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RATIONAL AND IRRATIONAL
NUMBERS

What is a rational number?

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 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.

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?

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

Why is it called rational? Each number of arithmetic has a name that indicates its ratio to 1, whch is its source. A whole number, such as "Five," is five times 1.  A fraction, such as "Two-thirds," is two thirds of 1. A mixed number, "Three and one-tenth," is three times 1 plus one tenth of 1.

We will see that language cannot express the relationship of an irrational number to 1.

What is more, we can in principle (by Euclid VI, 9) place any rational number exactly on the number line.

Rational, irrational numbers

We can say that we truly know a rational number.

2.  Which of the following numbers are rational?

1   −1   0   2
3
  2
3
    −5½   6.085   −6.085   3.1415926535897932384

To see the answer, pass your mouse over the colored area.
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.   A rational number can always be written in what form?

  As a fraction   a
b
, where a and b are integers (b not equal to 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:  b = 1.  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 fact that we even have the designation "rational" implies that there are numbers that are not. An example is Square root of 2 ("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.  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 = mn = 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 Square root of 2 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 Square root of 2 .  There really is a length that logically deserves the name, "Square root of 2 ."  Inasmuch as numbers name the lengths of lines, then Square root of 2 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.

Square root of 1 = 1  Rational

Square root of 2  Irrational

Square root of 3  Irrational

Square root of 4 = 2  Rational

Square root of 5,  Square root of 6,  Square root of 7, Square root of 8  Irrational

Square root of 9 = 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 Square root of 2 is irrational.

The existence of irrationals was first realized by Pythagoras in the 6th century B.C.  Rational, irrational numbersHe 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."

5.  Say the name of each number.

a)  Square root of 3   "Square root of 3."

b)  Rational, irrational numbers   "Square root of 5."

c)  Rational, irrational numbers  "2."  This is a rational -- nameable -- number.

d)  Rational, irrational numbers   "Square root of 3/5."

e)  Rational, irrational numbers  "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 one-fourth = .25, or it will not be, as one-third.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,

Square root of 2approximately1.4142135623730950488016887242097

Now, with rational numbers you sometimes see

one-11 = .090909. . .

By writing both the equal sign = and three dots (ellipsis) we mean:

"It is not possible to express one-11 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."

That illustrates the viewpoint that in the mathematics of computation and measuring, which includes calculus, we may say that something exists when it is caused to exist. A decimal approximation, then, will exist when someone calculates it. Nothing could possibly cause an infinity of anything to exist. (".090909 goes on forever.") What is more, actual infinities are not required to solve any problem in arithmetic or calculus; they have no consequences and therefore they are not even necessary.

However, if we imagine that the decimal did go on forever, then 1) it would never be complete and would never equal one-11; 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 one-11 is inexact.  But the decimal for one-fourth, which is .25, is exact.

The decimal for an irrational number is always inexact.  An example is the decimal for Rational, irrational numbers above.

If we write ellipsis --

Square root of 2 = 1.41421356237. . .

-- we mean:

No decimal for Square root of 2 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 Square root of 2 .

It is important to understand that no decimal that you or anyone will ever see  is equal to Square root of 2 , 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.

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 language is incapable of relating an irrational number to 1. Like Pythagoras, we cannot say. An irrational number and 1 are incommensurable.

Real numbers

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 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.     Rational, irrational numbers  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.

Rational, irrational numbers

*

See The Evolution of the Real Numbers starting with the natural numbers.

End of the lessson

Next Topic:  Functions


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