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RATIONAL AND IRRATIONAL NUMBERS

What is a rational number?

Which numbers have rational square roots?

The decimal representation of irrationals

What is a real number?

(At any rate, those are their numerals.)

If we include 0, we have the whole numbers:

0,  1,  2,  3,  and so on.

And if we include their algebraic negatives, we have the integers:

0,  ±1,  ±2,  ±3,  and so on.

± ("plus or minus") is called the double sign.

The following are the square numbers, or the perfect squares:

1   4   9   16   25   36   49   64, and so on.

They are the numbers 1· 1,  2· 2,  3· 3,  4· 4,  and so on.

What is more, we can in principle (by Euclid VI, 9) place any rational number exactly on the number line.

We can say that we truly know a rational number.

2.  Which of the following numbers are rational?

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").

All of them! All decimals are rational. That long one is an approximation to π, which, as we will see, is not equal to any decimal.

3.   A rational number can always be written in what form?

As a fraction

a b

, where a and b are integers (b 0).

An integer itself can be written as a fraction:  b = 1.  And from arithmetic, we know that we can write a decimal as a fraction.

When a and b are positive, that is, when they are natural numbers, then we can always name their ratio.  Hence the term, rational number.

At this point, the student might wonder, What is a number that is not rational?

An example of such a number is ("Square root of 2").  It is not possible to name any whole number, any fraction or any decimal whose

-- which is almost 2.

no common divisors except 1.  Therefore, m·m and n·n also have no common divisors -- they are relatively prime -- and it will be impossible to divide n·n into m·m and get 2.

There is no rational number -- no number of arithmetic -- whose square is 2.  Therefore we call an irrational number.

By recalling the Pythagorean theorem, we can see that irrational numbers are necessary.  For if the sides of an isosceles right triangle are called 1, then we will have  1² + 1² = 2, so that the hypotenuse is .  There really is a length that logically deserves the name, "."  Inasmuch as numbers name the lengths of lines, then is a number.

The existence of irrationals was first realized by Pythagoras in the 6th century B.C.  He realized that in the isosceles right triangle, the ratio of the hypotenuse to the side was not as two natural numbers. Their relationship, he said, was "without a name." Because if we ask, "What ratio has the hypotenuse to the side?" -- we cannot say.  We can express it only as "Square root of 2."

5.  Say the name of each number.

a)     "Square root of 3."              b)     "Square root of 5."

c)     "2."  This is a rational -- nameable -- number.

d)     "Square root of 3/5."         e)     "2/3."

In the same way we saw that only the square roots of square numbers are rational, we could prove that only the nth roots of nth powers are rational. Thus, the 5th root of 32 is rational, because 32 is a 5th power, namely the 5th power of 2. But the 5th root of 33 is irrational. 33 is not a perfect 5th power.

(That explanation is an example of mathematical realism. It asserts that in the mathematics of computation and measuring, which includes calculus, what exists is what we actually observe or name, now.  That .090909 never ends is a doctrine that need not concern us, because it serves no useful purpose. Such actual infinities have no practical effect on calculations in arithmetic or calculus.)

which is .25, is exact.

The symbol for decimal fractions was invented in the 16th century. Now, of course, we take decimals for granted, but at the time many thought it was not a very forward looking idea, because the decimals for only a very limited number of fractions are exact. Even the

As for the decimal for an irrational number, it is always inexact.  An example is the decimal for above.

If we write ellipsis --

= 1.41421356237. . .

-- we mean, "A decimal for will never be complete or exact.  Moreover, there will not be a predictable pattern of digits.  We could continue its rational approximation for as many decimal digits as we please by means of the algorithm, or method, for calculating each next digit (not the subject of these Topics); and again, the more digits we calculate, the closer we will be to ."

It is important to understand that no decimal that you or anyone will ever see  is equal to , or π, or any irrational number. We know an irrational number only as a rational approximation. And if we choose a decimal approximation, then the more decimal digits we calculate, the closer we will be to the value.

To sum up, a rational number is a number we can know exactly, either as a whole number, a fraction or a mixed number, but not always exactly as a decimal.  An irrational number we can never know exactly in any form.

The language of arithmetic is ratio. It is the language with which we relate each rational number to 1, which is their source. The whole numbers are the multiples of 1, the fractions are its parts: its halves, thirds, fourths, millionths. But we cannot relate an irrational number to 1. Like Pythagoras, we cannot say. An irrational number and 1 are incommensurable.

To put it another way, a rational number partakes of the essence of number, which is to answer the question "How many?" (whether apples or inches), for a rational number is composed of what is countable. An integer is a number of 1's (or −1's). A fraction or a decimal is a number of unit-fractions. But an irrational number is not a number of anything.

One often hears however that an irrational number is an infinite decimal.

= 1.41421356237. . .

But if a decimal, even as an idea, did not end, then it would not be a number. Why not? Because, to be useful, decimals have names, and therefore we can name their sum, their difference, their product and their quotient. But an infinite sequence of digits does not have a name. It is not that we will never finish naming it. We cannot even begin.

Finally, can what is infinite -- what does not have an end
to reach -- ever be equal to anything?

See The mathematical existence of numbers.

Real numbers

5.  What is a real number?

A real number is distinguished from an imaginary number. It is any rational or irrational number that we can name. They are the numbers we expect to find on the number line. They are the numbers we need for measuring.

(An actual measurement can result only in a rational number.
 An irrational number can result only from a theoretical  calculation; examples are the Pythagorean theorem, and solving
 an equation such as x³ = 5.
 Any serious theory of measurement must address the question:  Which irrational numbers are theoretically possible? Which ones  could be actually predictive of a measurement?)

Problem 1.   We have categorized numbers as real, rational, irrational, and integer.  Name all the categories to which each of the following belongs.

3  Real, rational, integer.		−3   Real, rational, integer.
−½   Real, rational.		Real, irrational.
5¾   Real, rational.		− 11/2   Real, rational.
1.732   Real, rational.		6.920920920. . .   Real, rational.
6.9205729744. . .   Real. And let us assume that it is irrational, that is, no matter how many digits are calculated, they do not repeat. In other words, we must assume that there is an effective procedure for computing each next digit. For if there were not, then that number would not have its position in the number system with respect to order; which is to say, it would not be a number.
6.9205729744    Real, rational. Every exact decimal is rational.

See The Evolution of the Real Numbers, starting with the natural numbers.

Next Topic:  Functions

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