Algebra Basics ബീജഗണിതം
Chapter 4 proves its tricks with letters instead of numbers. This page makes sure the letters never feel mysterious: a letter is simply a number whose value we have not fixed yet. You will learn to expand brackets, collect like terms, tell an equation from an identity, and write "any odd number" or "any two-digit number" in algebra.
1. A letter stands for any number
Write x for "whichever number you like". Then 2x + 1 means "double it and add one". Choosing x fixes the value of every expression that uses x.
An expression (വ്യഞ്ജകം) like x(x + 3) is a recipe; its value changes with x. In the chapter, x is often a date in a calendar or a digit of a number.
2. Expanding brackets: x(y + z) = xy + xz
This is the distributive law from page 1 written with letters. A rectangle x tall and (y + z) long splits into two rectangles of areas xy and xz.
Multiply the term outside by each term inside. Example: 3(a + 5) = 3a + 15; n(n + 1) = n² + n; (x + 7)·x = x² + 7x.
3. Like terms: what can be added together
x² + 3x + 2x + 6 has four terms. 3x and 2x are like terms (same letter, same power) and add to 5x. x² and 6 are different kinds and stay as they are: x² + 5x + 6. Think of tiles: squares, strips and units.
Blue tiles are x² (squares), amber tiles are x (strips), green tiles are 1 (units). Only tiles of the same kind are counted together.
4. Equation or identity? സമവാക്യം / സർവസമവാക്യം
An equation (സമവാക്യം) like 2x + 1 = 7 is true for some value of x (here x = 3) and false for the others. An identity (സർവസമവാക്യം, "equation for all") like x(x + 3) = x² + 3x is true for every x. The rules of this chapter – such as (x + 1)(y + 1) = xy + x + y + 1 – are identities: they never fail.
Green rows: both sides equal. An identity is green for every x you try; an equation is green only at its solution. Try x = 0, negative numbers and fractions too – the table uses −2 to 6.
Checking ten values and seeing a pattern is a good guess, not a proof. Famous trap: n² + n + 41 is a prime number for n = 0, 1, 2, …, 39 – forty checks in a row – and then fails at n = 40 (40² + 40 + 41 = 41 × 41). Another: joining points on a circle gives 1, 2, 4, 8, 16 regions… and then 31, not 32. That is why the textbook always ends a pattern with "ബീജഗണിതം ഉപയോഗിച്ച് തെളിയിക്കുക" – prove it with algebra. An identity proved with letters is true for all numbers at once.
5. General forms: writing "any number of a kind"
To prove something about all odd numbers you need a way to write any odd number. Every odd number is one more than an even number, so it is 2n + 1 for some whole number n. Similar formulas describe every kind of number the chapter uses.
Each formula produces the whole family as n runs through 0, 1, 2, 3, … Numbers that leave remainder (ശിഷ്ടം) 1 when divided by 3 are 1, 4, 7, 10, … = 3n + 1. Remainder 2 gives 3n + 2.
| Kind of number | Malayalam | General form | Examples |
|---|---|---|---|
| Even | ഇരട്ടസംഖ്യ | 2n | 0, 2, 4, 6, … |
| Odd | ഒറ്റസംഖ്യ | 2n + 1 | 1, 3, 5, 7, … |
| Multiple of 3 | മൂന്നിന്റെ ഗുണിതം | 3n | 0, 3, 6, 9, … |
| Remainder 1 on ÷ 3 | 3 കൊണ്ട് ഹരിച്ചാൽ ശിഷ്ടം 1 | 3n + 1 | 1, 4, 7, 10, … |
| Remainder 2 on ÷ 3 | 3n + 2 | 2, 5, 8, 11, … | |
| Remainder r on ÷ 4 | 4n + r (r = 0, 1, 2, 3) | 4n + 1: 1, 5, 9, … | |
| Consecutive numbers | അടുത്തടുത്ത സംഖ്യകൾ | x, x + 1, x + 2, … | 7, 8, 9 |
| Two-digit number | രണ്ടക്കസംഖ്യ | 10m + n (digits m, n) | 47 = 10·4 + 7 |
6. Two-digit numbers: 10m + n
The number with tens digit m and units digit n is 10m + n. Reversing the digits gives 10n + m. This is the key to the digit-reversal puzzle later in the chapter.
7. Neighbours in a calendar and in a multiplication table
If a calendar date is x, the date to its right is x + 1 and the date below it is x + 7 (a week later). In a multiplication table the cell in row m, column n holds m × n, so its right neighbour is m(n + 1) and the one below is (m + 1)n. The chapter uses exactly these facts.
Click any date (left) or any table cell (right). The highlighted 2 × 2 square is written in letters below the pictures.
A phone-bill formula "₹199 plus ₹2 per extra minute" is 199 + 2x. A bus fare table, a shop's "buy n items" cost, the dates of every Sunday in a month (x, x + 7, x + 14, …) – all are general forms. Algebra is a way to say "for every case at once", which is why engineers, programmers and the textbook all rely on it.
8. Check yourself
- A letter stands for any number; an expression's value depends on the letter's value.
- x(y + z) = xy + xz; collect like terms (3x + 2x = 5x), never unlike ones (x² + 5x stays).
- Equation: true for some x. Identity (സർവസമവാക്യം): true for all x. Patterns need proof, not just examples.
- General forms: even 2n, odd 2n + 1, remainder 1 on ÷ 3 is 3n + 1, two-digit number 10m + n, consecutive x, x + 1.
- Calendar: right neighbour x + 1, below x + 7. Table: cell (m, n) is mn.