1. Natural numbers, whole numbers, integer rational numbers, irrational numbers, real numbers.
1. Natural numbers, whole numbers, integer rational numbers, irrational numbers, real numbers. |
Learn and Remember :
1. Natural numbers : The group of numbers 1, 2, 3, 4, ... is called the set of counting numbers or natural numbers.
2 Whole numbers : The group of numbers 0, 1, 2, 3, 4, ... is called the set of whole numbers.
3. Integers : ..., -3, -2, – 1,0, 1, 2, 3, ... . The numbers in this group are called integers.
4. Rational numbers : If p is any integer and q is any non-zero integer, then -- is called a rational number.
5 ⁄ 7 , - 11 ⁄ 8, 15 ⁄ 17 are rational numbers.
5. The decimal form of a rational number :
(a) Terminating decimal fraction :
Let us find the decimal form of the rational number
In this example, the process of division comes to 45 ⁄4
an end. So 11.25 is the terminating decimal
fraction of the rational number 45 ⁄4
45 ÷4 = 11.45
(b) Non-terminating recurring decimal fraction :
Let us find the decimal form of the rational number 20 ⁄3
In this example, the process of division is unending. The digit 6 recurs.
The recurring digit (digits) is (are) marked with a dot (.) or a line above it (them).
20 ⁄3 =6.66... is written as 20 ⁄3 = 6.6
0.3737... is written as 0.37 and
5.72147214... is written as 5.7214.
6.Non-terminating recurring decimal forms and rational numbers :
Every number in the non-terminating recurring decimal form is a rational number.
7. Irrational numbers : The numbers whose decimal form is non-terminating and non-recurring are called irrational numbers.
e.g. √2=1.4142135623... i.e. the decimal form of √2 is non-terminating and non recurring. So √2 is an irrational number.
8. Real numbers : The collection of rational and irrational numbers together form real numbers.
Every rational number is a real number.
• Every irrational number is also a real number.
e.g. 2 ⁄7, -10 ⁄ 3 ,5.83, √2, – 11.34 are real numbers.
9. The place value of any digit in a number is determined by its position in the number as units, tens, hundreds, etc. e.g. the place value of 3 in the number 937865 is 30,000.
10. If the digits in the units and tens places of any number are interchanged, the difference between the number obtained and the original number is in multiples of nine. Consider the number 4597. If the digits 7 and 9 are interchanged, the difference between 4597 and 4579 is 18, which is a multiple of 9. It is also 9 times of the difference between the two digits. i.e. 9 x (9 – 7) =9 x 2 = 18.
11. (i) If the same digit occurs in the tens place and the units place of a number, then the difference between their place values is 9 times that digit. e.g. if the digit 5 is in the tens and units places, then the difference between their place values is 50 – 5 = 45. This is 9 times 5.
(ii) If the same digit occurs in the hundreds place and the tens place of a number, then the difference between their place values is 90 times that digit. e.g. if the digit 6 is in the hundreds and tens places, then the difference between their place values is 600 – 60 =540. This is 90 times 6.
(iii) If the same digit occurs in the hundreds place and the units place of a number, then the difference between their place values is 99 times that digit. e.g. in the number 5474, the difference between the place values of 4 is 400 – 4= 396 = 99 times 4.
(iv) If the same digit occurs in the thousands place and the tens place of a number, then the difference between their place values is 990 times that digit. e.g. in the number 75659, the difference between the place values of 5 is 5000 – 50 = 4950 = 990 times 5.
(v) If the same digit occurs in the thousands place and the units place of a number, then the difference between their place values is 999 times that digit. e.g. in the number 46326, the difference between the place values of 6 is 6000 - 6 =5994 = 999 times 6.
12. Given a number of certain digits, the greatest number is formed by using all its digits as '9'.
e.g. the greatest five-digit number is 99999.
13. Given a number of certain digits, the smallest number is formed by using the first digit as 'l' and the following digits as 'O'.
e.g. the smallest five-digit number is 10000.
14. Given certain digits, the greatest number is formed by arranging the digits in a descending order (Each digit used only once).
e.g. the greatest number formed by the digits 7, 8, 5, 9, 2 is 98752.
15. Given certain digits, the smallest number is formed by arranging the digits in an ascending order (Each digit used only once). However, if one of the digits is 'O' it should be placed second from left.
e.g. the smallest number formed by the digits 7, 8, 5, 9, 2 is 25789.
However, the smallest number formed by the digits 9, 5, 0, 2, 7 is 20579.
16.To form the greatest or the smallest four, five or six-digit numbers using the given 2, 3 or 4 digits only (each digit is to be used at least once) :
(i) While writing this type of greatest number, arrange the given digits in a descending order and write the greatest digit for maximum times.
e.g. the greatest six-digit number formed by using the digits 3, 5, 0 is 555530.
(ii) While writing this type of smallest number, arrange the given digits in an ascending order and write the smallest digit for maximum times.
e.g. the smallest five-digit number formed by using the digits 9, 2, 3 is 22239.
If one of the digits is ‘O', it should be placed second from the left and taken for maximum times.
e.g. the smallest six-digit number by using the digits 8,0,2 is 200008.
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