Number Systems

• All counting numbers together with 0 are called whole numbers.
• 0 is the smallest whole number and there is no largest whole number.
• Our number system is based on counting in tens i.e. it has base 10. Every whole number can be written by using the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called digits.
• The place value of a (non-zero) digit depends upon the place it occupies in the number; the place value of the digit 0 is always 0 regardless of the place it occupies in the number.
• The face value of a digit in a number is the digit itself, regardless of the place it occupies in the number.
• Addition properties of whole numbers
  Closure property: If a and b are any whole numbers then a +b is also a whole number.
  Commutative property: If a and b are any whole numbers then a+b= b+a.
  Associative law: If a, b, c are any whole numbers then (a+b)+c = a+(b+c).
• Multiplication properties of whole numbers
  Closure property: If a and b are any whole numbers then a × b is also a whole number.
  Commutative property: If a and b are any whole numbers then a × b = b × a.
  Associative law: If a, b, c are any whole numbers then (a × b) × c = a × (b × c).
  Distributive law: If a, b, c are any whole numbers then a × (b+c)=a × b + a × c.
• Division algorithm
  If a is any whole number and b is another smaller non-zero whole number then there exist unique whole numbers q and r such that
  a = b × q + r where 0 r < b.

Exercise

1. Write all possible numbers that can be formed by using the digits 0, 4, 7, using each digit at most once.
2. In two-digit numbers, how many times does the digit 5 occur in (i) the tens place (ii) the unit place?
3. Write all the number between 4 to 75 using the digits 6, 0, 7. The repetition of digits is (i) not allowed (ii) allowed.
4. Write the greatest 6-digit number formed by three different digits.
5. Write the smallest 8-digit number formed by four different digits.
6. Write the smallest 3-digit number which does not change if the digits are written in the reverse order.
7. Find the sum of 4-digit greatest number and the 5 digit smallest number, each number having three different digits.
8. What is the maximum and minimum number of digits in the sum on adding any two five digit numbers?
9. In the question below, some digits have been replaced by letters. Determine what digits the letters stand for. There may be more than one solution.
                                      1 a b
                                   +  a b
                                      1 b 8
10. The sum of the numbers in each row, each column and both diagonals of the magic number square below is 34. The numbers 1, 2, 3, 4, 5 and 6 have been left out. Write them in their correct places.
    

    16			10
    13		8	11
    	15	9
    	12	14	7

11. Divide 46087 by 356 and check the result by division algorithm.
12. State whether the following statements are true or false:
    (i) 59 -(25 -17) = (59 -25) -17
    (ii) 31 -(12 -5) = (31 -12) +5
    (iii) 639 -205 -157 = 639 -(205 -157)
13. Find the largest 4-digit number which is exactly divisible by 357.
14. Find the smallest 5-digit number which is exactly divisible by 279.

Answers