2
RATIONAL AND IRRATIONAL
NUMBERS
A proof that square root of 2 is not rational
Which natural numbers have rational square roots?
The decimal representation of irrationals
CALCULUS IS A THEORY OF MEASUREMENT. The necessary numbers are the rationals and irrationals. But let us start at the beginning.
The following numbers of arithmetic are the counting-numbers or, as they are called, the natural numbers:
1, 2, 3, 4, and so on.
(At any rate, those are their Arabic 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.
And these are their roots:
1 2 3 4 5 6 7 8, and so on.
The student is no doubt familiar with the radical sign: 
.
= 5.
"The square root of 25 is 5."
Rational and irrational numbers
1. What is a rational number?
A rational number is simply a number of arithmetic: a whole number, fraction, mixed number, or decimal; together with its negative image.
2. Which of the following numbers are rational?
| 1 | −1 | 0 | 2 3 |
− | 2 3 |
5½ | −5½ | 6.085 | −6.085 | 3.1415926535897 |
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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 shall see, is not equal to any decimal. For if it were, it would be rational.
3. Why the name rational?
Every number of arithmetic has a ratio—a relationship—to number 1, which is their source. The whole numbers are the multiples of 1, the fractions are its parts: its halves, thirds, fourths, fifths, millionths.
Ratio is the language of arithmetic. It is the language with which we relate rational numbers to one another and to 1. We will see that language cannot express the relationship of an irrational number to 1.
4. 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 we can always name the ratio that the fraction has to 1. It is the same as the numerator has to the denominator.
is to 1 as 2 is to 3. 2 is two thirds of 3. is two thirds of 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.
WE ARE ABOUT TO SEE that the square root of a number that is not a perfect square—√2, √3, √46—is not a rational number. It is not a number of arithmetic. Let us consider √2 ("Square root of 2"). 
is close because
| 7 5 |
· | 7 5 |
= | 49 25 |
—which is almost 2.
To see that there is no rational number whose square is 2, suppose there were. Obviously, it is not a whole number. It will be in the form of a fraction in lowest terms. But the square of a fraction in lowest terms is also in lowest terms.
No new factors are introduced and the denominator will never divide into the numerator to give 2—or any whole number.
There is no rational number whose square is 2 or any number that is not a perfect square. We say therefore that 
is 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.
5. 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.
This follows from the same proof that is irrational.
Not every length, then, can be named by a rational number. Pythagoras realized that in the 6th century B.C. 
He realized that the relationship of the diagonal of a square to the side was not as two natural numbers—which we can always name. That relationship, he said, was without a name. For if we ask, "What relationship has the diagonal to the side?"—we cannot say. Nowadays, of course, we call it "Square root of 2. ." But the idea of an irrational number had not yet occurred. It was not until many centuries after Pythagoras that the radical sign was created.
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. An irrational number and 1 are incommensurable.
6. 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
A number is not inherently a decimal. However, 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 if it were, the number would be rational.
Moreover, there will not be a predictable pattern of digits. For example,
≅ 1.4142135623730950488016887242097
(This symbol ≅ means "is approximately equal to.")
Now, with rational numbers, you sometimes see
= .090909. . .
By writing the equal sign = and three dots (ellipsis) rather than ≅, 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 ."
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. If we write ellipsis—
= 1.41421356237. . .
—we mean: "No decimal for will be exact. 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 ."
That is a fact. It is possible to produce a rational approximation and thus know it and utilize it. We have not said that the decimal for goes on forever, because we cannot produce an infinite sequence of digits nor could we be aware of one. It is nothing but a thought.
We are taught of course that certain decimals go on forever, and so we think that's mathematics—that's the way things are. We do not realize that it is a man made conceit.
This writer asserts that what we can actually bring into this world—1.41421356237—has more being for mathematics than what is only an idea. We can logically produce a decimal approximation.
What is more, infinite decimals are not required to solve any problem in calculus or arithmetic; they have no consequences and therefore they are not even necessary.
Finally, what would it even mean to say that a decimal that never ends "exists"? It certainly does not exist in this world. In quantum mechanics, it is not possible to say that even an electron exists until it is observed.
And so 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.
7. What is a real number?
A real number is what we call any rational or irrational number. They are the numbers we expect to find on the number line. The real numbers are the subject of calculus and of scientific measurement.
The term real number was coined by René Descartes in 1637. It was to distinguish it from an imaginary or complex number
(An actual measurement can result only in a rational number.
An irrational number is required logically or is the result of a definition. Logically, one is necessary upon applying the Pythagorean theorem or as the solution to an equation, such as
x3 = 5. The irrational number π results upon being 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.
| a) 3 Real, rational, integer. | b) −3 Real, rational, integer. | |
| c)−½ Real, rational. | d); Real, irrational. |
|
| e) 5¾ Real, rational. | f) − 11/2 Real, rational. | |
| g) 1.732 Real, rational. | h) 6.920920920. . . Real, rational. | |
| i) 6.92057263. . . Real. And let us assume that it is irrational, that is, no matter how many digits we calculate, they do not repeat. We must assume, therefore, that there is a procedure—a rule—for computing each next digit. For if there were not, then we would not know that symbol's position with respect to order. We would not be able to decide whether it is less than or greater than 6.920572635. Which is to say, it would not be a number. | ||
| (See Are the real numbers really numbers?) | ||
| j) 6.92057263 Real, rational. Every exact decimal is rational. | ||
8. 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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