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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: in the form a/b where a and b are integers, b nonzero.

Rational numbers are contrasted with irrational numbers - such like Pi and square roots and sines and logarithms of numbers. This article concentrates on rational numbers, and at the end of the article you can click on a link to continue studying about irrational numbers.

In mathematical terms a number is rational if you can write it in a form a/b where a and b are integers, b not zero.  Clearly all fractions are of that form.  Terminating decimal numbers can easily be written in that form: for example 0.67 is 67/100, 3.40938 = 340938/100000 etc.  You should review this with your child/students.

Mini-review: Write in the form a/b, where a, b are integers and b is not zero:
a) 0.7  b) 0.803  c) 1.902  d) 10.45  e) 4  f) 1 4/5  g) -100  h) -404.04  i) 405,700.3  j) 5.08497593

We can illustrate positive rational numbers with lines that go through the origin and another point with whole number coordinates.  For example the line y = 2x has the slope 2 and it goes through the point (1,2).  The line y = 3x goes through the point (1,3). The line y = 1/4x goes through the point (4,1).  The line y = 2 1/2 x goes through the point (2,5).  And, these points are the FIRST ones the lines go through after the origin.

Practice a little: Through which point do these lines go through first?
a) y = 9x   b) y = 243x   c) y = 5/6x   d) y = 8/3 x   e) y = 345/1039 x

What are the equations for these lines?

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 decimal numbers 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.

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... This will work, the decimals line up now!

Then we subtract the original, and the 1000a.

1000a = 135.135135135...
-   a = 0.135135135...
999a = 135 , from which a = 135/999.

Sometimes you have a decimal that has first couple of digits that are not part of a pattern, and then it takes on a pattern. For example, b = 5.65787878787... is such a number. The same trick works though, we multiply b times 10 enough times so that the repeating parts cancel each other in the subtraction.

b = 5.65787878787...
10b = 56.578787878...
100b = 565.7878787... This should work!

So subtracting

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: non-repeating non-terminating decimal numbers

After all this discussion about terminating decimal numbers and repeating decimal numbers we can then announce that the NON-repeating NON-terminating decimal numbers are exactly the IRRATIONAL NUMBERSContinue reading about the irrational numbers


See also:

Rational numbers are countable

and the following articles from Dr. Math at mathforum.org:

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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