- Level: International Baccalaureate
- Subject: Maths
- Word count: 1021
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 “” in other words it can be defined as a positive irrational number.
Thus, a number is a surd if and only if:
- 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, is not a surd. Again if is rational, then also is not a surd.
Surds have an infinite number of non-recurring decimal. Hence, surds are irrational numbers and are considered infinite surds.
Middle
→
→
x= OR x=
x = (x cannot have a negative value.)
Q: Considering another infinite surd where the first term is , to repeat the entire process and to find the exact value for this surd.
ANSWER:
= = 1.8477...
= OR = = 1.9615...
= OR = = 1.9903...
As it is observed that = , therefore it can be understood that = as the trend has been so until now.
Hence,
→ = = 1.9975...
→ = = 1.9993...
→ = = 1.9998...
→ = = 1.9999...
→ = = 1.9999...
→ = = 1.9999...
→ = = 1.9999...
[NOTE: VALUES OBTAINED USING THE GRAPHICAL DISPLAY CALCULATOR: CASIO fx- 9860G]
N | |
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 |
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
Conclusion
-x-30=0
→ -6x+5x -30 =0
→ x(x-6) +5(x-6) =0
→ (x+5) (x-6) =0
Thus, x=6 OR x=-5
Hence, proved.
- Supposing n =5
Then, on substituting 6 as the value of n in the general statement we get:-
= 6+6^2 = 42
k=42
Now substituting the value of k in the equation - x – k=0 we get:
- x – 42=0
→ -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.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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