What are rational and irrational numbers? Topics in precalculus

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P R E C A L C U L U S

RATIONAL AND IRRATIONAL
NUMBERS

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:

A rational number is simply any number of arithmetic: Any whole number, fraction, mixed number, or decimal; together with its negative image.

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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.

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.

( $fraction$ is to 1 as 2 is to 3. 2 is two thirds of 3. $fraction$ 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, fifths, 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.

WHY DO WE EVEN CALL numbers "rational"? Because there are numbers that are not rational. For example,

("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

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

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, 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 Rational, irrational numbers

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.

Only the square roots of the square numbers; that is, the square roots of the perfect squares.

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.

Not every length, then, can be named by a rational number. Pythagoras realized that in the 6th century B.C. Rational, irrational numbers

He realized that the ratio of the diagonal of a square to the side was not as two natural numbers—which we can always name. The ratio of the diagonal to the side, 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.

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.

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.

"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 one-11

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.

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.

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.

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 x³ = 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 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.

A variable is a symbol that takes on values. A value is a number.
A real variable takes on values that are real numbers.

Problem 2. Let x be a real variable, and let 3 < x < 4. Name five values that x might have.

RATIONAL AND IRRATIONAL NUMBERS

RATIONAL AND IRRATIONAL
NUMBERS