Y10M27012015NumbArith1

1. NUMBER SYSTEM AND BASIC
ARITHMETIC
REAL NUMBERS

2. Real Numbers
Real Numbers
Real Numbers include rational numbers and irrational numbers.
Rational Numbers
Numbers, which can be written as p/q (where p and q are integers and q is not equal to
zero), are called rational numbers.
Rational Numbers: {p/q: p and q are integers, q ≠ 0}
For example: If p=3 and q=4 then p/q = ¾ is a rational number.
Rational numbers can also be represented in decimal form with zero digits, terminating or
recurring digits after decimal.
So ¾ is a rational number. 5 is also a rational number because it can be written as 5/1.
Rational Number includes all integers and all fractions.

3. Real Numbers
Irrational Numbers
There are numbers, which cannot be written in form of p/q, where p and q are integers.
For example √2, √3, π etc. In decimal form, irrational number can be expressed as
numbers with non-recurring digits after the decimal.
For example √2 = 1.4142135623730950…. and so on,
but these digits after decimal would never repeat in pattern and never terminate (opposite
to rational numbers).
Examples:

4. Real Numbers
Theorems
Rational Number + Rational Number = Rational Number
Rational Number − Rational Number = Rational Number
Rational Number × Rational Number = Rational Number
Rational Number ÷ Rational Number = Rational Number
Rational Number + Irrational Number = Irrational Number
Rational Number − Irrational Number = Irrational Number
Rational Number × Irrational Number = Irrational Number
Rational Number ÷ Irrational Number = Irrational Number
Irrational Number + Irrational Number = Rational or Irrational Number
Irrational Number − Irrational Number = Rational or Irrational Number
Irrational Number × Irrational Number = Rational or Irrational Number
Irrational Number ÷ Irrational Number = Rational or Irrational Number

5. Real Numbers
EXAMPLE 1: Express each of the following decimals in the form
p
q
:
(i) 0. 35 (ii) 0. 585

6. Real Numbers
EXAMPLE 2: Express the following decimals in the form
p
q
.
(i) 0.32 (ii) 0.123

7. Real Numbers
EXAMPLE 3: Prove that 2 is not a rational number.

8. Real Numbers
EXAMPLE 4: Prove that 3 − 2 is an irrational number.

9. NUMBER SYSTEM AND BASIC
ARITHMETIC
APPROXIMATION

10. Approximation
Rounding Using Decimal Places
The digit is rounded up if the next digit is greater than or equal to 5. The digit is rounded
down if the next digit is less than 5.
Example 1: Round 87.3458 to (i) three decimal places (ii) two decimal places (iii) one
decimal place.
Example 2: Round 28.6738 to (i) three decimal places (ii) two decimal places (iii) one
decimal place.

11. Approximation
Rounding Using Significant Figures
When rounding to significant figures
the first significant figure in a number is the first non-zero digit.
– The first significant figure in 8 739 000 is 8
– The first significant figure in 0.000 573 is 5
Zeros at the end of a whole number or at the beginning of a decimal are not significant.
They are necessary place holders.
– The bold zeros in 6 240 000 and 0.007 032 are not significant.
Zeros between non-zero digits or zeros at the end of a decimal are significant.
– The bold zeros in 5 603 000 and 0.030 50 are significant.

12. Approximation
Example 1: Round each of the following to the number of significant figures indicated in
the brackets.
(i) 4.715 (1) (ii) 27.99 (3) (iii) 75 500 (2)
Example 2: Round each of the following to the number of significant figures indicated in
the brackets.
(i) 0.06737 (1) (ii) 0.405 (2) (iii) 0.7555 (2)

13. NUMBER SYSTEM AND BASIC
ARITHMETIC
PERCENTAGE

14. Percentage
Review of Percentage
PERCENT The word percent is an abbreviation of the Latin phrase 'per centum’ which
means per hundred or hundredths.
Review of Percentage
PERCENT The word percent is an abbreviation of the Latin phrase 'per centum’ which
means per hundred or hundredths.
PERCENT AS A FRACTION We have, 35% = 35 hundredths =
35
100
.
PERCENT AS A RATIO A percent can be expressed as a ratio with its second term 100
and first term equal to the given percent.
For example, 8% =
8
100
= 8: 100; 36%
36
100
=
9
25
= 9: 25
PERCENT IN DECIMAL FORM To convert a given percent in decimal form, we express it
as a fraction with denominator as 100 and then the fraction is written in decimal form.
For example, 65% =
65
100
= 0.65,7.4% =
7.4
100
= 0.074

15. Percentage
Example 1: A shopkeeper deposits $150 per month in his post office Savings Bank
account. If this is his 15% of his monthly income, find his monthly income.

16. Percentage
Example 2: 9. Ryan requires 40% to pass. If he gets 185 marks, falls short by 15 marks,
what were the maximum marks he could have got?

17. Percentage
Example 3: If A : B = 4 : 6 and B : C = 3 : 4, then by what percentage is C more than A.

18. Percentage
Example 4: Find the percentage increase in the area of a rectangle if the length is
increased by 20% and breadth is increase by 25%.

19. Percentage
Example 5: The price of a refrigerator, P is 5 times that of a cooler Q. If the price of P
increases by 20% and that of Q decreases by 40 %, then what is the change in the total
price of the refrigerator and cooler put together?