Real Numbers

 Real Numbers 

Consider the following pairs of integers:
17, 6;  5, 12;  20, 4

We can write the following relations for each such pair:

17 = 6 × 2 + 5 

5 = 12 × 0 + 5 

20 = 4 × 5 + 0 

That is, for each pair of positive integers `a` and `b`, we have found whole numbers `q` and `r`, satisfying the relation: 

`a = bq + r,    0 \leq r < b`

dividend = divisor × quotient + remainder 

Note that `q` or `r` can also be zero.

You may have also realised that this is nothing but a restatement of the long division process you have been doing all these years and that the integers q and r are called the quotient and remainder, respectively.

Euclid’s Division Lemma

Given positive integers `a` and `b`, there exist unique integers `q` and `r` satisfying `a = bq + r, 0 \leq r < b`.

This result was perhaps known for a long time but was first recorded in Book VII of Euclid’s Elements. Euclid’s division algorithm is based on this lemma.

➠  An algorithm is a series of well-defined steps which give a procedure for solving a type of problem.

➠ A lemma is a proven statement used for proving another statement.

➠ Euclid’s division algorithm is used to compute the Highest Common Factor (HCF) of two given positive integers.

Let us state Euclid’s division algorithm clearly.

To obtain the HCF of two positive integers, say `c` and `d`, with `c > d`, follow the steps below:

➤ Apply Euclid’s division lemma, to `c` and `d`. So, we find whole numbers, q and r such that `c = dq + r,  0 \le r < d`.

➤ If `r = 0`, `d` is the HCF of `c` and `d`. If `r \ne 0`, apply the division lemma to `d` and `r`.

 ➤ Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

This algorithm works because HCF `(c, d)` = HCF `(d, r)` where the symbol HCF `(c, d)` denotes the HCF of `c` and `d`, etc.

Example

Suppose we need to find the HCF of the integers 455 and 42.

Since 455 > 42, apply Euclid's lemma 

455 = 42 × 10 + 35

Since remainder 35 `\ne` 0, apply Euclid's lemma to divisor 42 and remainder 35.

42 = 35 × 1 + 7

Since remainder 7 `\ne` 0, apply Euclid's lemma to divisor 35 and remainder 7.

35 = 7 × 5 + 0

Now, remainder is 0, and divisor is 7

∴ HCF (455, 42) = 7 

➤ Let `x` be a rational number whose decimal expansion terminates. Then we can express `x` in the form `p/q` , where `p` and `q` are coprime, and the prime factorisation of `q` is of the form `2^n5^m`, where `n, m` are non-negative integers.

➤ Let `x = p/q` be a rational number, such that the prime factorisation of `q` is of the form `2^n5^m`, where `n, m` are non-negative integers. Then `x` has a decimal expansion which terminates.

➤ Let `x = p/q` be a rational number, such that the prime factorisation of `q` is not of the form `2^n5^m`, where `n, m` are non-negative integers. Then `x` has a decimal expansion which is non-terminating repeating (recurring).

References

1. NCERT Mathematics Class X
2. Mathematics (R.S. Aggarwal)
3. Mathematics (R.D. Sharma)