Saturday, September 15, 2012

Number classification


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:

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:
√3
1.7320508075688772935274463415059 (etc)
√99
9.9498743710661995473447982100121 (etc)
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…

= 1.414213…

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.


(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.


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.


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
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
 

 
, 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 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.

= 1 Rational

Irrational

Irrational

= 2 Rational

, , , Irrational

= 3 Rational

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 --

= 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.

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.

= 1.41421356237. . .

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.
Real, irrational.
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:
Note that zero is neither positive nor negative.

About integers

The 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 Web

Integers
6th Grade Math Class Integer Pictures - Oak Point Intermediate School









Rational 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:










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:

Irrational Number - Eric Weisstein's World of Mathematics



 




 











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:
2
=
(2k)2/b2
2
=
4k2/b2
2*b2
=
4k2
b2
=
2k2.




This means b

2 is even, from which follows again that b itself is an even number!!!
WHY is that a contradiction? Because we started the whole process saying that a/b is simplified to the lowest terms, and now it turns out that a and b would both be even. So √2 cannot be rational.

 


 

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?




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).



 

Is the product of two irrational numbers always an irrational number?




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?




Answer:

sqrt(2)*sqrt(3) is an irrational product.


 

Can you add two irrational numbers to get a irrational number?




 

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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