WHAT IS IRRATIONAL NUMBERS?

Math Adda

The invention of irrational numbers

We have seen  that when two squares are in the same ratio as two square numbers, then their sides will have the same ratio as the square roots.  Thus if CD were 1 meter, then we would like to say that AB is "Square root of 2" meters -- but there is no such rational number. For if there were, then again, AB, CD would have a common measure, which they do not.

If we insist, however, that there be a number to indicate the ratio of AB to CD, then we keep the name "Square root of 2," and we call it an irrational number.  Its numeral is .  That is,

 ×  = 2.

 is that number which when multiplied by itself -- when squared -- is 2.  "Square root of 2" is its name.  It is not any whole number, any fraction, any mixed number or any decimal -- it is not any rational number.  It is a pure creation.

Now,

7 5

is close to , because

7 5

×

7 5

=

49 25

-- which is almost 2.  But 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.  It will be impossible to divide n × n intom × m and get 2

There is no rational number whose square is 2.

Naturally, we wonder, "How much is ?"  We can only answer

approximately.  As a fraction, we saw it is almost

7 5

.   As a decimal, it is

approximately 1.414.  How would anyone know that?  Again, because 1.414 squared is almost 2.

1.414 × 1.414 = 1.999396.

We could come closer to  by approximating it to more decimal places.  But no decimal squared will ever be exactly 2.   is irrational.

It was argued for many centuries whether  "really" is a number. Mathematicians thought of  as a convenient symbol, but as for its being a number, that was something else.  Because again, if we ask, How much is it?. . . We cannot say.

In the following Topic, we will investigate in what sense irrational numbers "exist."  And we will return to our original inquiry:

If AB, CD are lengths, will there always be a number n -- rational or irrational -- such that, proportionally,

AB is to CD  as  1 is to n?

In any event,  is not the only irrational number.  Let us illustrate that by asking:

The square roots of which natural numbers are rational?

Answer.   Only the square roots of square numbers.  Thus,

 = 1  is rational.

,  are irrational.

 = 2  is rational.

,  ,  ,   are irrational.

 = 3  is rational.

And so on.

Problem 1.   Say the name of each number.

a)	Square root of 3.		b)	Square root of 5.
c)	Two. It's rational.		d)	Square root of 3/5.
e)	Three-fifths.		f)	One.

Problem 2.   Which of the following are rational numbers and which are irrational?

a)	Irrational.		b)	Rational.
c)	Rational.		d)	Irrational.

A common measure with 1

We have seen that every number has a ratio to 1.  A rational number is to 1 in the same ratio as two natural numbers.  That is, every rational number has a common measure with 1 (Topic 8).  We can say, then, that an irrational number is a number that has no common measure with 1.

1 and  are incommensurable.

Problem 3.   What number is a common measure of each pair?

a)   1 and 5   1

b)   1 and

5 8

1 8

c)   1 and 2.617    .001

d)   1 and     None.

Problem 4.   Why have irrational numbers been invented?

To express the ratios of incommensurable magnitudes; in particular, the ratio of a magnitude incommensurable with 1.

Problem 5.   The squares on the sides of triangle ABC are in the ratio 1 : 4 : 3.  Express the ratio of each pair of sides as a ratio of numbers, whether rational or irrational.

a)  AB : BC = 1 : 2         b)  BC : CA = 2 :          c)  CA : AB =  : 1

d)   is approximately 1

5 7

, or

12  7

.  How could you know that?

12  7

·

12  7

=

144  49

-- which is almost 3.

e)   Use that approximation to approximate the ratio of BC to CA as a
e)   ratio of natural numbers.

BC : CA  =  2 :

12  7

=  14 : 12 = 7 : 6

Problem 6.   One square is four fifths of another.

a)  Are the sides commensurable?   No. 4 and 5 are not both square
a) numbers.

b)  Express the ratio of the sides as a ratio of numbers.

2 : 

c)  Are the squares commensurable?

Yes. They are in the same ratio as natural numbers.

WHAT IS THE NATURAL NUMBERS?

Math Adda                           WHAT IS THE NATURAL NUMBERS?

MATHEMATICS IS NOT ONLY about numbers, it is about things that are not numbers; for example, length;  the distance from here to there,

A--------------------------B

C--------I------------------I-------------------D

from A to B.  Now a length is not a number,  yet we describe lengths by saying that they are like numbers.  For if CD is made up of three segments equal to AB, then we say,

AB is to CD  as  1 is to 3.

And if AB happens to be 1 centimeter, we would say that CD is 3 centimeters.

That is, we can say that CD is "three times" AB, and that is called theratio -- the relationship -- of CD to AB.  The eventual question will be:
                  1
A--------------------------B
                                 n
C---------------------------------------------D

If AB, CD are any lengths, will we always be able to express their ratio in words.  Will there always be a number n such that, proportionally,

AB is to CD  as  1 is to n?

That number will be called a real number, which is a number required to name the length of a line, relative to the line we have called 1.  We will see that there will be problems.  At the root of the problem is the difference between arithmetic and geometry.

The natural numbers

Arithmetic begins with counting, and a unit is whatever we would call one.  One apple, one orange, one person.  We count units -- which must have the same name. One apple, two apples, three apples. Therefore we must recognize whether things are the same or different. ("Here's one; here's one; these two are the same.  This is not one. Those two are different.")

A natural number is a collection of these indivisible ones.  5 people,
10 chairs, 32 names.  You cannot take half of any one.

The natural numbers have their everyday names and symbols:  1, 2,
3, 4, and so on.  Those symbols, however, are called numerals, and they represent the numbers.  They stand in for them.  Throughout history there have been many ways of representing numbers.  The student is surely familiar with the Roman numerals:  I, V, X, and so on.

The natural number is the actual collection of units, /////, whether strokes, apples, letters, or the idea of units.  For there is no '5' apart from five units, even though we do not say the word units.  The natural numbers are truly natural.  We find them in nature.

Nevertheless, it is conventional to refer to the numerals -- 1, 2, 3, 4, and so on -- as "numbers," which they are not.

Numerals are physical things. We write them on paper, and they obey rules of computation according to their physical appearance on the page. Mathematics at every level is concerned with physical form.

We are about to see that we can always name the ratio, the relationship, of any two natural numbers.

Cardinal and ordinal

The natural numbers have two forms, cardinal and ordinal.  The cardinal forms are

One, two, three, four,

and so on.  They answer the question How much? or How many?

The ordinal forms are

First, second, third, fourth,

and so on.  They answer the question Which one?

We will see that the ordinal numbers express division into equal parts. They will answer the question, Which part?

Parts of natural numbers

We say that a smaller number is a part of a larger number  if the smaller is contained in the larger an exact number of times. (That is called analiquot part.)  Equivalently, the larger number is a multiple of the smaller.

Consider these first few multiples of 5:

5, 10, 15, 20, 25, 30.

5 is the first multiple of 5.  10 is the second;  15, the third; and so on.

5 is a part of each multiple except itself.  It is a part of 10, of 15, of 20, and so on.

Now, since 15 is the third multiple of 5, we say that 5 is the third partof 15.  We use that same ordinal number to name the part.

The ordinal number names which part of fifteen 5 is.

5 is the fourth part of 20; it is the fifth part of 25; the sixth part of 30.  And so on.

5 is which part of 10?  We do not say the second part. We say half.  5 is half of 10.

It is extremely important to understand that we are not speaking here of proper fractions -- numbers that are less than 1 and that we need for measuring.  We are explaining how the ordinal numbers -- third, fourth, fifth, and so on -- name the parts of the cardinal numbers.  The names of the parts in fact are prior to the names of the fractions, as we will see. (Why do we call the number we write as 1 over 3 "one-third"? Because 1 is the third part of 3)

Note that 5 is not a part of itself.  There is no such thing as the first part.

So, with the exception of the name half, we name each part with ordinal number.  The ordinal number names which part.

What is a real number?

Math Adda                                      

Real numbers

What is a real number?

A real number is distinguished from an imaginary number.
It is a number we require for measuring rather than counting. Hence a real number is any rational or irrational number that we might name. They are the numbers we expect to find on the number line.

(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 solving
 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, andinteger.  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 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.

Class 10 Polynomial CBSE test Paper O1

Math Adda 

Polynomials sample test paper 

1. The graph of y=f(x) is given below. Find the number of zeroes of f(x)

2. Write the zeroes of the polynomial x2 -2x + 4.

3. Find a quadratic polynomial, the sum and product of whose zeroes are 0 and 5 respectively.

4. Find the quadratic polynomial, the sum and product of whose zeroes are 4 and 1, respectively

5. If a andb are the zeros of the quadratic polynomial f(x)= x2-5x+4, find the value of 1/a + 1/b-2a b

6. Find the zeroes of the quadratic polynomial 4 √3 x2 + 5 x - 2 √3 and verify the relationship between the zeroes and the coefficients.

7. Find the zeroes of the quadratic polynomial 4u2 + 8u and verify the relationship between the zeroes and the coefficients

8. Find the quadratic polynomial, the sum and product of whose zeroes are √2 and √3 respectively.

9. If a and b are the zeros of the given quadratic polynomial f(x)= 5x2 - 7x + 1, find the value 1/a + 1/b

10. Find the zeroes of the polynomial x2 – 3 and verify the relationship between the zeroes and the
Coefficients

11. Find the remainder when p(x)= x3-6x2+2x-4 when divided by 1 - 2x.

12. Find the remainder when x51+51 is divided by (x+1).

13. Find all the integral zeros of x3 -3x2 - 2x + 6

14. Obtain all zeros of 3x4 + 6x3 - 2x2 - 10x - 5, if two of its zeros are √5/√3 and - √5/√3

15. If (x - 2) and [x – ½ ] are the factors of the polynomials qx2 + 5x + r prove that q = r

16. If the zeroes of the polynomial are 3x2 − 5x + 2 are a+ b and a- b, find a and b.

17. On dividing 2x2 + 3x + 1 by a polynomial g(x), the quotient and the remainder were 2x-1 and 3 respectively. Find g (x).

What is a rational number?

Math Adda                     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.

Q. Which of the following numbers are rational?

1

−6

3½

−

2 3

0

5.8

3.1415926535897932384626433

A rational number can always be written 

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

·

7 5

=

49 25

-- which is almost 2.

cbse guide : Test Paper-2011-12

Math Adda                      Test Paper-2011-12

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cbse guide.weebly.com/classx2.html

Math Adda by cbse guide

Decimal Number

Fundamental Theorem of Arithmetic

Zero of a Quadratic

 The Quadratic Formula

Standard Form of Quadratic Equation

Volume of Material and shape of object

Frustum of a Cone

Jokes

Math Adda                                                 लतीफ़े

 

एक रेलवे स्टेशन पर प्रेमी-प्रेमिका को मना रहा था, वो अंत में बोला- अगर तुमने मुझसे शादी करना नामंजूर किया तो मैं आने वाली गाड़ी से कटकर मर जाऊंगा!

प्रेमिका- मुझे सोचने दो जल्दी क्या है गाड़ी तो हर आधे घंटे में आती है।

 

संता का ससुर उसकी पिटाई कर रहा था।

बंता- आप इसे क्यों मार रहे हो?

ससुर - मैंने इसे अस्पताल से एसएमएस किया.. तुम बाप बन गए हो, इसने अपने सारे दोस्तों को फारवर्ड कर दिया।

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पिता (पुत्र से)- परीक्षा निकट है, तुमको रात-दिन पढ़ना चाहिए।

पुत्र (पिता से)- जी, रात को तो मैं सोता हूं।

पिता- जागा करो।

पुत्र- आप तो कहते हैं रात को उल्लू जागा करता है।

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डॉक्टर (मरीज से)- तुम एक दिन में कितनी बीड़ी पीते हो?

मरीज (डॉक्टर से)- जी, एक दिन में करीब बीस बीड़ी।

डॉक्टर- यदि मुझसे इलाज कराना है, तो इतनी सारी बीड़ी पीने से परहेज रखना होगा। आज से ही एक नियम बना लो कि सिर्फ भोजन के पश्चात ही एक बीड़ी पियोगे। मरीज ने डॉक्टर की बात सुनकर इलाज कराना शुरु कर दिया। कुछ महीने के बाद मरीज का स्वास्थ्य एकदम सुधर गया।

डॉक्टर- देखा, मेरे बताए परहेज से तुम्हें कितना स्वास्थ्य लाभ हुआ।

मरीज- लेकिन डॉक्टर साहब, एक दिन में बीस बार भोजन करना भी कोई सरल कार्य नहीं है।

 

पत्नी ने दुकान के आगे लगा बोर्ड पढ़ा।

बनारसी साड़ी 10 रु.

नायलोन साड़ी 8 रु.

कॉटन साड़ी 5 रु.

पत्नी (पति से)- जल्दी से मुझे 500 रु. दो मुझे पचास साड़ी खरीदनी है।

पति- ठीक से पढ़ ये प्रेस की दुकान है।

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एक रेलवे स्टेशन पर प्रेमी-प्रेमिका को मना रहा था, वो अंत में बोला- अगर तुमने मुझसे शादी करना नामंजूर किया तो मैं आने वाली गाड़ी से कटकर मर जाऊंगा!

प्रेमिका- मुझे सोचने दो जल्दी क्या है गाड़ी तो हर आधे घंटे में आती है।

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वकील (अपराधी से)- चाकू पर तुम्हारी उंगलियों के निशान पाए गये हैं। खून तुमने ही किया है।

अपराधी- वकील साहिब आप ऐसा कैसे कह सकते है कि ये निशान मेरी उंगलियों के हैं? क्योंकि खून करते समय मैंने तो दस्ताने पहन रखे थे!

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मालकिन (नौकरानी से)- क्या हुआ तुम तीन दिन से काम पर नही आई।

नौकरानी- मैंने फेसबुक पर अपडेट तो किया था की मैं गांव जा रही हूं, आपके पति ने कमेंट भी दिया था मिस यू।

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दुकान पर आया ग्राहक कभी कोई चीज उठाता, उसे देखता, फिर उसे रखकर दूसरी चीज उठा लेता। कुछ पूछताछ भी की उसने, लेकिन कुछ खरीदा नही काफी देर तक उसने ऐसा ही किया तो झुंझलाकर दुकानदार ने पूछा- श्रीमान जी, आखिर आपको चाहिए क्या ?

मौका ! ग्राहक का सपाट सा जवाब दिया।

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मालकिन (नौकरानी से)- क्यों महारानी जी आज आने में इतनी देर क्यों लगा दी?

नौकरानी- मालकिन मैं सीढि़यों से गिर गई थी।

मालकिन- तो क्या उठने में इतनी देर लगती है।

बच्चे

 

पिता (पुत्र से)- परीक्षा निकट है, तुमको रात-दिन पढ़ना चाहिए।

पुत्र (पिता से)- जी, रात को तो मैं सोता हूं।

पिता- जागा करो।

पुत्र- आप तो कहते हैं रात को उल्लू जागा करता है।

 

पिता (पुत्र से)- बेटे तुम इतने महान बनो कि तुम्हारा नाम दुनिया के चारों कोनों में फैले।

पुत्र (पिता से)- पापा महान तो बन जाऊंगा पर एक समस्या है।

पिता- वह क्या?

पुत्र- दुनिया गोल है, उसके चार कोने हो ही नहीं सकते फिर नाम कैसे चारों कोनों में फैलेगा।

 

पिता (पुत्र से)- क्या मैं तुम्हारी पढ़ने में हेल्प करूं?

पुत्र - नही पापा मैं बिना आपकी हेल्प के ही फेल होना चाहता हूं।

 

मेहमान खाना खाते हुए बोले- ये तुम्हारा कुत्ता मुझे बहुत देर से घूर रहा है??

चिंटू- अंकल आप जल्दी से खाना खा लो, वो अपनी प्लेट पहचान गया है।

 

अध्यापिका (गोलू से)- तुम खाना खाने से पहले भगवान की प्रार्थना किया करो।

गोलू- मुझे इसकी जरूरत नही क्योंकि मेरी मम्मी बहुत अच्छा खाना बनाती है

पति पत्नी और वो

 

पत्नी ने दुकान के आगे लगा बोर्ड पढ़ा।

बनारसी साड़ी 10 रु.

नायलोन साड़ी 8 रु.

कॉटन साड़ी 5 रु.

पत्नी (पति से)- जल्दी से मुझे 500 रु. दो मुझे पचास साड़ी खरीदनी है।

पति- ठीक से पढ़ ये प्रेस की दुकान है।

 

पत्नी (पति से)- तुम्हें मेरे रिश्तेदार पसंद नही!

पति (पत्नी से)- क्या बात कर रही हो! मुझे अपनी सास से अच्छी तुम्हारी सास लगती है?

 

पत्नी (पति से)- तुम्हें नही लगता की जरा सी समझदाराी से लाखों तलाक के मामले रोके जा सकते हैं।

पति (पत्नी से)- जरा सी समझदारी से शादियां भी तो रोकी जा सकती है।

 

मरते समय पति ने अपनी पत्नी को सब कुछ सच बताना चाहा। उसने कहा- मैं तुम्हें जीवन भर धोखा देता रहा। सच तो यह है कि दर्जनों औरतों से मेरे संबंध रहे हैं।

पत्नी (पति से)- मैं भी सच बताना चाहूंगी। तुम बीमारी से नही मर रहे मैंने तुम्हें धीरे-धीरे असर करने वाला जहर दिया है।

 

पति (पत्नी से)- अरे, सुनती हो, डाक्टरो ंका कहना है कि अधिक बोलने से उम्र काफी कम हो जाती है।

पत्नी (मुस्कुराकर)- अब तो तुमको विश्वास हो गया ना कि मेरी उम्र पैंतालीस से घटकर पच्चीस कैसे हो गयी