Topics in

P R E C A L C U L U S

2

# 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.

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?

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 in the standard way we name whole numbers, fractions, and mixed numbers.  "Five." "Six thousand eight hundred nine." "Nine hundred twelve millionths." "Three and five-eighths."

What is more, 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.

2.  Which of the following numbers are rational?

 1 −1 0 23 − 23 5½ −5½ 6.08 −6.08 3.14159

All of them! All decimals are rational. That long one is an approximation to π, which, as we will see, is not equal to any decimal.

3.   A rational number can always be written in what form?

 As a fraction ab , where a and b are integers (b 0).

Numbers that can be written in that form, we call rational. That is their formal definition. That is how a rational number looks. As for what it is, is a different story.

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.  We can always put into words how a rational number is related to 1. Hence the term, rational number.

 ( 23 is to 1 as 2 is to 3.  2 is two thirds of 3. 23 is two thirds of 1.)

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. 75 is close, because
 75 · 75 = 4925

-- 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 mn in lowest terms.

That is, suppose

 mn · mn = m · m n · n = 2.
 But that is impossible.  Since mn 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; that is, the square roots of the perfect squares.

= 1  Rational

Irrational

Irrational

= 2  Rational

,  ,  ,  Irrational

= 3  Rational

And so on.

The square roots of the square numbers are the only square roots that we can name.

The existence of irrationals was first realized by Pythagoras in the 6th century B.C.  He realized that in the isosceles right triangle, the ratio of the hypotenuse to the side was not as two natural numbers. Their relationship, he said, was "without a name." 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 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

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 the number would be rational

Moreover, there will not be a predictable pattern of digits.  For example,

Now, with rational numbers you sometimes see

 1 11 = .090909. . .

By writing both the equal sign = and 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 ."

(That explanation is an example of mathematical positivism. It asserts that in the mathematics of computation and measuring, which includes calculus, what exists is what we actually see or name, now.  That .090909 never ends, is a metaphysical doctrine that need not concern us, because it serves no purpose. Such actual infinities are not required to solve any problem in arithmetic or calculus.  Besides, can what is infinite -- what does not reach an end -- ever be equal to anything?)

 We say that any decimal for 1 11 is inexact.  But the decimal for ¼,

which is .25, is exact.

The symbol for decimal fractions was invented in the 16th century. Now, of course, we take decimals for granted, but at the time many thought it was not a very forward looking idea, because the decimals for only a very limited number of fractions were exact. Even the

 decimal for as simple a fraction as 13 is inexact. See Lesson 24 of Arithmetic.

As for the decimal for an irrational number, it 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 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.

(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. Real, irrational. 5¾  Real, rational. − 11/2 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. In other words, we must assume that there is an effective procedure for computing each next digit. For if there were not, then that symbol would not have its 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.

(But see:  Are the real numbers really numbers?)

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.

Next Topic:  Functions

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