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Interesting Numbers

Friendly Numbers

Sociable Numbers

Quadratic Formula

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• 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.

Digital RootDigital Root

Example 1: Find the digital root of 69,794,6986 + 9 +7 + 9 + 4 + 6 + 9 + 8 = 585 + 8 = 131 + 3 = 4

The digital root is 4

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• 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:

Narcissistic NumbersNarcissistic Numbers

153 = 13 + 53 + 33

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• 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.

FactorialsFactorials

7! = 7 6 5 4 3 2 1 = 5,040

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• 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.

The Number eThe Number e

e = 27182818284590452…

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• 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

DivisorsDivisors

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• 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!

Friendly NumbersFriendly Numbers

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Friendly NumbersFriendly 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!

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ProjectProject

• 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

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Sociable NumbersSociable 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

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Quadratic formulaQuadratic formula

• The Quadratic formula works to factorise all equations of the form ax2 + bx + c = 0

• The roots are:

––––––––––––––

–b b2 – 4ac

2a

x =

–––––––

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a = 1 b = 9 c = 20

(9) 2 4(1)(20)

2(1)x =

(9)

x2 + 9x + 20 = 0

b 4ac2

2a x =

b

81 – 80

2 =

9 1

2 =

9

2 =

9 1 8

2 =

x = 4 or – 5

or10

2

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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.

Area = 77 cm2

x + 4

x

x2 + 4x – 77 = 0

x(x + 4) = 77

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a = 1 b = 4 c = – 77

(4) 2 4(1)(–77)

2(1)x =

(4)

x2 + 4x – 77 = 0

b 4ac2

2a x =

b

x = or – 11

or22

2

16 + 308

2 =

4 324

2 =

4

2 =

4

18 = 14

2

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