We grouped the negative numbers with the natural numbers to get the integers. Now we increase our set of numbers further to get the rational numbers.
The rational numbers include all of the fractions made by dividing one integer by another, except that you can't divide by zero. So positive and negative fractions, fractions smaller than one, fractions larger than one, and all the integers (you can have 1 on the bottom your fraction) make up the rational numbers.
You can remember that the RATIOnals are made up from RATIOs.
Rational numbers do not have to be written as a fraction, but can be in decimal form - they are still rational.
Some rational numbers:
1/2, 6, -4/5, 10/3, 54 3/4, 1.75, -0.33333
With the inclusion of fractions in the rational numbers we are better off for division that we were with the integers. We can now divide any two numbers and get another rational number, with only one exception. We cannot divide by zero.
Like the integers, we can add, subtract and multiply without restriction. But we still have to be careful about exponentiation.
In most cases we can only raise rationals to the power of an integer. 32 is OK, and, unlike the integers, we can now do 3-2, but we cannot do 31/2 as that is outside of the rational numbers. We can only raise numbers to a fraction when the answer lies in the rational numbers. For example 25(1/2) = 5 or (8/27)(1/3) = 2/3.
The rational numbers are given the symbol: ℚ
Showing posts with label fractions. Show all posts
Showing posts with label fractions. Show all posts
Friday, April 17, 2009
Thursday, April 9, 2009
Continued fractions
Any whole number or simple fraction can be written as a continued fraction. Continued fractions appear a little strange and look like this:
[a0 ; a1, a2, a3 ...]
For example, the continued fraction for 40/33 is:
You can check this on a calculator by starting at the bottom of the fraction and working upwards. First calculate 1/2, then add 2, then do 1/x , then add 1, then do 1/x again, then add 4, do 1/x again and finally add one to get 1.21212... = 40/33.
To work out the numbers that go in a continued fraction you go in the opposite direction. Start with 40/33. = 1.21212.
1.2121212
Subtract the whole number part, in this case 1
Take 1/x
Subtract the whole number part, in this case 4
Take 1/x
Subtract the whole number part, in this case 1
Take 1/x
Subtract the whole number part, in this case 2
Take 1/x
We are left with a whole number so put this number in our continued fraction and we have reached the end.
40/33 = 1 + 1/( 4 + 1/ (1 + 1/ (2 + 1/2))
You can use the same method to work out continued fractions for square roots. Then the continued fraction goes on forever (unless the square root is a whole number eg √4 = 2):
For square roots, the continued fraction soon forms a repeating pattern.
√2 = [1; 2, 2, ...]
√3 = [1; 1, 2, 1, 2, ...]
√5 = [2; 4, 4, ...]
√6 = [2; 2, 4, 2, 4, ...]
√7 = [2; 1, 1, 1, 4, 1, 1, 1, 4, ...]
For other numbers, such as π, the continued fraction does not form a pattern:
π = [3; 7, 15, 1, 292, 1, 1, 1, 2. ...]
By using the start of a continued fraction we can determine a fraction that is close to the irrational number. For the case of π:
The 292 in the continued fraction for π is quite large. This indicates that we will have to use much larger numbers in the faction to estimate π, before we find a better approximation than 355/113. So 355/113 will be quite a good approximation for π.
1
a0 + ----------------------
1
a1 + ---------------
1
a2 + --------
a3 + ...
To make it easier to write down, the above fraction is written as:[a0 ; a1, a2, a3 ...]
For example, the continued fraction for 40/33 is:
1
1 + ----------------------
1
4 + ---------------
1
1 + ---------
1
2 + ---
2
= [1; 4, 1, 2, 2]You can check this on a calculator by starting at the bottom of the fraction and working upwards. First calculate 1/2, then add 2, then do 1/x , then add 1, then do 1/x again, then add 4, do 1/x again and finally add one to get 1.21212... = 40/33.
To work out the numbers that go in a continued fraction you go in the opposite direction. Start with 40/33. = 1.21212.
1.2121212
Subtract the whole number part, in this case 1
| value remaining | continued fraction so far |
| 0.2121212 | 1 + ... |
Take 1/x
| value remaining | continued fraction so far |
| 4.7142857 | 1 + 1/ ... |
Subtract the whole number part, in this case 4
| value remaining | continued fraction so far |
| 0.7142857 | 1 + 1/( 4 + ...) |
Take 1/x
| value remaining | continued fraction so far |
| 1.4000000 | 1 + 1/( 4 + 1/ ...) |
Subtract the whole number part, in this case 1
| value remaining | continued fraction so far |
| 0.4000000 | 1 + 1/( 4 + 1/ (1 +...)) |
Take 1/x
| value remaining | continued fraction so far |
| 2.5000000 | 1 + 1/( 4 + 1/ (1 + 1/ ...)) |
Subtract the whole number part, in this case 2
| value remaining | continued fraction so far |
| 0.5000000 | 1 + 1/( 4 + 1/ (1 + 1/ (2 + ...)) |
Take 1/x
| value remaining | continued fraction so far |
| 2.0000000 | 1 + 1/( 4 + 1/ (1 + 1/ (2 + 1/...)) |
We are left with a whole number so put this number in our continued fraction and we have reached the end.
40/33 = 1 + 1/( 4 + 1/ (1 + 1/ (2 + 1/2))
You can use the same method to work out continued fractions for square roots. Then the continued fraction goes on forever (unless the square root is a whole number eg √4 = 2):
1
√2 = 1 + ---------------------
1
2 + --------------
1
2 + --------
2 + ...
= [1; 2, 2, 2, 2, ...]For square roots, the continued fraction soon forms a repeating pattern.
√2 = [1; 2, 2, ...]
√3 = [1; 1, 2, 1, 2, ...]
√5 = [2; 4, 4, ...]
√6 = [2; 2, 4, 2, 4, ...]
√7 = [2; 1, 1, 1, 4, 1, 1, 1, 4, ...]
For other numbers, such as π, the continued fraction does not form a pattern:
π = [3; 7, 15, 1, 292, 1, 1, 1, 2. ...]
By using the start of a continued fraction we can determine a fraction that is close to the irrational number. For the case of π:
| [3;] | = | 3/1 | = | 3 | = | π - 0.14 |
| [3; 7] | = | 22/7 | = | 3.1428 | = | π + 0.0012 |
| [3; 7, 15] | = | 333/106 | = | 3.141509 | = | π - 0.000083 |
| [3; 7, 15, 1] | = | 355/113 | = | 3.14159292 | = | π + 0.00000027 |
| [3; 7, 15, 1, 292] | = | 103993/33102 | = | 3.14159265301 | = | π - 0.00000000058 |
The 292 in the continued fraction for π is quite large. This indicates that we will have to use much larger numbers in the faction to estimate π, before we find a better approximation than 355/113. So 355/113 will be quite a good approximation for π.
Wednesday, April 8, 2009
Pi by fractions
The circumference of a circle divided by its diameter is always the same number, no matter the size of the circle. This number is called π (pi) and has the value 3.1415926535...
π is an irrational number, which means that it cannot be expressed as a simple fraction such as 1/5 or 3/4. However, you can get pretty close.
A fraction that is often used is 22/7. This is not all that good:
Although within 0.04% of the correct answer, 22/7 is only correct to 2 decimal places. We can do better than this.
355/113 is correct to 6 decimal places. It's within 0.000009% of π.
355/113 is such a good approximation to π, that there is not a more accurate fraction until 52163 / 16604, and that is only marginally closer to π, still only correct to 6 decimal places.
To be accurate to 7 decimal places we need to go as far as 86953 / 27678
The importance of 355/113 has been recognized, giving it the name Milü.
π is an irrational number, which means that it cannot be expressed as a simple fraction such as 1/5 or 3/4. However, you can get pretty close.
A fraction that is often used is 22/7. This is not all that good:
| 22/7 | = | 3.14285714... |
| π | = | 3.14159265... |
Although within 0.04% of the correct answer, 22/7 is only correct to 2 decimal places. We can do better than this.
355/113 is correct to 6 decimal places. It's within 0.000009% of π.
| 355/113 | = | 3.1415929203... |
| π | = | 3.1415926535... |
355/113 is such a good approximation to π, that there is not a more accurate fraction until 52163 / 16604, and that is only marginally closer to π, still only correct to 6 decimal places.
| 52163/16604 | = | 3.1415923874... |
| 355/113 | = | 3.1415929203... |
| π | = | 3.1415926535... |
To be accurate to 7 decimal places we need to go as far as 86953 / 27678
| 86953/27678 | = | 3.1415926006... |
| 52163/16604 | = | 3.1415923874... |
| 355/113 | = | 3.1415929203... |
| π | = | 3.1415926535... |
The importance of 355/113 has been recognized, giving it the name Milü.
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