2 RATIONAL AND IRRATIONAL

1  −1  0  2 3 
−  2 3 
5½  −5½  6.085  −6.085  3.1415926535897932384 
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All of them. All decimals are rational. That long one is an approximation to π, which, as we shall see, is not equal to any decimal. For if it were, it would be rational.
3. A rational number can always be written in what form?
As a fraction  a b 
, where a and b are integers (b 0). 
That is the formal definition of a rational number. That is how we can make any number of arithmetic look.
An integer itself can be written as a fraction: 5 = . And from arithmetic, we know that we can write a decimal as a fraction.
When a and b are natural numbers, then the fraction has the same ratio to 1 as the numerator has to the denominator. Hence the name, rational number.
( is to 1 as 2 is to 3. 2 is two thirds of 3. is two thirds of 1.)
The language of arithmetic is ratio. It is the language with which we relate rational numbers to one another, and 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 will see that language cannot express the relationship of an irrational number to 1
Finally, 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.
WHY DO WE EVEN CALL a number "rational"? Because there are numbers that are not rational. An example is ("Square root of 2"). It is not possible to name any number of arithmetic—any whole number, any fraction, or any decimal—whose square is 2. is close because
7 5 
·  7 5 
=  49 25 
—which is almost 2.
To prove that there is no rational number whose square is 2, suppose there were. Then we could express it as a fraction in lowest terms. That is, suppose
· = = 2.
But that is impossible. Since is in lowest terms, then m and n have 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 1^{2} + 1^{2} = 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.
4. Which natural numbers have rational square roots?
Only the square roots of the square numbers; that is, the square roots of the perfect squares.
= 1 Rational
Irrational
Irrational
= 2 Rational
, , , Irrational
= 3 Rational
And so on.
The square roots of the square numbers are the only square roots that we can name. That follows from the same proof that is irrational.
The existence of irrationals was first realized by Pythagoras in the 6th century B.C. He realized that, in a square of side 1, the ratio of the diagonal to the side was not as two natural numbers. Their relationship, he said, was "without a name." For if we ask, "What ratio has the diagonal to the side?"—we cannot say. We can express it only as "Square root of 2."
Irrational numbers have been called surds, after the Latin surdus, deaf or mute. Why deaf or mute? Because there is nothing we can hear. An irrational number cannot say how much it is, nor how it is related to 1. Language is incapable of relating an irrational number to 1. An irrational number and 1 are incommensurable.
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.
The decimal representation of irrationals
When we express a rational number as a decimal, then either the decimal will be exact, as = .25, or it will not be, as .3333. Nevertheless, there will be a predictable pattern of digits. But when we express an irrational number as a decimal, then clearly it will not be exact, because it were, the number would be rational.
Moreover, there will not be a predictable pattern of digits. For example,
1.4142135623730950488016887242097
(This symbolmeans "is approximately.")
Now, with rational numbers you sometimes see
= .090909. . .
By writing both the equal sign = and three dots (ellipsis) we mean:
"It is not possible to express exactly as a decimal. However, we can approximate it with as many decimal digits as we please according to the indicated pattern; and the more decimal digits we write, the closer we will be to ."
That is a fact. It is possisble to witness that decimal approximation and produce it. We have not said that .090909 goes on forever, because that is only an idea. We cannot witness that nor can we produce an infinite sequence of digits.
This writer asserts that what we can actually bring into this world—.090909—has more being for mathematics than what is only an idea. We can logically produce a decimal approximaiton.
We are taught, of course, that .090909 goes on forever, and so we think that's mathematics—that's the way things are. We do not realize that it is a human invention.
What is more, infinite decimals are not required to solve any problem in arithmetic or calculus; they have no consequences and therefore they are not even necessary.
Even if we imagine that the decimal did go on forever, then 1) it would never be complete and would never equal ; and 2) it would not be a number. Why not? Because, like any number, a decimal has a name. It is not that we will never finish naming an infinite sequence of digits. We cannot even begin.
We say that any decimal for is inexact. But the decimal for , which is .25, is exact.
The decimal for an irrational number is always inexact. An example is the decimal for above.
If we write ellipsis—
= 1.41421356237. . .
—we mean:
No decimal for will be 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.
(For a decimal approximation of π, see Topic 9 of Trigonometry.)
To sum up, a rational number is a number we can know and name 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.
5. What is a real number?
A real number is distinguished from an imaginary or complex number. It is what we call any rational or irrational number.
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 or a definition. Examples of calculations are the Pythagorean theorem, and the solution to an equation, such as
x^{3} = 5. The irrational number π is defined as the ratio of the circumference of a circle to the diameter.)
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. We must assume, however, that there is an effective procedure for computing each next digit. For if there were not, then that symbol would not have a position in the number system with respect to order; which is to say, it would not be a number.  
(See Are the real numbers really numbers?)  
6.9205729744 Real, rational. Every exact decimal is rational. 
7. What is a real variable?
A variable is a symbol that takes on values. A value is a number.
A real variable takes on values that are real numbers.
Calculus is the study of functions of a real variable.
Problem 2. Let x be a real variable, and let 3 < x < 4. Name five values that x might have.
*
See The Evolution of the Real Numbers starting with the natural numbers.
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