SIMPLE MATHEMATICAL PROOFS.....

SIMPLE MATHEMATICAL PROOFS

A NEW APPROACH TOWARDS MATHEMATICS

TABLE OF CONTENTS

11)    INTRODUCTION

22)    WHY (-VE) * (-VE) IS (+VE)

33)    WHY ANYTHING RAISED TO THE POWER 0 IS 1

54)    THINGS TO KNOW ABOUT MATHEMATICS

65)    CONCLUSION

 

The topic I have here is certainly very intriguing and weird to be presented among other blogs that you might have come across, filled with jargon terms and technical brilliance, possibly good enough to revolutionize the entire world. At the same time, I assure you dear friends, that next few minutes will help you to revolutionize yourself. I am going to present now a series of simple mathematical proofs, some of which you are quiet likely to have stumbled upon on few occasions and rest are quiet unique.

 Has anyone of you ever realized that we conveniently disregard the necessity of getting a convincing reason for few mathematical concepts like

1)    Why negative *negative is positive?

2)    Why alternate interior angles are equal?

3)    Why vertically opposite angles are equal?

4)    Why anything raised to the power 0 is 1?

And so forth…..

Even if someone does ask questions like this most teachers especially in schools and in some cases, even in colleges ridicule such questions saying it is very evident and needs no proof. It is as much the mark of an ignorant man, they say, to require persuasion of evident truths as to believe what is obscure without question.....

In my personal opinion, as a society, we can't do without mathematics. Mathematics is in our culture. It's a vehicle of progress of all other sciences. On an

individual level, a small percentage of professions require mathematical knowledge of various degree. Consumer mathematics, however useful, is a misnamed term for numeracy. An average person may be happy without a scruple of mathematical fluency.

1)  Why negative * negative is positive?

A thought occurred to me in my schooling….

If  3*2 is 6 ……. Then (-3)*(-2) = (-6)  right???? Yes, of course I know that it is a terrible mistake. But still I wasn’t quite convinced.

Now let me try to convince you that (-3)*(-2) is indeed equal to (+6)…..

When an arithmetic progression with a common difference d is multiplied with a constant x, we get another arithmetic progression of common difference xd.

Now I am going to take an arithmetic progression, say AP1.

3,2,1,0,-1,-2 and so on………

AP 1   *  constant     =  AP 2

3         *      2             =   6

2        *       2             =   4

1        *       2             =   2

0        *       2             =   0

    -1        *      2             =             ?

 

Now to find what comes in place of that “?” ….. Consider AP 2.

In AP 2…. First term, say a1 is 6, a2 is 4….. a4 is 0 then in place of a5 we have “?” .

We know c.d = a5 – a4= a2- a1…. This implies

a5 = a4+c.d….i.e.

a5= a4+a2-a1

a5= 0+4-6 = (-2)

Therefore, (-2) comes in place of “?”…..

AP 1 * constant   =  AP 2

3       *     2          =    6

2       *     2          =    4

1       *    2           =    2

0       *    2           =    0

-1     *    2           =    (-2)

Thus we have proved (-ve) * (+ve) is (-ve)…….

Now to prove that (-ve) * (-ve) is (+ve) we are going to follow a similar procedure.

Take an arithmetic progression, say AP1.

3, 2, 1, 0,-1,-2 and so on………

AP 1   *    constant     =   AP 2

3         *     - 2             = -6

(Since –ve * +ve is –ve)

2         *     - 2             =  -4

1         *     - 2             =  -2

0         *     - 2             =  0

-1        *     - 2             =  X

To find X, consider AP 2…..

First term a1= -6, a2= -4…   a4=0, a5= X

c.d= a5-a4=a2-a1…. This implies

a5= a4+c.d … here common difference is +2.

Therefore a5= 2….. This helps us to arrive at the result

(-1)*(-2) is (+2) and hence –ve * -ve is +ve………

Now let us discuss about a much commonly known proof for proving the above result……..

We know,

•         (-1) + (1) = 0

Now multiply “a” on both sides of the equation, such that a>0…..

•        a [ (-1) + (1) ] = a*0

•        a * (-1) +  a   = 0  ( by distributive property of multiplication and by identity      

that    a*0 = 0 )    

Now subtract ‘a ‘on both sides of the equation. 

•      a * (-1) + a - a = 0 – a

•      a * (-1) + 0      = -a

•      Therefore a*(-1)  = (-a)

mplication is that ….. +ve * -ve  is -ve   

We know,

•         (-1) + (1) = 0

Now multiply ‘ –a ‘  on both sides of the equation, such that a>0 and –a <0…..

•        -a [ (-1) + (1) ] = a*0

 -a * (-1) +  -a   = 0  ( since we have proved –ve * +ve is –ve we can write (-a)*1   is –a) 

Now add ‘a ‘on both sides of the equation. 

•      -a * (-1) + (-a) +  a = 0 + a

•      -a * (-1) + 0      =  +a

•      Therefore  -a*(-1)  = (+a)

Implication is that ….. -ve * -ve  is +ve

Thus indeed …..(-3) * (-2) is +6    

4) Anything raised to the power 0 is 1????

We very often use this property in mathematics…..

Anything to the power 0 is 1….. But most of us have no idea why is that so…

Here is a commonly known proof for the benefit of those who do not know…..

a^0  can be written as….. a^(m-m) 

             a^(m-m) = a^m * a^(-m)

                           = (a^m) / (a^m)

                           = 1

thus, anything raised to the power 0 is one…..

•      Things to know in mathematics

•      There are few kindergarden topics in mathematics that we often disregard.

Here is an example…..

if     x+3 = 6

then ,  x = 6 – 3

                     x = 3…..

Why does + on coming to the right hand side of the equation becomes minus (-)……???

Why not  x=  6*3 or x = 6/3????

•      Well… as some of you would have figured out…

The sign “ = “ is like a physical balance. Operations done on L.H.S should be done on R.H.S to maintain equality.

   x + 3 = 6

   x + 3 – 3 = 6 – 3

   x + 0   = 6 – 3

By identity x+ 0= x = 6-3 =3….

We actually subtract 3 on both sides rather than bringing the three from L.H.S to R.H.S……..

Same is the case when we right x/2 = 4 as x = 2 * 4… we actually multiply both sides by 4…….

In fact, math is not always friendly…. There are few fallacies that have no convincing answer….

For example:

    We are going to evaluate {(1/x)dx  by partial integration:

      {udv=uv− {vdu 

•      For this example, set u=1/x, v=x, then du=−1/x^2 dx,    dv=dx.

So we obtain,

•      {(1/x)dx=1−{(x*(-1/x^2))dx 

•      {(1/x)dx=1-{ (-1/x)dx 

•      {(1/x)dx = 1 + {(1/x)dx

•       A = 1 + A

•      0 = 1

•      Jack Glover wishes to add enough 50% antifreeze solution to 16 gallons of a 5% antifreeze solution to obtain a 20% antifreeze solution. How much of the 50% solution should he add?

•       

•      I talked to an employee at a service station. He did this kind of problem in mathematics classes, but on the job nobody does it this way. They test antifreeze in the car for the temperature at which it provides protection. Then they add enough antifreeze to get to the desired temperature.

•      While many people use some mathematics in their jobs, Dr. Appelbaum feels that mathematics should nonetheless be taught primarily for its own sake. Applications are fine if they are simple and appealing; otherwise they should be left to an applied course. She sees the current emphasis on applications as a response to anti-intellectualism among students. When students ask what good the mathematics is, she suspects that the students are really saying that they cannot understand the subject and so hope that it is no good. She has often met people who are glad that they studied mathematics, or wish that they had studied more, but never anyone who said that they were sorry to have learned mathematics.

•      Mathematics contains much that will neither hurt one if one does not know it nor help one if one does know it.

Stand firm in your refusal to remain conscious during algebra. In real life, I assure you, there is no such thing as algebra.

Though, mathematics is not entirely useful and essential to life, learn maths for its pure fun, satisfaction and not just because it is a 4 credit subject…!!!!

      *       Have a good day!!!!

              ….. the end…..