Rational and irrational numbers - Topics in precalculus

Topics in

P R E C A L C U L U S

Table of Contents | Home

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?

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 ordinary 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?

2
3

3.1415926535897932384626433

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

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

square is 2.

7
5

is close, because

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· 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 natural numbers have rational square roots?

Only the square roots of the square numbers.

We call those square numbers perfect squares.

= 1 Rational

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. In the isosceles right triangle, he called the ratio of the hypotenuse to the side "unnameable" or "speechless." 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 that we saw that only the square roots of square numbers are rational, we could prove that the only nth roots that are rational, are the nth roots of perfect nth powers. 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

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. . .

By writing three dots (ellipsis) we mean, "A decimal for

1
11

will never

be complete or exact. 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

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, "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 decimal digits we calculate, the closer we will be to ."

It is important to understand that no decimal 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, we could say that an irrational number is a number that we can never know exactly. While a rational number we can know exactly, either as a whole number or a fraction, but not always exactly as a decimal.

One sometimes hears that an irrational 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?

What is more, if the decimal really went on forever (whatever that means), then it would not be a number Why not? Because numbers have names. And an infinite decimal could never be the name of any measurement. It is not that we will never finish naming an infinite decimal. We cannot even begin

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 we need 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.
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, 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 we would not know which number we are computing. And that symbol would not refer to any "number"!

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 2. 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.

Next Topic: Functions

Table of Contents | Home

Please make a donation to keep TheMathPage online.
Even $1 will help.

Questions or comments?

E-mail: themathpage@nyc.rr.com