Transcript Chapter 16 - Mentor Books

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Interesting Numbers
Friendly Numbers
Sociable Numbers
Quadratic Formula
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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
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The digital root is 4
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
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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
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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…
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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
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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!
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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!
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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
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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
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Quadratic formula
• The Quadratic formula works to factorise all
equations of the form ax2 + bx + c = 0
• The roots are:
–––––––
– b  b2 – 4ac
x = ––––––––––––––
2a
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x2 + 9x + 20 = 0
a = 1 b = 9 c = 20
x=
x=
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- b b 2 - 4ac
2a
- (9)  (9) 2 - 4(1)(20)
2(1)
-9  1
-9  81 – 80
=
=
2
2
-9  1
- 10
-8
or
=
=
2
2
2
x = -4 or –5
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
Area = 77 cm2
x(x + 4) = 77
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x2 + 4x – 77 = 0
x
x2 + 4x – 77 = 0
a = 1 b = 4 c = –77
x=
x=
- b b 2 - 4ac
- (4)  (4) 2 - 4(1)(–77)
2(1)
16 + 308 - 4  324
=
=
2
2
14
- 22
- 4  18
or
=
=
2
2
2
x = 7 or –11
-4 
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2a
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