# Activities-Real numbers

From Karnataka Open Educational Resources

# Problem 1

- For every integer prove that x(x+1) is an even integer (Problem related to mathematical proofs in Chapter 1)

**Approaches to solutions**

## The concepts that a pupil must know are

- What is an integer?
- What is an even integer?
- What is an odd integer?
- X and ( x+1) are consecutive integers and x(x+1) is the representation of the product
- Pupil should have the concept of distributive property of integers
- The pupil must have an opportunity for an arguement that the proof is true even of negative integers
- Pupil must have a sound understanding of Euclid's lemma
- The difference between mathematical proofs and Verification/Justification-the scope of mathematical proofs is beyond verification-Higher order skill in problem solving

## Method -1: Solution by cases

Proposition: x is an integer. i.e., x=m, where m is an integer

Conclusion : x(x+1) is an even integer. i.e., x(x+1) =2K

Pupil can solve this in several ways viz., proofs by cases.

### Case -1: x is an even integer

Then x=2k+0 = 2K, by Euclid's lemma

Discussion on Euclid's lemma: What is division? Discuss on the process of division rather than on the procedure of division. Consider the example . Is it (12-5)=7 then (7-5)=2

Discussion on Euclid's lemma: What is division? Discuss on the process of division rather than on the procedure of division. Consider the example . Is it (12-5)=7 then (7-5)=2
**Abstract thinking:**

b) a (q

-bq

.........

(a - bq)

.........