Find the square root of 3 by division method. Hence, show that 3 is an irrational number.

Square root of 3 by division method, 1 . 7 3 1 3 - 1 27 2 0 0 - 1 8 9 343 1 1 0 0 - 1 0 2 9 7 1 0 0 3 = 1 . 7320508075688772 . . . The square root of a number is the number that when multiplied by itself gives the original number as the product. A rational number is defined as a number that can be expressed in the form of a division of two integers, i.e. p/q, where q is not equal to 0 . 3 = 1 . 7320508075688772 . . . and it keeps extending. Since it does not terminate or repeat after the decimal point, 3 is an irrational number. Hence, 3 is an irrational number.

Find the square root of 3 by division method. Hence, show that 3 is an irrational number.

Square root of 3 by division method, 1 . 7 3 1 3 - 1 27 2 0 0 - 1 8 9 343 1 1 0 0 - 1 0 2 9 7 1 0 0 3 = 1 . 7320508075688772 . . . The square root of a number is the number that when multiplied by itself gives the original number as the product. A rational number is defined as a number that can be expressed in the form of a division of two integers, i.e. p/q, where q is not equal to 0 . 3 = 1 . 7320508075688772 . . . and it keeps extending. Since it does not terminate or repeat after the decimal point, 3 is an irrational number. Hence, 3 is an irrational number.

Write the decimal expansion of the number 46 100 and say what kind of decimal expansion number has.

Given, 46 100 We need to write the decimal expansion of the given number, and we have to say what kind of decimal expansion number has. By actual division, we get, 0 . 4 6 100 4 6 - 0 0 4 6 0 - 4 0 0 6 0 0 - 6 0 0 0 The remainder becomes zero. We know that, Rational numbers are present in the form of p q , where q ≠ 0 , On dividing p by q, if the remainder becomes zero after few steps, and then the decimal expansion terminates or ends after few steps. Such decimal expansion is called terminating decimal expansion. On dividing p by q, if the remainder never becomes zero and the sets of digits repeats periodically or in the same interval, then the decimal expansion is called non-terminating repeating decimal expansion. Hence, on dividing 46 100 we get exact value 0 . 46 and remainder becomes zero. So, we say that 46 100 is a terminating decimal expansion.

Write the decimal expansion of the number 5 1 8 and say what kind of decimal expansion number has.

Given, 5 1 8 in improper fraction will be 41 8 . We need to write the decimal expansion of the given number, and we have to say what kind of decimal expansion number has. By actual division, we get, 5 . 1 2 5 8 4 1 - 4 0 1 0 - 8 2 0 - 1 6 4 0 - 4 0 0 The remainder becomes zero. We know that, Rational numbers are present in the form of p q , where q ≠ 0 , On dividing p by q, if the remainder becomes zero after few steps, and then the decimal expansion terminates or ends after few steps. Such decimal expansion is called terminating decimal expansion. On dividing p by q, if the remainder never becomes zero and the sets of digits repeats periodically or in the same interval, then the decimal expansion is called non-terminating repeating decimal expansion. Hence, on dividing 41 8 we get exact value 5 . 125 and remainder becomes zero. So, we say that 5 1 8 is a terminating decimal expansion.

Write the decimal expansion of the number 2 9 and say what kind of decimal expansion number has.

Given, 2 9 We need to write the decimal expansion of the given number, and we have to say what kind of decimal expansion number has. By actual division, we get, 0 . 2 2 9 2 - 0 2 0 - 1 8 2 0 - 1 8 0 Here, the remainder never becomes zero and remainders repeat after a certain stage. We know that, Rational numbers are present in the form of p q , where q ≠ 0 , On dividing p by q, if the remainder becomes zero after few steps, and then the decimal expansion terminates or ends after few steps. Such decimal expansion is called terminating decimal expansion. On dividing p by q, if the remainder never becomes zero and the sets of digits repeats periodically or in the same interval, then the decimal expansion is called non-terminating repeating decimal expansion. Hence, on dividing 2 9 we get the repeated number 3 and remainder never becomes zero. So, we say that 2 9 is a non- terminating repeating decimal expansion.

Write the decimal expansion of the number 2 11 and say what kind of decimal expansion number has.

Given, 2 11 We need to write the decimal expansion of the given number, and we have to say what kind of decimal expansion number has. By actual division, we get, 0 . 1 8 11 2 - 0 2 0 - 1 1 9 0 - 8 8 2 Here, the remainder never becomes zero and remainders repeat after a certain stage. We know that, Rational numbers are present in the form of p q , where q ≠ 0 , On dividing p by q, if the remainder becomes zero after few steps, and then the decimal expansion terminates or ends after few steps. Such decimal expansion is called terminating decimal expansion. On dividing p by q, if the remainder never becomes zero and the sets of digits repeats periodically or in the same interval, then the decimal expansion is called non-terminating repeating decimal expansion. Hence, on dividing 2 11 we get the repeated numbers 18 again and again after the decimal point and remainder never becomes zero. So, we say that 2 11 is a non- terminating repeating decimal expansion.

Write the decimal expansion of the number 5 13 and say what kind of decimal expansion number has.

Given, 5 13 We need to write the decimal expansion of the given number, and we have to say what kind of decimal expansion number has. By actual division, we get, 0 . 3 8 4 6 13 5 - 0 5 0 - 3 9 1 1 0 - 1 0 4 0 0 6 0 - 5 2 8 0 - 7 8 2 5 13 = 0 . 384615 ¯ Here, the remainder never becomes zero and remainders repeat after a certain stage. We know that, Rational numbers are present in the form of p q , where q ≠ 0 , On dividing p by q, if the remainder becomes zero after few steps, and then the decimal expansion terminates or ends after few steps. Such decimal expansion is called terminating decimal expansion. On dividing p by q, if the remainder never becomes zero and the sets of digits repeats periodically or in the same interval, then the decimal expansion is called non-terminating repeating decimal expansion. Hence, on dividing 5 13 we see that the numbers 384615 after the decimal point keeps repeating and remainder never becomes zero. So, we say that 5 13 is a non- terminating repeating decimal expansion.

Write the decimal expansion of the number 229 400 and say what kind of decimal expansion number has.

Given, 229 400 We need to write the decimal expansion of the given number, and we have to say what kind of decimal expansion number has. By actual division, we get, 0 . 5 7 2 5 400 2 2 9 - 0 0 0 2 2 9 0 - 2 0 0 0 2 9 0 0 - 2 8 0 0 1 0 0 0 - 8 0 0 2 0 0 0 - 2 0 0 0 0 The remainder becomes zero. We know that, Rational numbers are present in the form of p q , where q ≠ 0 , On dividing p by q, if the remainder becomes zero after few steps, and then the decimal expansion terminates or ends after few steps. Such decimal expansion is called terminating decimal expansion. On dividing p by q, if the remainder never becomes zero and the sets of digits repeats periodically or in the same interval, then the decimal expansion is called non-terminating repeating decimal expansion. Hence, on dividing 229 400 we get exact value 0 . 5725 and remainder becomes zero. So, we say that 229 400 is a terminating decimal expansion.

Without actual division, find the rational number 7 16 is terminating decimal or not.

First, we note that the given rational number 7 16 is in its lowest terms. The denominator of the given rational number 16 = 2 4 × 5 0 Therefore, 16 has no prime factors other than 2 . So, 7 16 is a terminating decimal. Hence, the given rational number i.e. 7 16 is a terminating decimal.

Without actual division, find the rational number 21 80 is terminating decimal or not.

First, we note that the given rational number 21 80 is in its lowest terms. The denominator of the given rational number 80 = 16 × 5 = 2 4 × 5 1 , which is of the form 2 m × 5 n , where m and n are whole numbers. So, 21 80 is a terminating decimal. Hence, the given rational number i.e. 21 80 is a terminating decimal.