Decimal rational numbers are quotients of integers, where the denominator is a power of ten. They are named after the Latin word decem, ten. (Why, then, is December the twelfth month of the year instead of the tenth? It's an interesting question, but we're not going in that direction just now.)
Of the three rational numbers 34/7 and 2/5 and -17/10000, 2/5 and -17/10000 can also be rewritten as decimals (0.4 and -0.0017) but 34/7 cannot.
Of course 34/7 can be approximated in decimal form - your calculator will do it, giving 4857/1000 as a reasonable approximation and 45871/10000 as an even better one - and the approximation raises more interesting questions. But, no matter how long the string of decimal digits gets, 34/7 can't be written exactly as one fraction with a tens-power denominator.
Some fractions can be expressed as an
''infinitely long'' decimals in a way that makes sense,
when the digits fall in repeating patterns and when we
have decided what we mean by adding up all those decimal-denominator
terms that go on indefinitely. An example would be
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So here is one conclusion: although every decimal number of a finite length is a rational number, not all rational numbers can be expressed in this way.
Assigning a ''size'' to an infinite set of objects seems like a squirrelly enterprise, but there are those who have done it. Suppose that we call INTCOUNT the ''size'' of the set of integers (the positive, and negative, natural numbers): that is, we would say that the family of all the integers in the world has INTCOUNT members.
Of course INTCOUNT is not a finite number, and we cannot expect it to follow the ordinary rules of arithmetic.
Just for interest (we are certainly not going to prove these statements here!) here are some facts about INTCOUNT:
By the way: what do you suppose that phrase ''far, far larger'' means? If by the time you are reading this page you have been discussing limits in calculus class, you will realize that we have no mathematical way of classifying any particular number as ''large'' or ''small.'' We can certainly say that 23 is a larger number than 18. But, is 23 a large number? (And is it far, far larger than 18, or just a tiny bit larger?)Among friends we often use more colorful and casual language: in this situation we would probably call a 500-digits-long integer a ''large'' number. We might even describe 0.000000001 as ''small.''
After all, it is rational numbers that are displayed on your calculator! If you enter p on your calculator keypad, for instance, what you see on its display is not really p (which is not a rational number) but a decimal approximation of p. The decimal approximation is (being a decimal number with only finitely many digits) a rational number.
The amazing thing is how extravagantly the rational numbers are sprinkled across the whole length of the ''real number line.''
For instance, picture p sitting on the real number line, minding its own business, somewhere between 3 and 4. Nearby it is a rational number, 3.14, which is a rough approximation. If that's not good enough, there is a nearer approximation, 3.1416. Not good enough? There's an even closer one: try 355/113.
The remarkable thing is, that no matter how close we wish the approximation to be, there's a rational number that will approximate p that well. There will always be a better approximation to p, or to any other irrational number. Any real number can be approximated, to any desired accuracy, by a rational number. Life is good.
In algebra (which is the language of calculus)
the word ''fraction'' is also taken to mean algebraic
fraction
- that is, the quotient of algebraic expressions. So,
not only is 2/3 a fraction, but we would
call
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Quotients of variable expressions obviously cannot be rewritten as decimals. Fortunately, all the arithmetic rules for combining rational numbers also work for algebraic fractions.
That means, of course, that as a calculus student you can not turn over all the fractions you meet to your calculator! We now have two reasons for this:
That's reassuring! All the possible values of an algebraic fraction are real numbers: so algebraic fractions ''work'' according to the real-number arithmetic that we know all about.
The real number system includes two ''neutral numbers,'' zero and one: one neutral number for each of the two common ways in which real numbers are combined arithmetically. Zero is the neutral number if we are adding numbers, and one is the neutral number if we are multiplying them. Here are some of the properties of these ''neutral'' numbers:
It's helpful to keep more than one interpretation of fractions in mind: we can think of the fraction a/b as either
Here are some ways that the ''neutral numbers'' zero and one work, when we see them as fractions.
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