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Interesting Numbers Friendly Numbers Sociable Numbers Quadratic Formula

Interesting Numbers Friendly Numbers Sociable Numbers Quadratic Formula. Digital Root. The Digital Root of a non negative integer is calculated by adding all the digits of the number. The digits of this number are then added. This process is repeated until it results in a single-digit.

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Interesting Numbers Friendly Numbers Sociable Numbers Quadratic Formula

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  1. Interesting Numbers Friendly Numbers Sociable Numbers Quadratic Formula

  2. Digital Root • The Digital Root of a non negative integer is calculated by adding all the digits of the number. The digits of this number are then added. • This process is repeated until it results in a single-digit. Example 1: Find the digital root of 69,794,698 6 + 9 +7 + 9 + 4 + 6 + 9 + 8 = 58 5 + 8 = 13 1 + 3 = 4 The digital root is 4

  3. Narcissistic Numbers • These are numbers which equal the sum of their digits raised to the power of the number of digits. • 153 is a narcissistic number because it is a 3 digit number which is the sum of the cubes of its digits: 153 = 13 + 53 + 33

  4. Factorials • These are straightforward numbers which you will need at senior cycle level. They are denoted by an exclamation mark: • n! (n factorial) is the product of all the integers less than or equal to n. 7! = 7  6  5  4  3  2  1 = 5,040

  5. The Number e • e is an irrational number. It appears in calculations where values increase exponentially and continually. • It helps create the formulas for exponential systems, like the growth of bacteria, the growth of money in a compound interest account or radioactive decay. e= 27182818284590452…

  6. Divisors • Divisors are all the numbers which divide evenly into a number including the number 1, but not including the number itself in the case of friendly and sociable numbers • Therefore the divisors of 24 are 1, 2, 3, 4, 6, 8 and 12 • They sum to 36

  7. Friendly Numbers • These are pairs of numbers such that each number is the sum of the divisors of the other number. • Talismen sold in the middle ages would be inscribed with these numbers, on the grounds that they would promote love. • An Arab mathematician claims that people would write one of the pair of numbers on one fruit and eat it, writing the second number on another fruit and give it to a lover as a mathematical aphrodisiac!

  8. Friendly Numbers • There was only one pair discovered until Fermat discovered the pair 17,296 and 18,416 in 1636 • Descartes discovered the pair 9,363,584 and 9,437,056 in 1638 • In 1866 a sixteen year old Italian found the pair 1184 and 1210 • Computers can be programmed to find larger ones now!

  9. Project • The divisors of 1184 are 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592 • Sum = 1210 • The divisors of 1210 are 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605 • Sum = 1184

  10. Sociable Numbers • These are groups of three or more numbers which form closed loops • The sum of the divisors of the first give the second • The sum of the divisors of the second give the third and so on until the divisors of the last give the first number

  11. ––––––– –b±b2– 4ac –––––––––––––– x = 2a Quadratic formula • The Quadratic formula works to factorise all equations of the form ax2+bx+c= 0 • The roots are:

  12. 2 b- 4ac -b± x = 2a -(9)± (9)2 -4(1)(20) x = -9 ± 81–80 2(1) = 2 -9 ± 1 = 2 -10 2 -8 2 -9 ± 1 or = = 2 x2+ 9x+ 20 = 0 a= 1 b= 9 c= 20 x=-4 or –5

  13. The area of a rectangle is 77 cm2. One side is 4 cm longer than the other. Find the length and breadth of the rectangle. x+ 4 x Area = 77 cm2 x(x + 4) = 77 x2+ 4x– 77 = 0

  14. 2 b- 4ac -b± x = 2a -(4)± (4)2 -4(1)(–77) x = - 4 ± 324 - 4 ± 16+308 2(1) = = 2 2 14 2 -22 2 - 4±18 = or = 2 x2+ 4x– 77 = 0 a= 1 b= 4 c=–77 x=7 or –11

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