Prerequisite 3 of 4Needed for pages 123, 135–136

Laws of Arithmetic and Expressions ക്രിയാനിയമങ്ങൾ

The chapter gives meanings to sums like 5 + (−3) and products like (−10) × (−2). It does this by insisting that the old laws of arithmetic must keep working. So here are those laws – order, grouping and the distributive law – plus how to put numbers into expressions like x² + 3x + 2.

📖 What you already know

Adding and multiplying whole numbers, fractions and square roots.
A letter stands for a number; x² means x × x.

1. Order does not matter (commutative law)

5 + 3 = 3 + 5, ½ + ¼ = ¼ + ½, √2 + √3 = √3 + √2; and 5 × 3 = 3 × 5, ½ × ¼ = ¼ × ½. The textbook uses exactly these examples to decide that 5 + (−3) should equal −3 + 5, and 10 × (−2) should equal (−2) × 10.

🎬 Animation: 3 + 5 and 5 + 3 on the line; 3 × 5 and 5 × 3 as arrays

2. Grouping does not matter (associative law)

(2 + 4) + 5 = 2 + (4 + 5) = 11. Practice set 1 asks you to check this when some numbers are negative.

🖐️ Group them either way

x y z

3. The distributive law

To multiply a sum by a number, multiply each part and add: (15 + 10) × 2 = (15 × 2) + (10 × 2). In algebra: (x + y)z = xz + yz. The area picture shows why.

🎬 Animation: a rectangle (x + y) by z cut in two

⭐ Key idea

(x + y)z = xz + yz, and likewise (x + y)(u + v) = xu + xv + yu + yv. On page 136 the textbook uses the first one to show why (−10) × (−2) must be 20.

4. Putting numbers into expressions

🧮 Evaluate x² + 3x + 2 and (x + 1)(x + 2)

Both give the same number for every x: they are the same expression written two ways. Practice set 3 asks for these with x = −1 and x = −2.

5. Keeping a pattern alive

Look at the 3-times table going down: each answer is 3 less than the one above. If the pattern is to continue below 0, what must 3 × (−1) be?

🎬 Animation: 3 × 2, 3 × 1, 3 × 0, 3 × (−1), …

💡 The chapter's big idea in one picture

We do not "prove" 3 × (−1) = −3 from nothing – we choose it so that the pattern (and the laws) keep working. Every new rule in the chapter is chosen in this spirit.

🌍 Real life

Mental maths uses these laws all the time: 25 × 4 × 7 = 100 × 7 (grouping), 15 × 12 = 15 × 10 + 15 × 2 (distributive).

6. Check yourself

🧭 Summary

Order and grouping of adding or multiplying do not change the answer.
(x + y)z = xz + yz.
To evaluate an expression, replace the letter by the number and follow the order of operations.
New meanings are chosen so that the old laws and patterns keep working.

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