1. Number classification
We were challended to count something such as rocks, trees, and a herd of animals. In response to the challenge, we created counting numbers, often namely natural numbers because of assumption that these numbers were produced naturally or given by the God. These are numbers like these: 1, 2,3.... Then, in ancient India, there was interesting number called zero, denoted as 0, recognized. O plus natural numbers called as the whole numbers. In other words, the whole numbers are like this: 0, 1,2, 3....
Moreover, we have been challenged to deal with various problems such as money exchange expression of temperature ans so on. Then,we created negative numbers. Negative numbers plus the whole numbers are called as the integers. These are like this: ....-3, -2, -1, 0, 1, 2, 3 ...
However, all integers are not sufficient to solve all problems such as sharing a piece of cakes with others and work out percents and percentages and so on. It enables to produce a fraction, i.e, rational numbers, is expressed as a quotient of two integers numbers. Those numbers are called rational numbers. Rational numbers are represented as a decimals. Some decimals terminate whereas others do exhibit a pattern of repetition such as 7/66, 2/3 ... However, there are decimals that do not fit into either of the above mentioned two categories. Such decimals are called irrational numbers. These numbers are naturally occurred.
Let us consider the isosceles right triangles whose legs are each of length. Then, its hypotenus is the positive number whose square is 2. This number is irrational because it does exhibit no pattern decimals such as 1.4142....
Other popular irrational number is a number that is equal to the ratio of circumstance to its diameter of any circle, called ''pi'.
There are many irrational numbers. Some are shown in a table. However, any saying to wonder whether any given number is irrational or not is not easy peasy math. A popular way to solve this sort of problem is a method of contradiction.
Rational plus irrational numbers are called as real numbers. The extension of the real numbers produces another new class of numbers, called complex numbers.
It is kindly requested to have a look at the following references if you like to know more about number properties. It is honestly acknowledged that the following information was detached from the websites such as:
- http://www.factmonster.com/ipka/A0876704.html
- http://www.mathsisfun.com/irrational-numbers.html
- http://mathworld.wolfram.com/IrrationalNumber.html
- http://www.newton.dep.anl.gov/askasci/math99/math99223.htm
- http://wiki.answers.com
Irrational Numbers
An
Irrational Number is a real number that cannot be written as a
simple fraction.
Irrational
means not Rational
Examples:
Rational
Numbers
OK. A Rational Number
can be written as a Ratio of two integers (ie a simple fraction).
Example: 1.5 is rational,
because it can be written as the ratio 3/2
Example: 7 is rational,
because it can be written as the ratio 7/1
Example 0.333... (3
repeating) is also rational, because it can be written as the ratio 1/3
Irrational
Numbers
But some numbers cannot be
written as a ratio of two integers ...
...they
are called Irrational Numbers.
It is irrational because it
cannot be written as a ratio (or fraction),
not because it is crazy!
|
Example: π (Pi) is a famous
irrational number.
π
= 3.1415926535897932384626433832795 (and more...)
You cannot write down a
simple fraction that equals Pi.
|
The popular approximation of 22/7
= 3.1428571428571... is close but not accurate.
Another clue is that the decimal
goes on forever without repeating.
Rational
vs Irrational
So you can tell if it is Rational or
Irrational by trying to write the number as a simple fraction.
Example:
9.5 can be written as a simple fraction like this:
9.5
= 19/2
So
it is a rational number (and so is not irrational)
Here are some more examples:
Number
|
As
a Fraction
|
Rational
or
Irrational?
|
1.75
|
7/4
|
Rational
|
.001
|
1/1000
|
Rational
|
√2
(square root of 2)
|
?
|
Irrational
!
|
Square
Root of 2
Let's look at the square root of 2
more closely.
If you draw a square (of size
"1"),
what is the distance across the diagonal?
|
The answer is the square root of 2,
which is 1.4142135623730950...(etc)
But it is not a number like 3, or
five-thirds, or anything like that ...
... in fact you cannot write
the square root of 2 using a ratio of two numbers
... I explain why on the Is It
Irrational? page,
... and so we know it is an
irrational number
Famous
Irrational Numbers
Pi is a famous irrational number. People have calculated Pi
to over one million decimal places and still there is no pattern. The first
few digits look like this:
3.1415926535897932384626433832795
(and more ...)
|
|||||
The number e (Euler's Number)
is another famous irrational number. People have also calculated e
to lots of decimal places without any pattern showing. The first few digits
look like this:
2.7182818284590452353602874713527
(and more ...)
|
|||||
The Golden Ratio
is an irrational number. The first few digits look like this:
1.61803398874989484820... (and
more ...)
|
|||||
Many square roots, cube roots, etc
are also irrational numbers. Examples:
|
|||||
But √4 = 2 (rational), and √9 = 3
(rational) ...
...
so not all roots are irrational.
|
Note
on Multiplying Irrational Numbers
Have a look at this:
- π × π = π2 is irrational
- But √2 × √2 = 2 is rational
So be careful ... multiplying
irrational numbers can result in a rational number!
History
of Irrational Numbers
Apparently Hippasus
(one of Pythagoras' students) discovered irrational numbers when trying
to represent the square root of 2 as a fraction (using geometry, it is
thought). Instead he proved you couldn't write the square root of 2 as a
fraction and so it was irrational.
However Pythagoras could not accept the existence of irrational
numbers, because he believed that all numbers had perfect values. But he could
not disprove Hippasus' "irrational numbers" and so Hippasus
was thrown overboard and drowned!
A rational number is a number that can be written as a ratio.
That means it can be written as a fraction, in which both the numerator (the
number on top) and the denominator (the number on the bottom) are whole
numbers.
- The number 8 is a rational
number because it can be written as the fraction 8/1.
- Likewise, 3/4 is a rational
number because it can be written as a fraction.
- Even a big, clunky fraction
like 7,324,908/56,003,492 is rational, simply because it can be written as
a fraction.
Every whole number is a rational number, because any whole
number can be written as a fraction. For example, 4 can be written as 4/1, 65
can be written as 65/1, and 3,867 can be written as 3,867/1.
Irrational Numbers
All numbers that are not rational are considered irrational. An
irrational number can be written as a decimal, but not as a fraction.
An irrational number has endless non-repeating digits to the
right of the decimal point. Here are some irrational numbers:
π = 3.141592…
Although irrational numbers are not often used in daily life,
they do exist on the number line. In fact, between 0 and 1 on the number line,
there are an infinite number of irrational numbers!
Fact Monster/Information Please® Database,
© 2008 Pearson Education, Inc. All rights reserved.
Irrational Number
An irrational number is a number that cannot be expressed
as a fraction
for any integers
and
. Irrational numbers have decimal expansions
that neither terminate nor become periodic. Every transcendental
number is irrational.
There is no standard notation for the set of irrational
numbers, but the notations
,
, or
, where the bar, minus sign,
or backslash indicates the set complement of the rational numbers
over the reals
, could all be used.
The most famous irrational number is
, sometimes called Pythagoras's
constant. Legend has it that the Pythagorean philosopher Hippasus used
geometric methods to demonstrate the irrationality of
while at sea and,
upon notifying his comrades of his great discovery, was immediately thrown
overboard by the fanatic Pythagoreans. Other examples include
,
,
, etc. The Erdős-Borwein
constant
(1)
|
|||
(2)
|
|||
(3)
|
(Sloane's A065442;
Erdős 1948, Guy 1994), where
is the numbers of
divisors of
, and a set of
generalizations (Borwein 1992) are also known to be irrational (Bailey and
Crandall 2002).
Numbers of the form
are irrational
unless
is the
th power of an integer. Numbers of the form
, where
is the logarithm, are
irrational if
and
are integers, one of which has
a prime factor
which the other lacks.
is irrational for
rational
.
is irrational for every
rational number
(Niven 1956, Stevens
1999), and
(for
measured in degrees) is
irrational for every rational
with
the exception of
(Niven
1956).
is irrational for every
rational
(Stevens 1999).
The irrationality of e was proven by Euler in
1737; for the general case, see Hardy and Wright (1979, p. 46).
is irrational for positive integral
. The irrationality of pi itself was proven by
Lambert in 1760; for the general case, see Hardy and Wright (1979, p. 47). Apéry's constant
(where
is the Riemann zeta
function) was proved irrational by Apéry (1979; van der Poorten 1979). In
addition, T. Rivoal (2000) recently proved that there are infinitely many
integers
such that
is irrational.
Subsequently, he also showed that at least one of
,
, ...,
is irrational
(Rivoal 2001).
From Gelfond's theorem,
a number of the form
is transcendental
(and therefore irrational) if
is algebraic
, 1 and
is irrational and algebraic. This
establishes the irrationality of Gelfond's constant
(since
),
and
. Nesterenko
(1996) proved that
is irrational. In
fact, he proved that
,
and
are algebraically
independent, but it was not previously known that
was irrational.
Given a polynomial equation
(4)
|
where
are integers, the roots
are either integral or
irrational. If
is irrational,
then so are
,
, and
.
Irrationality has not yet been established for
,
,
, or
(where
is the Euler-Mascheroni
constant).
Quadratic
surds are irrational numbers which have periodic continued fractions.
Hurwitz's
irrational number theoremgives bounds of the form
(5)
|
for the best rational approximation possible for an
arbitrary irrational number
, where the
are called Lagrange numbers
and get steadily larger for each "bad" set of irrational numbers
which is excluded.
The series
(6)
|
where
is the divisor function,
is irrational for
and 2.
SEE ALSO: Algebraic Integer,
Algebraic Number,
Almost Integer, Continuum, Decimal Expansion,
Dirichlet
Function, e, Ferguson-Forcade
Algorithm, Gelfond's
Theorem, Hurwitz's
Irrational Number Theorem, Near Noble Number,
Noble Number, Pi, Pythagoras's
Constant, Pythagoras's
Theorem, Regular
Number, Repeating
Decimal, q-Harmonic
Series, Quadratic
Surd, Rational
Number, Segre's
Theorem, Transcendental
Number
REFERENCES:
Apéry,
R. "Irrationalité de
et
." Astérisque
61, 11-13, 1979.
Bailey,
D. H. and Crandall, R. E. "Random Generators and Normal Numbers."Exper.
Math. 11, 527-546, 2002.
Preprint
dated Feb. 22, 2003 available at http://www.nersc.gov/~dhbailey/dhbpapers/bcnormal.pdf.
Borwein,
P. "On the Irrationality of Certain Series." Math. Proc. Cambridge
Philos. Soc. 112, 141-146, 1992.
Courant,
R. and Robbins, H. "Incommensurable Segments, Irrational Numbers, and the
Concept of Limit." §2.2 in What
Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.
Oxford, England: Oxford University Press, pp. 58-61, 1996.
Erdős,
P. "On Arithmetical Properties of Lambert Series." J. Indian Math.
Soc. 12, 63-66, 1948.
Gourdon,
X. and Sebah, P. "Irrationality Proofs." http://numbers.computation.free.fr/Constants/Miscellaneous/irrationality.html.
Guy,
R. K. "Some Irrational Series." §B14 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 69,
1994.
Hardy,
G. H. and Wright, E. M. An
Introduction to the Theory of Numbers, 5th ed. Oxford, England:
Clarendon Press, 1979.
Huylebrouck,
D. "Similarities in Irrationality Proofs for
,
,
, and
." Amer.
Math. Monthly 108, 222-231, 2001.
Manning,
H. P. Irrational
Numbers and Their Representation by Sequences and Series. New York:
Wiley, 1906.
Nagell,
T. "Irrational Numbers" and "Irrationality of the numbers
and
." §12-13 in Introduction
to Number Theory. New York: Wiley, pp. 38-40, 1951.
Nesterenko,
Yu. "Modular Functions and Transcendence Problems." C. R. Acad.
Sci. Paris Sér. I Math. 322, 909-914, 1996.
Nesterenko,
Yu. V. "Modular Functions and Transcendence Questions."Mat. Sb.
187, 65-96, 1996.
Niven,
I. M. Irrational
Numbers. New York: Wiley, 1956.
Niven,
I. M. Numbers:
Rational and Irrational. New York: Random House, 1961.
Pappas,
T. "Irrational Numbers & the Pythagoras Theorem." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 98-99,
1989.
Rivoal,
T. "La fonction Zeta de Riemann prend une infinité de valeurs
irrationnelles aux entiers impairs." Comptes Rendus Acad. Sci. Paris
331, 267-270, 2000.
Rivoal,
T. "Irrationalité d'au moins un des neuf nombres
,
, ...,
." 25 Apr
2001. http://arxiv.org/abs/math.NT/0104221.
Sloane,
N. J. A. Sequence A065442in "The
On-Line Encyclopedia of Integer Sequences."
Stevens,
J. "Zur Irrationalität von
." Mitt. Math.
Ges. Hamburg 18, 151-158, 1999.
van
der Poorten, A. "A Proof that Euler Missed... Apéry's Proof of the
Irrationality of
." Math.
Intel. 1, 196-203, 1979.
Weisstein,
E. W. "Books about Irrational Numbers." http://www.ericweisstein.com/encyclopedias/books/IrrationalNumbers.html.
Referenced on Wolfram|Alpha:
Irrational Number
CITE THIS AS:
Weisstein, Eric W.
"Irrational Number." From MathWorld--A
Wolfram Web Resource. http://mathworld.wolfram.com/IrrationalNumber.html
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:
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?
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
|
2
3
|
−
|
2
3
|
5½
|
−5½
|
6.08
|
−6.08
|
3.1415926535897932384626
|
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 will see, is not equal to any decimal.
3. A rational number can always be
written in what form?
As a fraction
|
a
b
|
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 positive, when they are natural numbers, then a rational number has the
same ratio to 1 as two natural
numbers, that is, as the numerator has to the denominator. Hence the term, rational
number.
(
|
2
3
|
is to 1 as 2 is to
3. 2 is two thirds of 3.
|
2
3
|
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.
|
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
|
=
|
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; that is, the square roots of the perfect squares.
And so on.
Only the square roots
of square numbers are rational.
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 realism. It
asserts that in the mathematics of computation and measuring, which includes
calculus, what exists is what we actually observe or name, now. That .090909
never ends is a doctrine that need not concern us, because it serves no useful
purpose. Such actual
infinities have no practical effect on calculations in arithmetic or
calculus.)
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
|
1
3
|
is inexact.
|
As for the decimal
for an irrational number, it is always inexact. An example is the decimal for
above.
If we write ellipsis
--
-- 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.
To sum up, a rational
number is a number we can know 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 each rational number 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 we cannot relate an
irrational number to 1. Like Pythagoras, we cannot say. An irrational
number and 1 are incommensurable.
To
put it another way, a rational number partakes of the essence of number,
which is to answer the question "How many?" (whether apples or
inches), for a rational number is composed of what is countable. An integer is
a number of 1's (or −1's). A fraction or a decimal is a number of unit-fractions.
But an irrational number is not a number of anything.
One
often hears however that an irrational number is an infinite decimal.
But
if a decimal, even as an idea, did not end, then it would not be a number. Why
not? Because, to be useful, numbers -- decimals -- have names, and
therefore we can name their sum, their difference, their product and their
quotient. But an infinite sequence of digits does not have a name. It is not
that we will never finish naming it. We cannot even begin.
Finally,
can what is infinite -- what does not reach an end
-- ever be equal to anything?
5. What is a real
number?
A real number is distinguished
from an imaginary
number. It is any rational or irrational number that we can name. 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; examples
are the Pythagorean theorem, and the
solution to an equation, such as x³ = 5.
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.
|
||
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.
|
||
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.
What is an integer? { ... -3, -2, -1, 0, 1, 2, 3, ... }Integers are the whole numbers, negative whole numbers, and zero. For example, 43434235, 28, 2, 0, -28, and -3030 are integers, but numbers like 1/2, 4.00032, 2.5, Pi, and -9.90 are not. We can say that an integer is in the set: {...3,-2,-1,0,1,2,3,...} (the three dots mean you keep going in both directions.)It is often useful to think of the integers as points along a 'number line', like this: About integersThe terms even and odd only apply to integers; 2.5 is neither even nor odd. Zero, on the other hand, is even since it is 2 times some integer: it's 2 times 0. To check whether a number is odd, see whether it's one more than some even number: 7 is odd since it's one more than 6, which is even.Another way to say this is that zero is even since it can be written in the form 2*n, where n is an integer. Odd numbers can be written in the form 2*n + 1. Again, this lets us talk about whether negative numbers are even and odd: -9 is odd since it's one more than -10, which is even. Every positive integer can be factored into the product of prime numbers, and there's only one way to do it for every number. For instance, 280 = 2x2x2x5x7, and there's only one way to factor 280 into prime numbers. This is an important theorem: the Fundamental Theorem of Arithmetic. See Notes and Literature on Prime Numbers from Understanding Mathematics by Peter Alfeld, and the Largest Known Primes page. Most mathematicians, at least when they're talking to each other, use Z to refer to the set of integers. In German the word "zahlen" means "to count" and "Zahl" means "number." Mathematicians also use the letter N to talk about the set of positive integers, in other words the set {1,2,3,4,5,6, ...}. From the Dr. Math archives:
Adding and
Subtracting Integers (Elementary/Addition)
Consecutive
Integers (High School/Algebra)
Introduction
to Negative Numbers (Elementary/Subtraction)
Is Zero
Even, Odd, or Neither? (Elementary/About Numbers)
Sets
and Integer Pairs (High School/Discrete Math)
Sets and
Subsets (Middle School/Algebra)
Partitioning
the Integers (High School/Discrete Math)
Why is 1 not
considered prime? (Middle School/About Numbers)
On the WebRational Numbers 5/1, 1/2, 1.75, -97/3 ...A rational number is any number that can be written as a ratio of two integers (hence the name!). In other words, a number is rational if we can write it as a fraction where the numerator and denominator are both integers.The term "rational" comes from the word "ratio," because the rational numbers are the ones that can be written in the ratio form p/q where p and q are integers. Irrational, then, just means all the numbers that aren't rational. Every integer is a rational number, since each integer n can be written in the form n/1. For example 5 = 5/1 and thus 5 is a rational number. However, numbers like 1/2, 45454737/2424242, and -3/7 are also rational, since they are fractions whose numerator and denominator are integers. So the set of all rational numbers will contain the numbers 4/5, -8, 1.75 (which is 7/4), -97/3, and so on. Is .999 repeating a rational number? Well, a number is rational if it can be written as A/B (A over B): .3 = 3/10 and .55555..... = 5/9, so these are both rational numbers. Now look at .99999999..... which is equal to 9/9 = 1. We have just written down 1 and .9999999 in the form A/B where A and B are both 9, so 1 and .9999999 are both rational numbers. In fact all repeating decimals like .575757575757... , all integers like 46, and all finite decimals like .472 are rational. From the Dr. Math Archives:
Rational and
Irrational Numbers (Elementary/About Numbers)
About
Rational Numbers (Middle School/About Numbers)
Is a
Ratio Rational or Irrational? (High School/Analysis)
On the Web:
Concrete Algebra: Numbers
Rational Number - Eric Weisstein's World of Mathematics Real Numbers Complex Numbers Irrational Numbers sqrt(2), pi, e, the Golden Ratio ...Irrational numbers are numbers that can be written as decimals but not as fractions.An irrational number is any real number that is not rational. By real number we mean, loosely, a number that we can conceive of in this world, one with no square roots of negative numbers (such a number is called complex.) A real number is a number that is somewhere on a number line, so any number on a number line that isn't a rational number is irrational. The square root of 2 is an irrational number because it can't be written as a ratio of two integers. Other irrational numbers include the square root of 3, the square root of 5, pi, e, and the golden ratio. (For more information about pi and e, see Pi = 3.14159... and E = 2.71828..., also from the Dr. Math FAQ.) Pi is an irrational number because it cannot be expressed as a ratio (fraction) of two integers: it has no exact decimal equivalent, although 3.1415926 is good enough for many applications. The square root of 2 is another irrational number that cannot be written as a fraction. In mathematics, a name can be used with a very precise meaning that may have little to do with the meaning of the English word. ("Irrational" numbers are NOT numbers that can't argue logically!) From the Dr. Math Archives:
Venn Diagram of
Our Number System (Middle School/About Numbers)
Golden
Ratio and Golden Rectangle (Elementary/Golden Rectangle)
Irrational
Pi (High School/Transcendental Numbers)
The Number e
(High School/Transcendental Numbers)
Meaning
of Irrational Exponents (High School/Algebra)
Irrational
Powers (College/Modern Algebra)
Proof that
Sqrt(2) is Irrational (High School/Square Roots)
Proving the
Square Root of 3 Irrational (High School/Square Roots)
Are
Transcendentals Irrational? (High School/Transcendental Numbers)
On the Web:More information can be found by searching the Dr. Math archives or the Math Forum site for "integer" or "rational" or "irrational" (just the word, not the quotes). |
A proof that the square root
of 2 is irrational
How do we know that square root of 2 is an irrational number? In other words, how do we know that √2 wouldn't have a pattern in the decimal sequence? Maybe the pattern is very well hidden and is really long, billions of digits? Even if you check it till million first digits, maybe the pattern is just longer than you were able to print the digits out with your computer?
Here is where mathematical proof comes in. The proof that √2 is indeed irrational is usually found in college level texts, but it isn't that difficult to follow. It does not rely on computers at all, but instead it is a 'proof by contradiction' - if √2 WERE a rational number, then we'd get a contradiction. I encourage you to let your high school students study this proof since it is very illustrative of a typical proof in mathematics and is not very hard to follow.
The proof that
square root of 2 is irrational:
Let's suppose √2 were a rational number. Then we can write it √2
= a/b where a, b are whole numbers, b not zero. We additionally make it so
that this a/b is simplified to the lowest terms, since that can obviously be
done with any fraction.
It
follows that 2 = a2/b2, or a2 = 2 * b2. So the square of a is
an even number since it is two times something. From this we can know that a
itself is also an even number. Why? Because it can't be odd; if a itself was
odd, then a * a would be odd too. Odd number times odd number
is always odd. Check if you don't believe that! Okay, if a itself is an even number, then a is 2 times some other whole number, or a = 2k where k is this other number. We don't need to know exactly what k is; it won't matter. Soon is coming the contradiction: If we substitute a = 2k into the original equation 2 = a2/b2, this is what we get:
This means b
2 is even, from which follows again that b itself is an even
number!!!
|
Name: Jehanzeb
Status: student
Age: N/A
Location: N/A
Country: N/A
Date: 9/25/2005
Question:
Is the sum of two irrational numbers equal to a
rational number?
Replies:
The sum of two rational
numbers, p/q + n/m = (pm +qn)/qm. If either p/q or n/m = X or Y, respectively,
and either X or Y cannot be expressed as a ratio, then the sum cannot be
expressed as a ratio. However, an irrational number can be expressed as an
INFINITE sum of rational numbers. This was first noted by Newton (1676), but
Euler gave the first proof (1774). So the sum of two IRRATIONAL numbers can be
expressed as an INFINITE SUM of rational numbers.
Vince Calder
Jehanzeb,
Two irrational numbers very RARELY add up to a rational number.
Any integer, finite decimal, or repeating decimal is a rational number. A
rational number can be represented as a fraction. An irrational number cannot.
It IS true that two RATIONAL numbers add up to a rational number: two fractions
always add up to a fraction. Here are two examples: (1/2)+(1/3)=(5/6), 1+3=4.
The latter applies because you can represent it as (1/1)+(3/1)=(4/1).
However, sqrt(2) is not rational because there is no fraction, no ratio of
integers, that will equal sqrt(2). It calculates to be a decimal that never
repeats and never ends. The same can be said for sqrt(3). Also, there is no way
to write sqrt(2)+sqrt(3) as a fraction. In fact, the representation is already
in its simplest form. To get two irrational numbers to add up to a rational
number, you need to add irrational numbers such as [1+sqrt(2)] and [1-sqrt(2)].
In this case, the irrational portions just happen to cancel out, leaving:
[1+sqrt(2)]+[1-sqrt(2)]=2. 2 is a rational number (i.e. 2/1).
Dr. Ken Mellendorf
Physics Instructor
Illinois Central College
Is the product of two
irrational numbers irrational?
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Answer:
Sometimes
it is and sometimes it isn't. The square root of 2 and the square root of 3 are
both irrational, as is their product, the square root of 6. The square root of
2 and the square root of 8 are both irrational, but their product, the square
root of 16, is rational (in fact, it equals 4).
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Is the product of two
irrational numbers always an irrational number?
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Answer:
No.
The square root of two is an irrational number. If you multiply the square root
of two by the square root of two, you get two which is a rational number.
Example of two
irrational numbers the product of which is an irrational number?
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Answer:
sqrt(2)*sqrt(3) is an
irrational product.
Can you add two
irrational numbers to get a irrational number?
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Answer:
Sure. The sum of two
irrational numbers will USUALLY be irrational - in some cases it will be
rational. For a start, if you add the irrational number with itself, you get
another irrational number.
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