# 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

- Write all possible numbers that can be formed by using the digits 0, 4, 7, using each digit at most once.
- In two-digit numbers, how many times does the digit 5 occur in (i) the tens place (ii) the unit place?
- Write all the number between 4 to 75 using the digits 6, 0, 7. The repetition of digits is (i) not allowed (ii) allowed.
- Write the greatest 6-digit number formed by three different digits.
- Write the smallest 8-digit number formed by four different digits.
- Write the smallest 3-digit number which does not change if the digits are written in the reverse order.
- Find the sum of 4-digit greatest number and the 5 digit smallest number, each number having three different digits.
- What is the maximum and minimum number of digits in the sum on adding any two five digit numbers?
- In the question below, some digits have been replaced by letters.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 - 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 - Divide 46087 by 356 and check the result by division algorithm.
- 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) - Find the largest 4-digit number which is exactly divisible by 357.
- Find the smallest 5-digit number which is exactly divisible by 279.

## Answers

**1.**0, 4, 7, 40, 70, 47, 74, 407, 704, 470, 740

**2.**(i) 10 (ii) 9.

**3.**(i) 6, 7, 60, 70, 67 (ii) 6, 7, 60, 66, 70, 67

**4.**999987.

**5.**10000023.

**6.**101

**7.**19989

**8.**6; 5

**9**a = 2, b = 4; a= 4, b = 9

**10.**

16 | 5 | 3 | 10 |

13 | 2 | 8 | 11 |

4 | 15 | 9 | 6 |

1 | 12 | 14 | 7 |

**11.**Quotient = 129, remainder = 163

**12.**(i) False (ii) True (iii) False

**13.**9996

**14.**10044