Classification Of Real Numbers

Real Number are classified into rational and irrational numbers.

RATIONAL NUMBERS:

A number which can be expressed in the form p/q where p and q are integers and q ≠ 0 is called a rational number.

For example, 4 is a rational number since 4 can be written as 4/1 where 4 and 1 are integers and the denominator 1≠0. Similarly, the numbers ¾, - 2/5 etc. are also rational numbers.

Recurring decimals are also rational numbers. A recurring decimal is a number in which one or more digits at the end of a number after the decimal point repeats endlessly ( For example, 0.333….., 0.111111…, 0.166666…., etc. are all recurring decimals). Any recurring decimal can be expressed as a fraction of the form p/q and hence it is a rational number. We will study in another section in this chapter the way to convert recurring decimals into fractions.

Between any two numbers, there can be infinite number of other rational numbers.

IRRATIONAL NUMBERS:

Numbers which are not rational but which can be represented by points on the number line are called irrational numbers. Examples for irrational numbers are √2,√3,∜5, ∛8etc.

Numbers like π, ε are also irrational numbers.

Between any two numbers, there are infinite numbers of irrational numbers.

Another way of looking at rational and irrational numbers is

Any terminating or recurring decimal is a rational number.

Any non – terminating non – recurring decimal is an irrational number.

RULES FOR DIVISIBILITY:

In a number of situations, we will need to find the factors of a given number. Some of the factors of a given number can, in a number of situations, be found very easily either by observation or by applying simple rules. We will look at some rules for divisibility of numbers.

Divisibility by 2: A number divisible by 2 will have an even number as its last digit ( For example 128, 246, 2346 etc)

Divisibility by 3: A number is divisible by 3 if the sum of its digits is a multiple of 3.

For example, take the number 9123, the sum of the digits is 9 + 1 + 2 + 3 = 15 which is a multiple of 3. Hence, the given number 9123 is divisible by 3. Similarly 342, 789 etc are all divisible by 3. If we take the number 74549, the sum of the digits is 29 which is not a multiple of 3. Hence the number 74549 is not divisible by 3.

Divisibility by 4: A number is divisible by 4 if the number formed with its last two digits is divisible by 4. For example, if we take the number 178564, the last two digits form 64 since the number 64 is divisible by 4 the number 178564 is divisible by 4.

If we take the number 476854, the last two digits form 54 which is not divisible by 4 and hence the number 476854 is not divisible by 4.

Divisibility by 5: A number is divisible by 5 if its last digit is 5 or zero ( eg. 15, 40 etc)

Divisibility by 6: A number is divisible by 6 if it is divisible both by 2 and 3 ( 18, 42, 96 etc)

Divisibility by 7: If the difference between the number of tens in the number and twice the units digit is divisible by 7, then the given number is divisible by 7. Otherwise, it is not divisible by 7.

Take the units digit of the number, double it and subtract this figure from the remaining part of the number. If the result so obtained is divisible by 7, then the original number is divisible by 7. If that result is not divisible by 7, then the number is not divisible by 7.

For example, let us take the number 595. The units digit is 5 and when it is doubled, we get 10. The remaining part of the number is 59. If 10(which is the units digit double) is subtracted form 59 we get 49. Since this result 49 is divisible by 7, the original number 595 is also divisible by 7.

Similarly, if we take 967,doubing the units digit give 14 which when subtracted from 96 gives a result of 82. Since 82 is not divisible by 7, the number 967 is not divisible by 7.

If we take a larger number, the same rule may have to be repeatedly applied till the result comes to a number which we can make out by observation. Whether it is divisible by 7. For example, take 456745. We will write down the figures in various steps as shown below.

Since 25 in the last step is not divisible by 7, the original number 456745 is not divisible by 7

Divisibility by 8: A number is divisible by 8, if the number formed by the last 3 digits of the number is divisible by 8.

For example, the number 3816 is divisible because the last three digits form the number 816. which is divisible by 8. Similarly, the numbers 14328, 18864 etc. are divisible by 8. If we take the number 48764, it is not divisible by 8 because the last three digits number 764 is not divisible by 8.

Divisibility by 9: A number is divisible by 9 if the sum of its digits is a multiple of 9.

For example, if we take the number 6318, the sum of the digits of this number is 6 + 3 + 1 + 8 which is 18. Since this sum 18 is a multiple of 9. Similarly, the numbers 729, 981, etc are divisible by 9. If we take the number 4763, the sum of the digits of this number is 20 which is not divisible by 9. Hence the number 4763 is not divisible by 9.

Divisibility by 10: A number divisible by 10 should be end in zero.

Divisibility by 11: A number is divisible by 11 if the sum of the alternate digits is the same or they differ by multiples of 11- that is, the difference between the sum of digits in odd places in the number and the sum of the digits in the even places in the number should be equal to zero or a multiple of 11.

For example, if we take the number 132, the sum of the digits in odd places is 1 + 2 = 3 and the sum of the digits in even places is 3. Since these two sums are equal, the given number is divisible by 11.

If we take the number 785345, the sum of the digits in odd places is 16 and the sum of the digits in even places is also 16. Since these two sums are equal, the given number is divisible by 11. Hence the number is divisible by 11.

If we take the number 89394811, the sum of the digits is odd places is 8 + 3 + 4 + 1, which is equal to 16. The sum of the digits in even places is 9 + 9 + 8 + 1, which is equal to 27. The difference between these two figures is 11 ( 27 –16), which is a multiple of 11. Hence the given number 89394811 is divisible by 11.

The number 74537 is not divisible by 11 because the sum of the digits in odd places is 19 and the sum of the digits in even places is 7 and the difference of these two figures is 12 is not a multiple of 11.

Divisibility by numbers like 12, 14, 15 can be checked out by taking factors of the number which are relatively prime and checking the divisibility of the given number by each of the factors. For example, a number is divisible by 12 if it is divisible both by 3 and 4.

The next number that is of interest to us from divisibility point of view is 19.

Divisibility by 19: If the sum of the number of tens in the number and twice the units digit is divisible by 19, then the given number is divisible by 19. Otherwise it is not.

Take the units digit of the number, double it and add this figure to the remaining part of the number. If the result so obtained is divisible by 19, then the original number is divisible by 19. If that result is not divisible by 19, then number is not divisible by 19.

For example let us take the number 665. The units digit is 5 and when it is doubled, we get 10. The remaining part of the number is 66. if 10 ( which is the units digit doubled) is added to 66, we get 76. Since this result 76 is divisible by 19, the number 969 is divisible by 19.

For example let us take the number 665. The units digit is 5 and when it is doubled, we get 10. The remaining part of the number is 66. if 10 ( which is the units digit doubled) is added to 66, we get 76. Since this result 76 is divisible by 19, the number 969 is divisible by 19.

If we take 873, double the units digit (2 x 3 = 6) added to the remaining part of the number (87),we get 93 which is not divisible by 19. Hence the original number 873 is not divisible by 19.

If we take a larger number, the same rule may have to be repeatedly applied till the result comes to a number which we can make out by observation whether it is divisible by 19. For example, take 456760. We will write down the figures in various steps as shown below.

Since 57 in the last step is divisible by 19, the original number 456760 is divisible by 19.

Let us take another example, the number 37895. Let us follow the above process step by step till we reach a manageable number

37895 Double the units digit 5 and add the 10 so obtained to 3789. We get 3799 Double the units digit 9 and add the 18 so obtained to 379. We get 397 Double the units digit 7 and add the 14 so obtained to 39. We get 53.

Since 53 is not divisible by 19, 37895 is not divisible by 19.