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- Herausgeber: Books on Demand
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
In this book I present a unique formula for quadratic equations, which is a rewriting of the p-q-formula. This rewriting resulted in the equations being solved almost twice as fast by the new formula, when it was compared in a test with the p-q-formula. Another test also showed that the new formula was much faster than the Vedic formula. The new formula is unique because the equations in the test were solved by mere mental calculation, which improves the memory and increases mental agility and intelligence. When I discovered that the middle coefficient in a quadratic equation contains all information about its origin, it led to a rule, that simplified the solving of of all equations. In a quadratic equation the origin could be located, and then it became possible to create a rule how the coefficients were to be split up into factors. By means of this rule and some exercises the answer to an equation can be both calculated and checked regardless of how large the coefficients are.This universal method is intended to be used before the equation is solved by a formula. Since the origin of a quadratic equation could be located, it was also simple to find the origin to other types of equations, and therefore new methods could be created. This led to the fact that a cubic equation could be solved without taking detours like polynomial division, a guess or a test of a root. When the origin of an equation can be located it is as easy to solve a fifth degree equation as a quadratic equation, in the same simple way as unlocking a safe with a key. The purpose of the book is mainly to make it as simple as possible for the students to solve equations, but also to give them a good insight into the origin of an equation.
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CONTENTS
1. Introduction
The new quadratic formula
Derivation of the formula
The difference of two squares
Squaring of numbers ending in five
Table for squaring and square roots
The new formula was tested by students
Exercises during the test
The result of the test
In favour of mental mathematics
2. Quadratic equations
Multiplication of algebraic expressions
Table how to split up the coefficients into factors
The split up rule
Calculation and check of the answers
Summary
Quadratic equations are solved by factorising
Quadratic equations can be solved mentally
The core of a polynomial
When one outer coefficient is 1
Identical binomials
Completing the square
Exercises with answers
Multiplication of numbers
3. Cubic equations
The coefficients are split into factors
Equations when the first coefficient is 1
Equations when the first coefficient is larger than 1
The factor rule
A general method for cubic-, fourth- and fifth degree equations
Prime factorising
4. Fourth degree equations
Comparison with a conventional method
5 Fifth degree equations
Comparison with a conventional method
6 The greatest mathematician in Northern Europe
The Abel Prize
References
INTRODUCTION
In this book I present a unique quadratic formula, which turned out to be a rewriting of the p-q-formula. This rewriting resulted in the equations being solved, almost twice as fast with the new formula when it was compared in a test with the p-q-formula. Another test also showed that the new formula was much faster than the Vedic formula, see p.11. The new formula is unique because the equations in the test were solved by mere mental calculation, which improves the memory and increases mental agility and intelligence.
When I discovered that the middle coefficient in a quadratic equation contains all information about its origin, it led to a rule that simplified the solving of all equations. In a quadratic equation the origin could be located, and it became then possible to create a rule for how the coefficients were to be split up into factors. By means of this rule and some exercises the answer to an equation can both be calculated and checked regardless how large the coefficients are. This universal method should be used before the equation is solved by a formula, see pages 20-23.
Since the origin of a quadratic equation could be located, it was also simple to find the origin to other types of equations, and therefore new methods could be created. This led to the fact that a cubic equation could be solved without taking detours, like polynomial division, a guess or a test of a root. When the origin of an equation can be located, it is as easy to solve a fifth degree equation as a quadratic equation, in the same simple way as unlocking a safe with a key. The purpose of the book is mainly to make it as easy as possible for the students to solve equations, but also to give them a good insight into the origin of an equation.
During a visit to USA I met Dr. Anne Dow, Professor of Mathematics at MUM in Iowa, and the new quadratic formula was then verified and approved.
I was born in 1932 in the Norwegian city of Risör, and when I was 21years old I studied in Sweden to become a Chemical Engineer.
Chapter 1
THE NEW QUADRATIC FORMULA
Using the new formula we have seen that the equations can be solved mentally, and also almost twice as fast as by the p-q-formula. The result of a comparative test of the two formulas and a Vedic formula is accounted on page 18.
I had long tried to find a simple formula, but when I could solve several equations with irrational numbers, I realized that I had succeeded. Later it turned out that the formula is a rewrite of the p-q-formula.
The formulas are equal but have different properties. They can be compared to two equal cars, one of them with a stronger motor and almost twice as fast as the other one.
Derivation of the new quadratic formula
* Squaring of numbers ending in 5, see page 9.
The conjugate rule
The new formula for quadratic equations is based on the conjugate rule, which is often used to rewrite a difference into a product. If a and b are two numbers we have:
This identity is valid for any numbers a and b. The conjugate rule can be rewritten into a product, where one factor has plus and the other one has minus between its parentheses.
The conjugate rule can often be used for swift and elegant solutions, which is illustrated in the following examples.
Example 1
Example 2
Squaring of numbers ending in 5
As the middle coefficient is halved in the new formula, odd numbers will end in 5. An easy way to square such numbers by mental calculation, is to use a word formula : ” By one more than the one before”. By squaring of 5.5, 5 is multiplied by 6, i.e. 5 is multiplied by the number following 5. We multiply 5 by 6 and get 30, to which we add 25 in appropriate decimal position.
Table for squaring and square roots
This table contains numbers which are often needed to solve quadratic equations by the new formula. If you are familiar with the small multiplication table, it should not be any problem to square and calculate the square root by mental calculation. Let us determine the square root of 12.25. We realize that the number is more than 3 and less than 4. It is between 3 and 4, i.e. 3.5. The square root of 72.25 should be between 8 and 9 and can be verified to be 8.5. With some exercise you will soon solve equations faster by mental calculation than by means of a technical device.
The new formula is tested by students
A test was performed by the students at Blackeberg Gymnasium, a senior high school, which is ranked as one of the best schools in Stockholm, Sweden. Before the students received the exercises a comparison was made, which turned out that the equations were solved faster by the Vedic formula than by the p-q-formula. Therefore the new formula was only compared to the Vedic formula in the test performed by the students. The test was performed by 21 students and consisted of 20 equations.
To show how cumbersome and time-consuming the p-q-formula is compared with the new formula, each equation was solved by the p-q-formula.
Formula A: the new formula
Formula B: the Vedic formula
Formula C: the p-q-formula
Exercises during the test
The result of the test
After the test all the students liked the new formula, because it was so simple to square the small numbers by mental calculation. As the students were not used to the formulas, the time for solving the equations varied. Therefore it was not easy to evaluate the result of the test but, on average the students solved the equations faster with the new formula than with the Vedic formula. A few students solved the equations on average 16% faster with the new formula than with the Vedic formula.
When the same test was performed later by two experienced persons, all equations were solved by mental calculation when they used the new formula. The result of the test was that they solved the equations on average 30 % faster with the new formula compared with the Vedic formula and 75 % faster compared with the p-q-formula.
The new formula has shown to be the best, and is also able to solve almost all equations by mental calculation.
In favour of mental mathematics
The following points outline the benefits of a mental approach to mathematics.
1. Mental calculation sharpens the mind and increases intelligence and mental agility. This will be evident to anyone who has practiced mental calculation or who has seen its effect.
2. Mental calculation enhances the precision of thought. Numbers and other mathematical objects are neutral, giving only one correct answer to which everyone will agree and there is never a contradiction. This absolute precision is unique to mathematics so dealing intimately with numbers as we do by mental calculation, we cultivate fine and careful thinking.
3. Mental calculation leads naturally to the search for, and discernment of, constancy and law, which are very necessary attributes in a swiftly changing world.
4. Our mind has the ability to retain several ideas at once so that they can be compared and combined. This facility is enhanced by mental calculation as we practice holding the sum in the mind whilst operating with some of the figures.
5. Mental calculation improves the memory and depreciates if it is not exercised. Short term, medium term and long term memory are all stimulated by mental calculation.
6. Since numbers are absolutely reliable, mental calculation promotes confidence. In particular mental calculation creates confidence in oneself and in one's capabilities. To solve problems, a difficult one, by mental arithmetic without having to rely on artificial aid is a source of great satisfaction and encouragement.
7. Mental calculation is a delight to the mind: the intrinsic qualities, relationships and beauty of numbers, the way they create new numbers out of themselves is a source of great enjoyment.
8. Through mental calculation one becomes familiar with numbers and appreciates their various properties. This leads to real understanding of numbers.
9. Calculating mentally reveals subtle properties of numbers and their relationships more readily than if the calculation was written down and thereby fixed. Thus mental calculation leads naturally to innovation and to the invention of new methods, thereby developing the student's natural creativity.
10. Practical uses of mental calculation are many, as we all need to make quick on the spot, calculations from time to time.
