Numbers as Sums & the Distributive Law സ്ഥാനവിലയും വിതരണനിയമവും
The whole of Chapter 4 rests on one habit: seeing a number as a sum of convenient parts (26 = 20 + 6) and one rule for multiplying such sums. This page builds that habit from scratch with pictures of rectangles, and shows that the multiplication you already do in columns is secretly the same picture.
1. A number is a sum of its place values
When we write 26, the 2 does not mean two – it means two tens. The place value (സ്ഥാനവില) of a digit depends on where it stands. So 26 = 20 + 6, and 347 = 300 + 40 + 7. This is called the expanded form of the number.
Hundreds are big squares (100 small squares each), tens are strips of 10, ones are single squares. The number of squares in the picture is the number itself.
Every trick in Chapter 4 begins by rewriting a number as a sum (26 = 20 + 6) or as a difference (26 = 30 − 4). Convenient parts are usually round numbers – tens, hundreds – because multiplying by them is easy.
2. Area of a rectangle പരപ്പളവ്
A rectangle 7 cm long and 4 cm wide can be covered by small squares of side 1 cm: 4 rows of 7 squares, that is 7 × 4 = 28 squares. Area = length × breadth. The same number of squares appears whichever way you count – 7 rows of 4 or 4 rows of 7 – which is why 7 × 4 = 4 × 7.
3. The distributive law: a(b + c) = ab + ac വിതരണനിയമം
Multiply 5 by the sum 20 + 6. You may add first (5 × 26 = 130) or multiply each part and then add (5 × 20 + 5 × 6 = 100 + 30 = 130). The answer is the same – and the picture shows why.
The multiplier is shared out to every term inside the bracket. It works for any numbers – whole numbers, fractions, decimals – because a rectangle can always be cut.
4. Mental multiplication: split the bigger number
7 × 23 looks hard; 7 × 20 and 7 × 3 are easy: 140 + 21 = 161. That is the distributive law doing the work.
Try to say the answer before reading the working. Splitting at the tens is the standard mental trick.
5. The column method is the area picture
You have multiplied 26 × 15 in columns for years. Each row you write down is one "sum times a number", and the whole thing is four small rectangles added up.
Row 1 is 26 × 5 = (20 + 6) × 5 = 100 + 30. Row 2 is 26 × 10 = (20 + 6) × 10 = 200 + 60 (the zero is written as a shift). Adding the rows adds all four pieces: 100 + 30 + 200 + 60 = 390.
6. Multiplying by 10 and 100
Multiplying by 10 moves every digit one place to the left: 26 × 10 = 260; by 100, two places: 26 × 100 = 2600. That is why round numbers make such convenient parts: 20 × 30 is just 2 × 3 followed by two zeros, 600.
7. Break into convenient parts: two-digit × two-digit
Now combine everything: split both numbers at the tens and you get four partial products. (The next page of the chapter turns this into an identity.)
Say the four partial products aloud, then add. With practice, products like 23 × 14 become mental arithmetic.
A vegetable seller charging ₹23 per kilogram for 7 kg thinks "7 twenties is 140, 7 threes is 21 – ₹161". A tiler covering a 26-tile by 15-tile floor counts 20 × 15 and 6 × 15 separately. Splitting numbers into convenient parts is how shopkeepers, carpenters and engineers calculate without a calculator.
8. Check yourself
- Expanded form: 26 = 20 + 6, 347 = 300 + 40 + 7 (place value, സ്ഥാനവില).
- Area of a rectangle = length × breadth; cutting a rectangle splits its area into parts.
- a(b + c) = ab + ac – the distributive law – is just a rectangle cut into two.
- Mental maths: 7 × 23 = 7 × 20 + 7 × 3 = 161; 26 × 15 = 200 + 100 + 60 + 30 = 390.
- The column method writes these same partial products row by row.
Next: Algebra basics – letters, expressions and general forms ▶