Number system
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This document provides an introduction to different types of number systems including integers, decimals, rational and irrational numbers. It also discusses place value and face value. Several formulas are presented for calculating sums of natural numbers, odd numbers, even numbers, squares and cubes. Some tips and shortcuts are given for multiplication and finding remainders when dividing large numbers.
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* Let A's age be x and B's age be y
* Then average of A and B is (x + y)/2 = 20
* So, x + y = 40
* If C replaces A, average is (y + c)/2 = 19
* So, y + c = 38
* If C replaces B, average is (x + c)/2 = 21
* So, x + c = 42
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x = 22, y = 18, c = 20
The ages are:
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Number system
- 2. Introduction • Real Numbers : Decimals (fractions), Integers (No Decimals) • Decimals : RATIONAL – Finite (Terminating Decimal 2.7, 3.4) Infinite (Non Terminating Repeating Decimal 2.1212, 2.66) IRRATIONAL – Non terminating Non repeating decimal 2.5813, ∏
- 3. Introduction • Integers : Negative, Zero, Positive • Natural Numbers : Positive Integers 1,2,3,4.. • Whole Numbers : Includes 0.. 0,1,2,3… • 1 = Neither prime Nor Composite
- 4. Introduction • Face Value : 1, 2, 3 is face value. In 34562 the face value of 2 and 3 is 2 and 3 respectively. • Place Value : In 34562 the place value of 2 and 3 is 2 and 30000 respectively and 5 is 500.
- 5. Formulae • Sum of first N natural numbers : Sn = n(n+1) / 2 • Sum of squares first N natural numbers : Sn = n(n+1)(2n +1) / 6 • Sum of cubes first N natural numbers : Sn = [n(n+1) / 2]2
- 6. • Sum of first N odd natural numbers : Sn = [n]2 • Sum of first N even natural numbers : Sn = n(n+1) Formulae
- 7. Tips • (xn – an) is completely divisible by (x + a) when n is even number. • (xn + an) is completely divisible by (x + a) when n is odd number. • What is the remainder when 17200 is divided by 18? (17200 – 1200) is completely divisible by (17 + 1) as 200 is even number. So when 17200 is divided by 18 remainder is 1.
- 8. Contd.. • If 232 + 1 is completely divisible by a whole number, which number is completely divisible by this number? • Lets assume 232 + 1 = x + 1 296 + 1 = {(232 ) 3 + 1} = (x 3 + 1) = (x + 1)(x2 – x + 1)
- 9. Contd.. • What is the common factor of (22121 + 19121 ) and (22231 + 19231 ) • Answer will be 22+19 = 41. • Since 121 and 231 are odd.
- 10. Some shortcuts • Multiplication with base 100 : • 98*115 =? • -2*15=-30 • 115-2=98+15=113 • (113*100)-30 = 11300 – 30 = 11270
- 11. Some shortcuts • Multiplication with base 300 : • 297*292 =? • -3*-8=24 • 297-8=292-3=289 • (289 * 300) + 24 = 86700 + 24 = 86724
- 12. Some shortcuts • Multiplication with base 1000 : • 997*983 =? • -3*-17=51 • 997-17=983-3=980 • (980*1000)+51 = 980000 + 51 = 980051
- 13. Problem • What will be the remainder when (6767 + 67) is divided by 68? • (xn + an) is completely divisible by (x + a) when n is odd number. • So (6767 + 1) + 66 is when divided by 67+1 (68) • So remainder will be 66 as (6767 + 1) is completely divided by 68.
Editor's Notes
- n2