 C. 9988 D. 8888 E. None of these Quick yak: Use concept of Q 1.3 PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja P

  28. Solution P1.1 Answer: Option A Explanation: Largest 4-digit number = 9999 Reminder of 9999 / 88 = 55 Required number = (9999 - 55) = 9944. PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja P

  29. Practice P1.2 Find the unit digit of 999 Quick yak: Use concept of Q 1.4 PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja P

  30. Solution P1.2 The cyclicity of 9 is 2. 91 = 9 92 = 81 Divide 99 by 2. Remainder is 1. So, the unit digit is 9. PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja P

  31. Next Class: HCF & LCM PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja P

More Related