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Which is smaller? which is larger?

Today let us look at the following two tricky questions:

Question 1: Among these two numbers 4444 x 55555 and 5555 x 44444, which one is smaller? which one is larger?

Question 2: How about  5000 x 5000 and 4999 x 5001, which one is smaller? which one is larger?

The answer can be found below.

Question 1: Among these two numbers $4444 \times 55555$ and $5555 \times 44444$, which one is smaller? which one is larger?

We can do a survey and ask people around us to see how many people actually think $4444 \times 55555$ is the larger number, and how many people think otherwise.

It is likely that people would pick 4444 x 55555 as the larger one, because 55555 seems to be much bigger than 44444.

The answer is, these two numbers are actually equal to each other!

The reason is
$$4444 = 4 \times 1111$$ $$55555 = 5 \times 11111$$ so $$4444 \times 55555 = 4 \times 1111 \times 5 \times 11111$$
Similarly,
$$5555 \times 44444 = 5 \times 1111 \times 4 \times 11111$$
So they are equal!

Question 2: Among these two numbers $5000 \times 5000$ and $4999 \times 5001$, which one is smaller? which one is larger?

This is hard to decide! We can see that 4999 is one less than 5000, whereas, 5001 is one more than 5000. So which one is larger, $5000 \times 5000$ or $4999 \times 5001$?

The purpose of this question is to introduce the following beautiful algebraic identity.

Difference of two squares identity: $$a^2 - b^2 = (a + b)(a - b)$$

We will use the identity as follows:
$$4999 = 5000 - 1$$ $$5001 = 5000 + 1$$ so $$4999 \times 5001 = (5000 - 1)(5000 + 1)$$ $$= 5000 ^ 2 - 1^2 = 5000 \times 5000 - 1$$
This shows that $4999 \times 5001$ is smaller than $5000 \times 5000$.

Let us stop here for now. In the next post, we will learn some surprising and beautiful facts about the "difference of two squares" identity. Hope to see you again then.

Homework.

1. Among these two numbers 1212 x 343434 and 3434 x 121212, which one is smaller? which one is larger?

2. How about 4999 x 5001 and 4998 x 5002, which one is smaller? which one is larger?

3. 44 x 45 and 43 x 46, which one is smaller? which one is larger?