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Rational numbers

Rational numbers are whole numbers, fractions, and decimals - the numbers we use in our daily lives. They can be written as a ratio of two integers. Rational numbers are contrasted with irrational numbers - numbers such as Pi, √2, √7, other roots, sines, cosines, and logarithms of numbers. This article concentrates on rational numbers.

The definition says that a number is rational if you can write it in a form a/b where a and b are integers, and b is not zero. Clearly all fractions are of that form, so fractions are rational numbers. Terminating decimal numbers can also easily be written in that form: for example 0.67 = 67/100, 3.40938 = 340938/100000, and so on.

We can illustrate positive rational numbers in the coordinate plane with lines that go through the origin and another point with whole number coordinates. For example:



Line y = 5x goes through the point (1, 5).


Line y = (1/3)x goes through the point (3, 1).


Line y = (9/2)x goes through the point (2, 9).

And so on. Also, the points I listed are the FIRST points with whole-number coordinates these lines go through (after the origin).



Practice a little: What is the first point with whole-number coordinates that these lines go through?

a) y = 9x   b) y = 243x   c) y = 5/6x   d) y = 8/3 x   e) y = 345/1039 x


Now, can you imagine a line through origin that does NOT touch ANY of these points with whole number coordinates?????  It's hard, but those kind of lines do exist.  They just avoid touching any of the points with whole number coordinates, and their slope is an irrational number!!!  Difficult to fathom.  Of course when you are drawing lines on paper or on computer, you are limited in your accuracy and even a line y = Pi*x probably to go through a point with whole number coordinates, namely the point (7,22).  It really wouldn't go throuhg it if we could draw extremely accurately, it would just go close.  But since it goes close, 22/7 is a nice approximation to Pi.

Line y = Pi*x goes through the point (7,22)

The line y = Pi * x indeed looks like it goes through the point (7,22) since the graphics program cannot draw accurately enough.

Non-terminating repeating decimals are rational

We talked how terminating decimal numbers are obviously rational numbers. How about non-terminating decimal numbers? You might have never heard of those, though I hope you have. They are plentiful, too. Take for example 1/9 and convert it into a decimal number with long division algorithm. What do you get? How about 2/9?  3/9?  1/11?  2/13?  7/15? Can you find more fractions that turn into non-terminating decimal numbers?

Since 0.11111... = 1/9, then the decimal number 0.11111... is a rational number. In fact, every non-terminating decimal number that REPEATS a certain pattern of digits is a rational number. For example, let's make up a decimal number 0.135135135135135... that never ends. Do you believe we CAN write it as a fraction, in the form a/b? This sounds like it would be pure guesswork, but no, there is a method, a nice and clever one, in my opinion.

How to convert a repeating decimal into a fraction

Let's name our number a = 0.135135135... and multiply it by a power of 10, then subtract the original a and the new number so that the repeating decimal parts cancel each other in the subtraction.

a = 0.135135135...
10a = 1.35135135135...
100a = 13.5135135135...
1000a = 135.135135135...

Okay, using 1000a and a will work, the decimals will line up! So now we subtract 1000a and a:

1000a = 135.135135135...
-   a = 0.135135135...
999a = 135 ,

from which a = 135/999.


Another example of writing a repeating decimal as a fraction

Sometimes the first couple decimal digits are not part of the repeating pattern. For example, b = 5.65787878787... is such a number. The same trick works though: we multiply b by such power of ten that the repeating parts cancel each other in the subtraction.

b = 5.65787878787...
10b = 56.578787878...
100b = 565.7878787...

As you can see, the decimal parts of b and 100b are identical! So we can subtract them:

100b = 565.78787878...
−   b = − 5.65787878...
99b = 560.13

from which b = 560.13/99 = 56013/9900.



Practice a little: Can you convert these decimals into fractions by using the same idea?
a) 0.186186186186...
b) 4.1515151515...
c) 0.139999999....
d) 4.50398989898...
e) 0.30458304583045830458....

Irrational numbers

We've been discussing terminating decimal numbers and repeating decimal numbers. Guess what? NON-repeating and NON-terminating decimal numbers are the IRRATIONAL NUMBERS.

 


See also:

Rational numbers are countable

About Rational Numbers
Can you explain "rational numbers" to me? How do you express them?

Converting Repeating Decimals to Fractions
I know .333333333333 is 1/3, but what is the trick to it?



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