Real Numbers (10th)

Euclid’s division algorithm, as the name suggests, has to do with divisibility of integers. Stated simply, it says any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is than b.

Many of you probably recognise this as the usual long division process. Although this result is quite easy to state and understand, it has many applications related to the divisibility properties of integers. We touch upon a few of them, and use it mainly to compute the HCF of two positive integers.

The Fundamental Theorem of Arithmetic, on the other hand, has to do something with multiplication of positive integers. You already know that every composite number can be expressed as a product of primes in a unique way — this important fact is the Fundamental Theorem of Arithmetic. Again, while it is a result that is easy to state and understand, it has some very deep and significant applications in the field of mathematics.

We use the Fundamental Theorem of Arithmetic for two main applications. First, we use it to prove the irrationality of many of the numbers, such as 2 , 3 and 5 . Second, we apply this theorem to explore when exactly the decimal expansion of a rational number, say pq(q≠0), is terminating and when it is non- terminating repeating. We do so by looking at the prime factorisation of the denominator q of pq . You will see that the prime factorisation of q will completely reveal the nature of the decimal expansion of pq.

For example, Consider the numbers 4n, where n is a natural number. Check whether there is any value of n for which 4n ends with the digit zero.

Solution : If the number 4n, for any n, were to end with the digit zero, then it would be divisible by the prime number .

That is, the prime factorisation of 4n would contain the prime 5. This is not possible because 4n=22n; so the only prime in the factorisation of 4n is 2. So, the uniqueness of the Fundamental Theorem of Arithmetic guarantees that there are no other primes in the factorisation of 4n . So, there is no natural number n for which 4n ends with the digit zero.

So we took the uniqueness part of the theorem and applied it to quickly deduct the point that 4n will never end with zero for all n belongs to natural numbers.

HCF and LCM

We have already learnt how to find HCF and LCM earlier. The prime factorization method actually uses the Fundamental Theorem of Arithmetic to gets its result. Let us quick;y revisit with an example.

Find the HCF and LCM of 6 and 20.

Solution:

Prime factors of 6=*.

6= 21·3

Prime factors of 20=***

20= 22·5

So HCF(6,20)= and LCM(6,20)=

HCF(6, 20) = 21 = Product of the smallest power of each common prime factor in the numbers.

LCM (6, 20) = 22×31×51 = Product of the greatest power of each prime factor,involved in the numbers.

In the example above we can see that HCF(6, 20) × LCM(6, 20) = 2*60=120.

But 6*20 is also equal to 120.

In fact, it can be proven that for any a,b, HCF(a,b)LCM(a,b)=ab. This will easily enable us to find the HCF or the LCM if the other is given.

Summary

We have studied earlier that rational numbers have either a terminating decimal expansion or a non-terminating repeating decimal expansion. In this section, we are going to consider a rational number, say pq, q ≠ 0, and explore exactly when the decimal expansion of pq is terminating and when it is non-terminating repeating(or recurring).

To come up with some hypothesis we first start with some concrete examples. Let us take some decimals and convert them into pq form and see if we can identify some patterns.

0.375=3751000=375103

0.0875=87510000=875104

23.3456=23345610000=233456104

We can see a clear pattern. All the denominators are powers of .

If we want to rewrite 10 as a product of prime factors we can use the primes and .

Even though, we have worked only with a few examples, you can see that any real number which has a decimal expansion that terminates can be expressed as a rational number whose denominator is a power of 10. Also the only prime factors of 10 are 2 and 5. So, cancelling out the common factors between the numerator and the denominator, we find that this real number is a rational number of the form ,pq where the prime factorisation of q is of the form 2n5m, and n, m are some non-negative integers.

Formally we can define the theorem as:

Now, if we consider any non terminating and recurring rational nu,bers like 1/7, 2/3, 7/11 etc we see that the denominators are not of the form 2n·5m.

So we can define the theorem as

With the help of these theorems we can quickly find out if a rational number of the form p/q is terminating or non terminating. You can do so using the following steps.

1. Find out if q can be represented as a power of the prime factors 2 or 5.
2. If yes, its terminating. If not, it's non terminating.

For example, consider 3/8. Here 8 can be represented as 23. So it is terminating. We can chek it easily as below

3/8=323=3·5323·53=375103=0.375.