NUMBER SYSTEM AND BASIC ARITHMETIC

REAL NUMBERS

Real Numbers

Real NumbersReal Numbers include rational numbers and irrational numbers.

Rational NumbersNumbers, 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.

Real Numbers

Irrational NumbersThere 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:

Real Numbers

Theorems

Rational Number Rational Number = Rational NumberRational Number Rational Number = Rational NumberRational Number Rational Number = Rational NumberRational Number Rational Number = Rational Number

Rational Number Irrational Number = Irrational NumberRational Number Irrational Number = Irrational NumberRational Number Irrational Number = Irrational NumberRational Number Irrational Number = Irrational Number

Irrational Number Irrational Number = Rational or Irrational NumberIrrational Number Irrational Number = Rational or Irrational NumberIrrational Number Irrational Number = Rational or Irrational Number Irrational Number Irrational Number = Rational or Irrational Number

Real Numbers

EXAMPLE 1: Express each of the following decimals in the form :(i) (ii)

Real Numbers

EXAMPLE 2: Express the following decimals in the form. (i) (ii)

Real Numbers

EXAMPLE 3: Prove that is not a rational number.

Real Numbers

EXAMPLE 4: Prove that is an irrational number.

NUMBER SYSTEM AND BASIC ARITHMETIC

APPROXIMATION

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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.

Approximation

Rounding Using Significant Figures

When rounding to significant figuresthe 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.

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)

NUMBER SYSTEM AND BASIC ARITHMETIC

PERCENTAGE

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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 .

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% ; 36%

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,

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.

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?

Percentage

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

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%.

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?