Math's: Number System: Rational Numbers Between Two Rational Numbers

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Rational Numbers between Two Rational Numbers

  • To find one rational number between two rational numbers, the denominators of the two given rational numbers are made equal and then their sum is divided by . The average computed lies in the middle of the two given numbers.

  • To find two or more rational numbers between two rational numbers, the denominators of the two given rational numbers are made equal and then the rational numbers having the common denominator and the numerators lying between the numerators of the given two rational numbers lie between them.

1. Find a rational number between and .

Solution:

2. Find three rational numbers between and .

Solution:

Three rational numbers lying between and are and

Irrational Numbers

A rational number the decimal expansion of which is neither terminating nor repeating is an irrational number.

Irrational Numbers

Irrational Numbers

Ex- 0.12345…., 0.1010010001…

3. State which of the following numbers are irrational?

(i) (ii) (iii)

Solution: (i) is an irrational number because its value in decimal number which is equal to 3.145926….and more is neither terminating nor repeating.

(ii) is not an irrational number. It’s a rational number because it is a repeating decimal.

(iii) It’s a non-terminating decimal. So, it is an irrational number.

Finding an Irrational Number between Two Given Numbers

4. Find an irrational number between 5 and 7

Solution: Take the number

is an irrational number since it is the square root of a non-perfect square and it lies between 5 and 7?

Rounding off Decimal Numbers

  • Rounding off a decimal number is approximating an expanded decimal number and making it simpler to understand.

  • For rounding off a decimal number up to nth number of decimal places, we first look at the digit at decimal place.

  • If it is less than 5, the decimal number is stated as it is up to nth decimal places.

  • If it is equal to or more than 5, 1 is added to the digit at nth decimal place.

5. Round off 0.29999 up to two decimal places.

Solution: The digit at third decimal place = 9 (which is more than 5). So, we will add 1 to the 2nd digit and get the required number as 0.30 or 0.3

6. Round off 37.28929 up to three decimal places.

Solution: The digit at fourth decimal place = 2 (which is less than 5). So, we will just state the number as 37.289 as correct to three decimal places.

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