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Infinite Surds

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Introduction

MATH PORTFOLIO

Infinite Surds


Introduction

Surds: a surd is a number that can only be expressed exactly using the root sign “image00.pngimage00.png” in other words it can be defined as a positive irrational number.

Thus, a number image48.pngimage48.png is a surd if and only if:

  1.   It is an irrational number

      (b ) It is the root of a positive rational number.

The symbol  is called the radical sign. The index n is called the order of the surd and x is called the radicand.

For example:

√2 = (1.414.......) is a surd.

√3= (1.732......) is a surd.

√4= (2) is not a surd.

√5= (2.236....) is a surd.

Note: if n is a positive integer and a be a real number, then if a is irrational, image84.pngimage84.png is not a surd. Again if image84.pngimage84.png is rational, then also image84.pngimage84.png is not a surd.

Surds have an infinite number of non-recurring decimal. Hence, surds are irrational numbers and are considered infinite surds.

...read more.

Middle

image09.pngimage09.png

image10.pngimage10.png

image11.pngimage11.png

image02.pngimage02.png  x= image12.pngimage12.png  OR x= image13.pngimage13.png

image02.pngimage02.png x = image14.pngimage14.png      (x cannot have a negative value.)

Q: Considering another infinite surd image15.pngimage15.png where the first term is image17.pngimage17.png , to repeat the entire process and to find the exact value for this surd.

ANSWER:

image18.pngimage18.png= image19.pngimage19.png = 1.8477...

image20.pngimage20.png= image21.pngimage21.png  OR   image20.pngimage20.png= image22.pngimage22.png = 1.9615...

image23.pngimage23.png= image24.pngimage24.png   OR image25.pngimage25.png= image27.pngimage27.png = 1.9903...

As it is observed that image23.pngimage23.png=image27.pngimage27.png , therefore it can be understood that image25.pngimage25.png = image28.pngimage28.png as the trend has been so until now.

Hence,

→  image25.pngimage25.png = image28.pngimage28.png = 1.9975...

image29.pngimage29.png = image30.pngimage30.png  = 1.9993...

image31.pngimage31.png = image33.pngimage33.png = 1.9998...

image34.pngimage34.png = image35.pngimage35.png = 1.9999...

image36.pngimage36.png = image37.pngimage37.png = 1.9999...

image38.pngimage38.png = image39.pngimage39.png = 1.9999...

image40.pngimage40.png = image41.pngimage41.png = 1.9999...

[NOTE: VALUES OBTAINED USING THE GRAPHICAL DISPLAY CALCULATOR: CASIO fx- 9860G]

N

image42.png

1

1.8477

2

1.9615

3

1.9903

4

1.9975

5

1.9993

6

1.9998

7

1.9999

8

1.9999

10

1.9999

image44.png

From the above graph we realize as the value starts to increase after a certain number (1.999) it stabilizes and remains almost the same through out the graph. As we can see after 5 the value continues to remain the same till 10 and onwards. The general idea in both the question

...read more.

Conclusion

image06.pngimage06.png - x – k=0 we get:

image06.pngimage06.png -x-30=0

image06.pngimage06.png -6x+5x -30 =0

→ x(x-6) +5(x-6) =0

→ (x+5) (x-6) =0

Thus, x=6 OR x=-5

Hence, proved.

  1. Supposing  n =5

Then, on substituting 6 as the value of n in the general statement we get:-

image78.pngimage78.png = 6+6^2 = 42

image02.pngimage02.png k=42

Now substituting the value of k in the equation image06.pngimage06.png - x – k=0 we get:

image06.pngimage06.png - x – 42=0

image06.pngimage06.png -7x+6x -42 =0

→ x(x-7) +6(x-7) =0

→ (x+6) (x-7) =0

Thus, x=7 OR x=-6

Hence, proved.

HENCE, IT CAN BE CONCLUDED THAT EVEN AFTER USING OTHER VALUES OF “k” THE ANSWER OBTAINED IS AN INTEGER WHICH PROVES THE OBTAINED GENERAL STATEMENT VALID.

The scope of the general statement is that the need to always obtain an integer value as the answer is always satisfied by this general statement. The limitation of the general statement is that the use of a negative integer is invalid.

CONCLUSION:

By this I conclude that them I learnt how to use the surds and the correct time to use them.

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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