Maths Marvels

8/14/2019 Maths Marvels

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Some InterestingSome InterestingMaths MarvelsMaths Marvels

Math Magic


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1.1.

ARMSTRONG NUMBERS


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DEFINITIONDEFINITIONAn Armstrong number or a plus perfect number is anumber that is the sum of its own digits each raised tothe power of the number of digits.

153 = 1 3 + 5 3 + 3 3

370 = 3 3 + 7 3 + 0 3

371 = 3 3 + 7 3 + 1 3

EXAMPLESEXAMPLES


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2.

Dudeney Number Dudeney Number rfect cube such that the sum of its decimal digits is equal to the cube root

The name derives from Henry Dudeney


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1 x 1 x 1 ; 1 = 1 = 8 x 8 x 8 ; 8 = 5 + 1 + 2

3 = 1 7 x 1 7 x 1 7 ; 1 7 = 4 + 9 + 1 + 3

2 = 1 8 x 1 8 x 1 8 ; 1 8 = 5 + 8 + 3 + 2 7 6 = 2 6 x 2 6 x 2 6 ; 2 6 = 1 + 7 + 5 + 7 + 6 8 3 = 2 7 x 2 7 x 2 7 ; 2 7 = 1 + 9 + 6 + 8 + 3

EXAMPLESEXAMPLES


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FACTORIANFACTORIAN

factor ion is

natural number that quals the sum of he factor ials of

.ts decimal digits

3.3.


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EXAMPLESEXAMPLESThere are just four factorions and theyare 1, 2, 145 and 40585.

1! = 1

2! = 2*1 = 2145! = 1! + 4! + 5!40585! = 4!+0!+8!+0!+5!


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FRIEDMAN NUMBERFRIEDMAN NUMBER

4.4.

A Friedman number is an integer which, in a given

base, is the result of an expression using all its owndigits in combination with any of the four basicarithmetic operators (+, , , ) and sometimesexponentiation.


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EXAMPLESEXAMPLES

25, 121, 125, 126, 127, 128, 153, 216, 289, 343,

347, 625, 688, 736, 1022, 1024, 1206, 1255, 1260,1285, 1296, 1395, 1435, 1503, 1530, 1792, 1827,2048, 2187, 2349, 2500, 2501, 2502, 2503, 2504,2505, 2506, 2507, 2508, 2509, 2592, 2737, 2916,3125, 3159 etc.


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MUNCHAUSEN NUMBERS

3435 = 3 3 a+ 4 4 + 3 3 + 5 5438579088 = 4 4 + 3 3 + 8 8 + 5 5 + 7 7 + 9 9 + 0 0 + 8 8 + 8 8

5.

A Munchausen number is a number that is equal to the sum of its digits eachraised to a power equal to the digit. 0 and 1 are Munchausen numbers in anybase (using the convention that 0 0 = 0). Apart from 0 and 1 there are only two

other Munchausen numbers in the decimal system:


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PERFECT NUMBERSPERFECT NUMBERS

6.6.

In mathematics, a perfect number is a positive integer thatis the sum of its proper positive factors, that is, the sum of

the positive factors excluding the number itself.


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EXAMPLES1 6 1 ?

2 28 2 ?

3 496 3 ?

4 8128 4 ?

5 33550336 8 1456 ?

6 8589869056 10 1588 Cataldi

7 137438691328

12 1588 Cataldi


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FIBONACCI SERIESFIBONACCI SERIESthe 12th century, Leonardo Fibonacci discovered a simple numerical series

tarting with 0 and 1, each new number in the series is simply the sum of the two be, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .

7.

F 0 F 1 F 2 F 3 F 4 F 5 F 6 F 7 F 8 F 9 F 10 F 11 F 12 F 13 F 14 F 15 F 16 F 17 F 18 F 19 F 20

0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597

2584

4181

6765


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0+1 = 10+1 = 11+1 = 21+1 = 2

1+2 = 31+2 = 3And so onAnd so on


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FIBONNACI SEQUNCE IN NATURE

The distribution of seeds in sunflower is spiral. Theseeds of the sunflower spiral outwards in bothclockwise and counterclockwise directions from thecenter of the flower. The number of clockwise andcounterclockwise spirals are two consecutivenumbers in the Fibonacci sequence.

The shells of the snails follow the Fibonaccisequence. In the same way, the shells of thenautilus follow the same rule. The only differencebetween these two is that nautilus shells grow ina three-dimensional spiral, whereas snails shellsgrow in a two-dimensional spiral.


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PHI OR GOLDEN RATIOPHI OR GOLDEN RATIO8.8.

In mathematics, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is

equal to (=) the ratio of the larger quantity to the smaller one.The golden ratio is approximately 1.618033988


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P H I I N O U R LI F E

P H I I N O U R LI F E


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In human body, the ratio of the length of forearm to the length of the hand is equal to 1.618, that is, Golden Ratio. Another well-known examples on human body are:

A.The ratio between the length and width of faceB.C. Ratio of the distance between the lips and where the eyebrows

meet to the length of noseD.E. Ratio of the length of mouth to the width of noseF.G. Ratio of the distance between the shoulder line and the top of

the head to the head lengthH.I.J. Ratio of the distance between the finger tip and the elbow to the

distance between the wrist and the elbow

IN HUMAN BODYIN HUMAN BODY


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QUESTIONQUESTION

HOURHOUR


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BY:BY:KARTIKAY MITTALKARTIKAY MITTAL

IX-AIX-A1616

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Maths Marvels