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RATIONAL AND IRRATIONAL NUMBERS
Which numbers have rational square roots?
The decimal representation of irrationals
LET US BEGIN by recalling that a variable is a symbol that takes on values. A value is a number.
Thus, if x is a variable, then x might have the value 2, or −3, or 5.2, and so on.
Next, the following numbers of arithmetic are called the natural numbers:
1, 2, 3, 4, and so on.
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.
These are the square numbers, or the perfect squares:
1 4 9 16 25 49 64 . . .
They are the numbers 1· 1, 2· 2, 3· 3, 4· 4, and so on.
Rational and irrational numbers
1. What is a rational number?
Any number of arithmetic: Any whole number, fraction, mixed number or decimal; together with its negative image.
A rational number is a nameable number, in the sense that we can name it according to the standard way of naming whole numbers, fractions, and mixed numbers. "Five," "Six thousand eight hundred nine," "Nine hundred twelve millionths," "Three and one-quarter," and so on.
2. Which of the following numbers are rational?
| 1 | −6 | 3½ | − | 2 3 |
0 | 5.8 | 3.1415926535897932384626433 |
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All of them! All decimals are rational. That long one is an approximation to π; see Topic 11 of Trigonometry.
3. A rational number can always be written in what form?
| As a fraction | a b |
, where a and b are integers (b  0). |
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
| square is 2. | 7 5 |
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 | m n |
in lowest terms. |
That is, suppose
| m n |
· | m n |
= | m· m n· n |
= 2. |
| But that is impossible. Since | m n |
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 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.
4. Which numbers have rational square roots?
Only the square roots of the square numbers.
= 1 Rational
Irrational
Irrational
= 2 Rational
, 
, 
, 
Irrational
= 3 Rational
And so on.
Only the square roots of square numbers are rational.
The existence of these irrationals was first realized by Pythagoras in the 6th century B.C. He called them "unnameable" or "speechless." 
Because if we ask, "In the isosceles right triangle, 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."
The decimal representation of irrationals
When we express a rational number as a decimal, then either the decimal will
be a predictable pattern of digits. But if we attempted to express an irrational number as an exact decimal, then, clearly, we could not, because if we could then the number would be rational
Moreover, there will not be a predictable pattern of digits. For example,
1.4142135623730950488016887242097
Now, with rational numbers you sometimes see
| 1 11 |
= | .090909. . . |
| The three dots (ellipsis) mean, "It is not possible to express | 1 11 |
exactly as a decimal. However we can approximate it to as many decimal places as we please according to the indicated pattern; and the more
| decimal places we take, the closer we will come to | 1 11 |
." |
| We say that any decimal for | 1 11 |
is inexact. But the decimal for ¼, |
which is .25, is exact.
The decimal for any irrational number, however, is always inexact. An example is the decimal for 
above.
If we write ellipsis --
= 1.41421356237. . .
-- we mean, "It is not possible to express exactly as a decimal. What you see is a rational approximation. We could continue the approximation for as many decimal places as we please according to the procedure, or algorithm, for calculating the next digit (not the subject of these Topics); and again, the more decimal places we calculate, the closer we will come to ."
It is important to understand that no one has ever beheld the value of any irrational number. (π? What is its value? 3.14159? Not quite.) Our knowledge of any irrational number is only as a rational approximation. And if we choose a decimal approximation, then the more decimal places we calculate, the closer we will be to the value.
One sometimes hears that a real number, such as 
, "is"
an infinite decimal:
= 1.41421356237. . .
But anything we imagine to be actually infinite is never complete, never whole. And can something that is never whole ever be equal to anything?
In calculus, the student will see that an irrational number "exists" only in the sense that it is defined as the limit of the sequence of its rational approximations; assuming, of course, that such a sequence can be produced.
See The mathematical existence of numbers.
Real numbers
5. What is a real number?
Any number that you would expect to find on the number line. It is a number whose name will be the "address" of a point on the number line. Its absolute value will name the distance of that point from 0. The real numbers therefore are the numbers needed for measuring.
6. What are the two main categories of real numbers?
Rational and irrational.
(An actual measurement can result only in a rational number.
An irrational number can result only from a theoretical calculation.)
Problem. 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, that the digits do not repeat. Moreover, we must assume that there is an effective procedure for computing each next digit. For if there were not, then such a "number" would in no sense exist! | ||
| 6.9205729744 Real, rational. Every exact decimal is rational. | ||
7. What is a real variable?
A variable whose values are real numbers.
Calculus is the study of functions of a real variable.
Problem. Let x be a real variable, and let 3 < x < 4. Name five values that x might have.
*
To learn about the evolution of the real numbers starting with the natural numbers, click here.
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