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Numbers and Numerals: A Weekend Exploration
Part 3. Monday
Rhea and Dhruv continued to discuss numbers and numerals as they walked to school. Both of them had made some progress in their thinking after Sunday’s lunch and a long nap.
R: I can’t wait for Math class today! I am so excited to share with Ronak what we discovered!
D: But we still haven’t figured out why the proof he suggested was a trick.
R: The discussion with Socrates really helped! Let me ask you a question, what if we use Roman numerals to denote the consecutive numbers instead of using Arabic numerals: I-II, IV-V, VIII-IX? Does the digit-based proof work in this case?
R: Exactly! So our proof would just be restricted to Arabic numerals. But does that mean III and IV don’t yield an odd sum? And what does it mean to add the last digits of III and IV? It’s meaningless.
D: You’re right. Ok, I have a question for you now. When we began discussing even numbers, we defined even numbers as those with 0, 2, 4, 6 or 8 as their last digits. That means, none of the Roman numerals are even numbers, right?
R: Yeah, we do need a different definition of even. We need to go beyond these symbols, and think of the underlying concept of even. I guess this is what Anu Ma’am meant.
D: I think so too. How about this? When would you say that you have an even number of oranges in a basket?
R: I guess, when I can divide the oranges in the basket into two groups with equal numbers of oranges, and none are left over in the basket.
D: Hmm. Another way of looking at it is, you can form pairs of oranges from the oranges in the basket and keep doing this until you have used up all the oranges in the basket. If you started with an even number of oranges, you would never end up with an orange that is not part of a pair.
R: Cool! And if you began with an odd number of oranges, you would always end up with an orange that is not part of a pair.
D: Exactly! We can use the same concept of ‘even’ for our proof too, right?
R: This is great, Dhruv! So it doesn’t matter if we are using Arabic numerals or Roman numerals or the binary number system! What is common to 2 (Arabic), II (Roman), 10 (Arabic), 10 (Binary-2), and ८ (Hindi-8), is that they all represent numbers that have this property of being even.
D: This is so cool, Rhea! It’s just like how the words ‘cupboard‘ in English, ‘कपाट’ in Marathi, and ‘अलमारी’ in Hindi all represent the same thing.
R: Yeah, or ‘sugar’ in English and ‘चीनी’ in Hindi too!
D: Exactly. Wow! I hadn’t even thought of these differences before!
R (squinting): Hey I think that’s our bus -- Bus number 8.
D: Does that represent an even number? ;)
D: You seem a lot happier than you did on Friday! :)
Pause and Reflect:
Try to use the new concepts of ‘even’ and ‘odd’ numbers to prove that the sum of any two consecutive numbers is odd.
If you found this interesting, also think about Dhruv’s final question: Does the number 8 on the bus represent an even number? How about 6 o’clock on the clock? Or Rhea’s roll number: 26?