Download Number Systems

Number Systems
Main Concepts and Results:
Rational numbers
Irrational numbers
Locating irrational numbers on the number line
Real numbers and their decimal expansions
Representing real numbers on the number line
Operations on real numbers
Rationalisation of denominator
Laws of exponents for real numbers
• A number is called a rational number, if it can be written in the form
q are integers and q ≠ 0.
• A number which cannot be expressed in the form
q ≠ 0) is called an irrational number.
, where p and
, (where p and q are integers and
• All rational numbers and all irrational numbers together make the collection of real
numbers.
• Decimal expansion of a rational number is either terminating or non-terminating
recurring, while the decimal expansion of an irrational number is non-terminating nonrecurring.
• If r is a rational number and s is an irrational number, then r + s and r- s are irrationals.
Further, if r is a non-zero rational, then rs and r/ s are irrationals.
• For positive real numbers a and b :
• If p and q are rational numbers and a is a positive real number, then
Exercise:
1)
2)
3)
4)
5)
6)
What are natural numbers?
What are whole numbers?
What are integers?
What do you mean by the word “Zahlen” ?
What are rational numbers?
Are the following statements true or false? Give reasons for your answers.
(i)
Every whole number is a natural number.
(ii)
Every natural number is a whole number.
(iii)
Every integer is a whole number
(iv)
Every integer is a rational number.
(v)
Every rational number is an integer.
(vi)
Every rational number is a whole number.
(vii) There are infinitely many rational numbers between any two given rational
numbers.
(viii) Every irrational number is a real number
(ix)
Every point on the number line is of the form √ , where m is a natural
number.
(x)
Every real number is an irrational number.
7) Find five rational numbers between 1 and 2.
8) Find six rational numbers between 3 and 4.
9) Is zero a rational number? Can you write it in the form p/q , where p and q are
integers and q ≠ 0?
10) What are irrational numbers?
11) Every real number is represented by a unique point on the number line. Also, every
point on the number line represents a unique real number. True or false?
12) Locate √2 on the number line.
13) Locate √3 on the number line.
14) Are the square roots of all positive integers irrational? If not, give an example of the
square root of a number that is a rational number.
15) Show how √5 can be represented on the number line.
16) Find the decimal expansions of
, , .
17) Show that 3.142678 is a rational number. In other words, express 3.142678 in the
form , where p and q are integers and q
0.
18) Show that 0.3333... = 0. 3 can be expressed in the form
integers and q
0.
integers and q
0.
integers and q
0.
, where p and q are
19) Show that 1.272727... = 1.27 can be expressed in the form
, where p and q are
20) Show that 0.2353535... = 0 .235 be expressed in the form
, where p and q are
21) Find an irrational number between
and .
22) Write the following in decimal form and say what kind of decimal expansion each
has :
23) You know that = 0.142857 . Can you predict what the decimal expansions of
, , , , are, without actually doing the long division? If so, how?
24) Express the following in the form , where p and q are integers and q
0.
25) Express 0.99999 .... in the form . Are you surprised by your answer? With your
teacher and classmates discuss why the answer makes sense.
26) What can the maximum number of digits be in the repeating block of digits in the
decimal expansion of
? Perform the division to check your answer.
27) Look at several examples of rational numbers in the form p/q q ≠ 0), where p and q
are integers with no common factors other than 1 and having terminating decimal
representations (expansions). Can you guess what property q must satisfy?
28) Write three numbers whose decimal expansions are non-terminating non-recurring.
29) Find three different irrational numbers between the rational numbers and
30) Classify the following numbers as rational or irrational :
.
31) Visualize the representation of 5.37 on the number line upto 5 decimal places, that
is, up to 5.37777.
32) Visualise 3.765 on the number line, using successive magnification.
33) Visualise 4 .26 on the number line, up to 4 decimal places.
34) Check whether below numbers are irrational or not.
35) Add 2√2 + 5 √3 and √2 - 3√3 .
36) Multiply 6√5 by 2√5 .
37) Simplify the following expressions:
38) Rationalise the denominator of 1/√2 .
39) Rationalise the denominator of
40) Rationalise the denominator of
41) Rationalise the denominator of
.
√
√
√
√
.
.
42) Classify the following numbers as rational or irrational:
43) Simplify each of the following expressions:
44) Recall, π is defined as the ratio of the circumference (say c ) of a circle to its diameter
(say d). That is, π = c/d⋅ This seems to contradict the fact that π is irrational. How will
you resolve this contradiction?
45) Represent √9.3 on the number line.
46) Rationalise the denominators of the following:
47) Simplify:
48) Find:
49) Find:
50) Simplify :
Multiple Choice Questions:
51) Which of the following is not equal to
52) Every rational number is
(A) a natural number
(B) an integer
(C) a real number
(D) a whole number
53) Between two rational numbers
(A) there is no rational number
(B) there is exactly one rational number
(C) there are infinitely many rational numbers
(D) there are only rational numbers and no irrational numbers
54) Decimal representation of a rational number cannot be
(A) terminating
(B) non-terminating
(C) non-terminating repeating
(D) non-terminating non-repeating
55) The product of any two irrational numbers is
(A) always an irrational number
(B) always a rational number
(C) always an integer
(D) sometimes rational, sometimes irrational
56) .The decimal expansion of the number √2 is
(A) a finite decimal
(B) 1.41421
(C) non-terminating recurring
(D) non-terminating non-recurring
57) Which of the following is irrational?
58) Which of the following is irrational
59) A rational number between √2 and √3 is
60) The value of 1.999... in the form
, where p and q are integers and q ≠ 0 , is
61) 2√ 3 + √3 is equal to
62) √10 x √15 is equal to
63) The number obtained on rationalising the denominator of
64)
√
√
is
is equal to
65) After rationalising the denominator of
(A) 13
66) The value of
√
√
(B) 19
√
√
67) If√ 2 = 1.4142, then
68)
√
(A) 2.4142
√2 equals to
is equal to
√
√
(C) 5
is equal to
(B) 5.8282
69) The product √2 √2 √32 equals
√
√
, we get the denominator as
(C) 0.4142
(D) 35
(D) 0.1718
70) Value of
81
is
71) Value of (256)0.16 × (256)0.09 is
(A) 4
(B) 16
(C) 64
72) Which of the following is equal to x?
(D) 256.25