The Cubic Difference Formula (Mi)

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THE CUBIC DIFFERENCE FORMULA

(FINDING THE DIFFERENCE OF NEGATIVE CONSECUTIVE CUBED INTEGERS)

A Math InvestigationIn partial fulfillment of Statistics,

Analytic Geometry and Pre-Calculus

Blacano, Janine R.Dizon, Julius Blake C.

Rusiana, Kate Nicole S.

RAMON TEVES PASTOR MEMORIALDUMAGUETE SCIENCE HIGH SCHOOL

Mathematical InvestigationSY 2010-2011


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iABSTRACT

Mathematics has been especially hard for students everywhere. It is considered an elite subjectexcept for a chosen few. Most of the students are intimidated by this subject because not all can excel

because of some complicated equations or problems. Students dedicate themselves to find a shorter wayto solve for long and difficult mathematical equations. But nowadays, most of the students are dependenton using calculators, cell phones, adding machines and some enthralling devices. These devices havereplaced the accurateness of manual solving or counting problems. Too much dependence on theseobjects, can also affect ones motivation in learning math proving to be harmful. The researchers thenthought of a study of a mathematical way of solving the difference of negative consecutive cubedintegers. The success of this study can cover the problem of solving and acquiring the cube of largenumbers especially three-digit numbers. This study does not focus merely on solving the difference of

negative consecutive integers, but rather it focuses on presenting a better and convenient way of solvingthe difference of these negative consecutive cubes.

After several trials done, it has been proven that the tests were successful with all number ranges,like 5-digit numbers. Average students can comprehend this study, thereby bridging the gap betweenthem and math. No errors have been found and it can be applied to math problems anytime and anywhere.All you need is a paper and a pen.


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iiACKNOWLEDGEMENT

All glory and praise goes back to God, without whose guidance, all of math would be butrandom guesses in a chaotic universe.

Some of the people who helped make this study a humble success include the following:

For their helpful advice: Mrs. Joan Dolino

Ms. Mary Lou Tumapon

For patiently answering our questions: Mrs. Haide Duran

For tirelessly directing our efforts: Mr. Alvin Leo Suasin

For keeping a watchful eyes on us: Mrs. Melinda Favor

Others who helped in the completion of this study include Vaughn Randy Evero and EmilioTecson.


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iiiTABLE OF CONTENTS

Title Page

Abstract

Acknowledgement

Table of Contents

Introduction

Background of the Study

Statement of the Problem

Significance of the Study

Scope and Limitations

RRL

Methodology

Results and Discussion

Summary and Conclusions

Recommendations

Bibliography


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INTRODUCTION

Students nowadays are already very dependable on electronic devices and gadgets to

solve Mathematical problems and calculations. They depend so much on using these electronic

devices like cell phones, calculators, computers, adding machines and others. They use these

machines because of the convenience it offers to them, but they might have technical difficulties

thats why manual solving is still a better way for calculating and solv ing mathematical

problems. And because it uses the human brain, it also enhances the ability of the brain to solve

and understand.

Background of the Study

The difference obtained from the cubes of negative consecutive integers is a long

and difficult process that may take a long time especially if the numbers are 2 digit numbers.

So the researchers found a shortcut, or an easier way to solve the differences of these integers to

Offer convenience to the student.

Significance of the Study


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Students nowadays depend too much on calculators to solve a certain equation in just

seconds. Some cubes are too large and needs a lot of time to compute. The researchers found a

way to compute the difference of two negative consecutive integers if the minuend is greater

than the subtrahend. Hence, this study is very useful for computing the difference of two

negative consecutive integers easier, fast, and it is not time- costly, and most of all, it doesnt

need the use of calculators.

Statement of the Problem

This study aims to find and easier and faster way in solving and computing the difference

of negative consecutive cubed integers especially if it has a high value. And it can answer the

following questions:

1. Is the study an answer to a fast computation in solving the difference of negative

consecutive cubed integers?

2. Is the study practical and simple to use, even for an average student?

Scope and Limitations

The main purpose of this study is to find a fast and easier way of solving the differences

of negative consecutive integers. This study is also limited to the use of consecutive negative

integers and when a > b.


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Review of Related Literature

Cube (algebra )y=x, for integer values of 1x25.

In arithmetic and algebra, the cube of a number n is its third power the result of the number

multiplying by itself three times:

n3 = n n n.

This is also the volume formula for a geometric cube with sides of length n, giving rise to the

name.

The inverse operation of finding a number whose cube is n is called extracting the cube root of

n. It determines the side of the cube of a given volume. It is also n raised to the one-third power.

A perfect cube (also called a cube number , or sometimes just a cube ) is a number which is the

cube of an integer.

The sequence of non-negative perfect cubes starts (sequence A000578 in OEIS) :

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832,

6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389, 27000, 29791,

32768, 35937, 39304, 42875, 46656, 50653, 54872, 59319, 64000, 68921, 74088, 79507, 85184,

91125, 97736, 103823, 110592, 117649, 125000, 132651, 140608, 148877, 157464, 166375,

175616, 185193, 195112, 205379, 216000, 226981, 238328...

Geometrically speaking, a positive number m is a perfect cube if and only if one can arrange m

solid unit cubes into a larger, solid cube. For example, 27 small cubes can be arranged into one

larger one with the appearance of a Rubik's Cube, since 3 3 3 = 27.
http://en.wikipedia.org/wiki/File:CubeChart.svghttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Exponentiationhttp://en.wikipedia.org/wiki/Volumehttp://en.wikipedia.org/wiki/Cube_%28geometry%29http://en.wikipedia.org/wiki/Inverse_functionhttp://en.wikipedia.org/wiki/Cube_roothttp://en.wikipedia.org/wiki/Integerhttp://oeis.org/A000578http://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://en.wikipedia.org/wiki/Rubik%27s_Cubehttp://en.wikipedia.org/wiki/Rubik%27s_Cubehttp://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://oeis.org/A000578http://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Cube_roothttp://en.wikipedia.org/wiki/Inverse_functionhttp://en.wikipedia.org/wiki/Cube_%28geometry%29http://en.wikipedia.org/wiki/Volumehttp://en.wikipedia.org/wiki/Exponentiationhttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/File:CubeChart.svg


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The pattern between every perfect cube from negative infinity to positive infinity is as follows,

n3 = (n 1) 3 + (3 n 3) n + 1.Cubes in number theory

There is no smallest perfect cube, since negative integers are included. For example,

(4) (4) (4) = 64. For any n, ( n)3 = ( n3).

Base ten

Unlike perfect squares, perfect cubes do not have a small number of possibilities for the last two

digits. Except for cubes divisible by 5, where only 25 , 75 and 00 can be the last two digits, any

pair of digits with the last digit odd can be a perfect cube. With even cubes, there is considerable

restriction, for only 00 , o2 , e4 , o6 and e8 can be the last two digits of a perfect cube (where o

stands for any odd digit and e for any even digit). Some cube numbers are also square numbers,

for example 64 is a square number (8 8) and a cube number (4 4 4); this happens if and

only if the number is a perfect sixth power.

It is, however, easy to show that most numbers are not perfect cubes because all perfect cubes

must have digital root 1, 8 or 9. Moreover, the digital root of any number's cube can be

determined by the remainder the number gives when divided by 3:

If the number is divisible by 3, its cube has digital root 9;

If it has a remainder of 1 when divided by 3, its cube has digital root 1;

If it has a remainder of 2 when divided by 3, its cube has digital root 8.

Waring's problem for cubes

Main article: Waring's problem

Every positive integer can be written as the sum of nine (or fewer) positive cubes. This upper
http://en.wikipedia.org/wiki/Square_numberhttp://en.wikipedia.org/wiki/Even_and_odd_numbershttp://en.wikipedia.org/wiki/Digital_roothttp://en.wikipedia.org/wiki/Waring%27s_problemhttp://en.wikipedia.org/wiki/Waring%27s_problemhttp://en.wikipedia.org/wiki/Digital_roothttp://en.wikipedia.org/wiki/Even_and_odd_numbershttp://en.wikipedia.org/wiki/Square_number


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limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of

fewer than nine positive cubes:

23 = 2 3 + 2 3 + 1 3 + 1 3 + 1 3 + 1 3 + 1 3 + 1 3 + 1 3.

Fermat's last theorem for cubes

Main article: Fermat's last theorem

The equation x3 + y3 = z 3 has no non-trivial (i.e. xyz 0) solutions in integers. In fact, it has none

in Eisenstein integers .[1]

Both of these statements are also true for the equation [2] x3 + y3 = 3 z 3.

Sums of rational cubes

Every positive rational number is the sum of three positive rational cubes ,[3] and there are

rationals that are not the sum of two rational cubes .[4]

Sum of first n cubes

The sum of the first n cubes is the nth triangle number squared:

For example, the sum of the first 5 cubes is the square of the 5th triangular number,

A similar result can be given for the sum of the first y odd cubes,

but { x,y} must satisfy the negative Pell equation x2 2 y2 = 1 . For example, for y = 5 and 29,

then,
http://en.wikipedia.org/wiki/Fermat%27s_last_theoremhttp://en.wikipedia.org/wiki/Eisenstein_integershttp://en.wikipedia.org/wiki/Cube_%28algebra%29#cite_note-0http://en.wikipedia.org/wiki/Cube_%28algebra%29#cite_note-0http://en.wikipedia.org/wiki/Cube_%28algebra%29#cite_note-0http://en.wikipedia.org/wiki/Cube_%28algebra%29#cite_note-1http://en.wikipedia.org/wiki/Cube_%28algebra%29#cite_note-1http://en.wikipedia.org/wiki/Rational_numberhttp://en.wikipedia.org/wiki/Cube_%28algebra%29#cite_note-2http://en.wikipedia.org/wiki/Cube_%28algebra%29#cite_note-2http://en.wikipedia.org/wiki/Cube_%28algebra%29#cite_note-2http://en.wikipedia.org/wiki/Cube_%28algebra%29#cite_note-3http://en.wikipedia.org/wiki/Cube_%28algebra%29#cite_note-3http://en.wikipedia.org/wiki/Cube_%28algebra%29#cite_note-3http://en.wikipedia.org/wiki/Triangle_numberhttp://en.wikipedia.org/wiki/Odd_numberhttp://en.wikipedia.org/wiki/Pell_equationhttp://en.wikipedia.org/wiki/Pell_equationhttp://en.wikipedia.org/wiki/Odd_numberhttp://en.wikipedia.org/wiki/Triangle_numberhttp://en.wikipedia.org/wiki/Cube_%28algebra%29#cite_note-3http://en.wikipedia.org/wiki/Cube_%28algebra%29#cite_note-2http://en.wikipedia.org/wiki/Rational_numberhttp://en.wikipedia.org/wiki/Cube_%28algebra%29#cite_note-1http://en.wikipedia.org/wiki/Cube_%28algebra%29#cite_note-0http://en.wikipedia.org/wiki/Eisenstein_integershttp://en.wikipedia.org/wiki/Fermat%27s_last_theorem


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and so on. Also, every even perfect number, except the first one, is the sum of the first 2 ( p1)/2

odd cubes,

28 = 2 2(23 1) = 1 3 + 3 3

496 = 2 4(25 1) = 1 3 + 3 3 + 5 3 + 7 3

8128 = 2 6(27 1) = 1 3 + 3 3 + 5 3 + 7 3 + 9 3 + 11 3 + 13 3 + 15 3

Sum of cubes in arithmetic progression

There are examples of cubes in arithmetic progression whose sum is a cube,

33 + 4 3 + 5 3 = 6 3

11 3 + 12 3 + 13 3 + 14 3 = 20 3

313 + 33 3 + 35 3 + 37 3 + 39 3 + 41 3 = 66 3

with the first one also known as Plato's number. The formula F for finding the sum of an n

number of cubes in arithmetic progression with common difference d and initial cube a3,

F (d ,a ,n) = a 3 + ( a + d )3 + ( a + 2 d )3 + ... + ( a + dn d )3

is given by,

F (d ,a ,n) = ( n / 4)(2 a d + dn)(2a2 2ad + 2 adn d 2n + d 2n2)

A parametric solution to,

F (d ,a ,n) = y3

is known for the special case of d = 1, or consecutive cubes, but only sporadic solutions are

known for integer d > 1, such as d = {2,3,5,7,11,13,37,39}, etc.
http://en.wikipedia.org/wiki/Even_numberhttp://en.wikipedia.org/wiki/Perfect_numberhttp://en.wikipedia.org/wiki/Odd_numberhttp://en.wikipedia.org/wiki/Arithmetic_progressionhttp://en.wikipedia.org/wiki/Plato%27s_numberhttp://en.wikipedia.org/wiki/Plato%27s_numberhttp://en.wikipedia.org/wiki/Arithmetic_progressionhttp://en.wikipedia.org/wiki/Odd_numberhttp://en.wikipedia.org/wiki/Perfect_numberhttp://en.wikipedia.org/wiki/Even_number


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A cube number (or a cube) is a number you can write as a product of three equal factors of

natural numbers.

Formula: k=a*a*a=a (k and a stand for integers.)

On the other hand a cube number results by multiplying an integer by itself three times.

Formula: a*a*a=a=k (a and k stand for integers.)

The same factor is called the base.

After this a negative number like (-2)= -8 or a fraction number like (2/3)=8/27 are suspended.

If it is appropriate, the number 0 is also a cubic number.

These are the first 100 cube numbers.

Cube Root

It is easy to find a cube number. It is more difficult, to find the base of a cube number.

This procedure is called extracting the cube root of n .

The cube root of a natural number can be written as .

Waring's Problem

The English mathematician Eduard Waring (1734-1798) maintained the following statement


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among others.

"Every natural number is either a cube number or the sum of 2,3,4,5,6,7,8 or 9 cube numbers."

(2), page 37ff.

That means that 9 is a smallest number.

It can be more than 9 as the following sum of 180 with 64 (!) cubic numbers shows.

180 = 6+7+8+...+67+68+69 (1). Already 4 summands will do, 180=1+3+3+5.

The first numbers

1=12=1+1

3=1+1+1

4=1+1+1+1

5=1+1+1+1+1

6=1+1+1+1+1+1

7=1+1+1+1+1+1+1

15=2+1+1+1+1+1+1+1

23=2+2+1+1+1+1+1+1+1

Special Cube Numbers

Square numbers among the cubes

There are cubes, which also are squares.

You can construct them step by step by squaring cube numbers.


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23 leads to 2 6=64 , 33 to 3 6=729 , 4 to 4 6=4 096 , ...

The next cube numbers and at the same time square numbers upto 1 million are 15 625, 46 656,

117 649, 262 144 und 531 441. I must not forget 1.

A cube number is the third power of its digit sum.

512=8=(5+1+2)

4913=17=(4+9+1+3)

5832=18=(5+8+3+2)

17576=26=(1+7+5+7+6)

19683=17=(1+9+6+8+3)A number is the sum of the third power of its digits.

153=1+5+3

370=3+5+0

371=3+5+1

407=4+0+7

The simple program on the right found

this.

for x=0 to 9for y=0 to 9for z=0 to 9if x*x*x+y*y*y+z*z*z=1000*x+100*y+z then print x,y,znext znext ynext x

Variations

22+2=2+2+2

12*3=1+2+3

32*5=3+2+550*5=5+0+551*3=5+1+3

151+3=1+5+1+3 und 153+1370+1=3+7+0+1371+1=3+7+1+1400+7=4+0+0+7401+7=4+0+1+7 und 407+1=4+0+1+7464+5=4+6+4+5 und 465+4=4+6+4+5624+7=6+2+4+7 und 627+4=6+2+4+7643+7=6+4+3+7 und 647+3=6+4+3+7733+7=7+3+3+7 und 737+3=7+3+3+7773+4=7+7+3+4 und 774+3=7+7+3+4914+5=9+1+4+5 und 915+4=9+1+4+5

12+32=1+2+3+220+23=2+0+2+321+23=2+1+2+330+32=3+0+3+231+32=3+1+3+2

107*8=1+0+7+8180*3=1+8+0+3989*2=9+8+9+218*30=1+8+3+0


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Terms with equal digits

3+7=37*(3+7)

4+8=48*(4+8)

14+7=147*(14+7)

14+8=148*(14+8)

Cube numbers are written in the digits 1 to 9. No digit is twice or more.

125*438976=380

8*24137569=578

8*32461759=628Two numbers with common features

Example24=13824 and 76=43897625=15625 and 75=42187549=117649 and 51=132651125=1953 125 and 875=669921875251=15813251 and 749=420189749

0624 = 242970624 and 9376 =8242383093760625 = 244140625 and 9375 = 823974609375

21952 = (6+8+5+9) and 6859 =(2+1+9+5+2)

Explanation24+76=100, 13824 and 43897625+75=100, 15625 and 42187549+51=100, 117649 and 132651125+875=1000, 1953125 and 669921875251+749=1000, 15813251 and 420189749

0624+9376=10000, 242970624 and 8242383093760625+9375=10000, 244140625 and 823974609375

21952 and 2+1+9+5+2, plus 6+8+5+9 and 6859

The bases form an arithmetic progression

180 = 6+7+8+...+67+68+69

540 = 34+35+ ... +158

2856 = 213+214+ ... +555

5544 = 406+407+ ... +917


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16834 = 1134+1135+ ... +2133

3990 = 290+293+ ... +935

29880 = 2108+2111+ ... +4292

408 = 149+256+363

440 = 230+243+265+269+282

1155 = 435+506+577+648+719+790

2128 = 553+710+867+1024+1181+1338+1475

168 = 28+41+54+67+80+93+106+119

64085 = 935+5868+10801+15734+20667+25600+30533+35466+40399+45332495 = 15+52+89+126+163+200+237+274+311+248


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METHODOLOGY

A. Materials/ Equipments:

Pen

Paper

B. General Procedure

First, all the terms of simple multiplication of integers are fully understood. Even

if the process is similar to multiplication; you still need the knowledge about

multiplication of integers. Then we proceed to the proper steps in finding the difference

of negative consecutive integers. And most importantly, we will use our formula for

finding the difference faster and easier.

So, first, we choose any integer, it must be negative consecutive and a is greater

Than b. Example:

Second, we will find the difference of their cubes. Here:

-23 and -24 , -23 is greater than -24 and they are both

Consecutive negative.

Multiplicand

Multiplier

-23 3 -24 3, To solve for the difference, we will use our own formula:


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SUMMARY AND CONCLUSION

We have applied our formula using several negative consecutive integers. All the test

results did not have a single miscalculation or a single mistake. As far as this study has reached,

there has been no disapproval of answer . Thus the researchers conclude that this study is

successful and can be officially used, even for average students.


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The Cubic Difference Formula (Mi)

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