1.4. What have we discussed?

1. Rational numbers are closed under the operations of addition, subtraction and multiplication.

Closure means that when we perform an operation (like addition, subtraction, or multiplication) on two rational numbers, we will always get another rational number as a result. For example, if we take two rational numbers like 1/2 and 3/4, and add them, we will still get a rational number: 1/2 + 3/4 = 5/4.

Think of closure like a jar that can only hold marbles of certain sizes. If you put in two marbles, as long as they fit inside the jar (like two rational numbers), you will always get another marble that also fits (a new rational number).

2. The operations addition and multiplication are (i) commutative for rational numbers.

Commutativity means that the order in which we add or multiply rational numbers doesn't change the result. For addition: a + b = b + a. For example, 1/2 + 1/3 is the same as 1/3 + 1/2. The result will still be 5/6.

It's like having two friends, Alex and Jamie. If you invite Alex first and then Jamie, it's the same as inviting Jamie first and then Alex. The end result is still the same, just like the sum or product of two numbers.

2. The operations addition and multiplication are (ii) associative for rational numbers.

Associativity means that when three or more numbers are added or multiplied, the way in which we group them does not matter. For example, (a + b) + c = a + (b + c). If we take 1/2, 1/3, and 1/4, we can group them differently and still get the same result.

Imagine a group of friends who want to share snacks. Whether they decide to share the first two friends' snacks before third friend (like (A + B) + C) or all together in a different order (like A + (B + C)), they still end up sharing the same amount of snacks.

3. The rational number 0 is the additive identity for rational numbers.

The additive identity means that when you add 0 to any rational number, the result is the same rational number. For example, 5 + 0 = 5.

Think of an empty plate. Adding no food (0) to your plate doesn't change the fact that your plate still holds the same amount of food it had before!

4. The rational number 1 is the multiplicative identity for rational numbers.

The multiplicative identity means that when you multiply any rational number by 1, the number stays the same. For instance, 7 × 1 = 7.

Imagine a magic mirror that, when you stand in front of it, reflects your exact image. Multiplying by 1 acts like that mirror: it doesn't change anything; you are still you!

5. Distributivity of rational numbers: For all rational numbers a, b and c, a(b + c) = ab + ac and a(b – c) = ab – ac.

Distributivity means that multiplying a number by the sum (or difference) of two other numbers is the same as doing the multiplication separately and then adding (or subtracting) the results. For example, if a = 2, b = 3, and c = 4, then 2(3 + 4) = 23 + 24 = 6 + 8 = 14.

It's like if you are buying apples and bananas together. If you buy 3 apples and 4 bananas, you can think of it as buying 3 apples for a certain price and 4 bananas for another price. Multiply and add in one step or do each separately, either way, the total cost will be the same!

6. Between any two given rational numbers there are countless rational numbers. The idea of mean helps us to find rational numbers between two rational numbers.

Between any two rational numbers, you can always find more rational numbers. For example, if we take 1/2 and 1/3, there are many rational numbers between them like 5/12. You can also find the average of two numbers to discover a rational number in between.

Imagine a road where every time you look, there’s another car. No matter how precise you are, there are always more cars between any two you choose, just like rational numbers.

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