Chapter 7 · Sections 1–2Textbook pages 119–126

Positions and Addition സ്ഥാനവും സംഖ്യയും

In Class 8 you met negative numbers as temperatures below zero, and learned to work out sums like 3 − 7 = −4. Now we give negative numbers a new job – telling which side of a fixed point something is – and discover what a sum like 5 + (−3) must mean so that one simple equation works for every move.

📖 What you already know (pages 1–4 of this guide)

For positive x, y: if x < y then x − y = −(y − x); −x + y = y − x; −x − y = −(x + y) (textbook page 119).
Adding positive numbers in either order gives the same answer: 5 + 3 = 3 + 5.
On a number line, positives are to the right of 0 and negatives to the left.

1. Warm-up: the three Class 8 rules അളവുകളും സംഖ്യകളും

⭐ Key box (textbook page 119)

x, y എന്ന ഏതു രണ്ട് അധിസംഖ്യകളെടുത്താലും – for any two positive numbers x and y:

(i) if x < y, then x − y = −(y − x) (ii) −x + y = y − x (iii) −x − y = −(x + y)

🎬 Animation: 3 °C falls by 7 °C

The six warm-up sums on page 119 (6 − 8, −6 + 8, −6 − 8 and the same with 2½ and 3½) are solved with animations on page 6.

2. A position needs a direction

A point P moves along a straight line. We describe where it is by its distance from a fixed point O. But "3 metres from O" could be on either side! So we also need to say which side.

🎬 Animation: right is positive, left is negative

💡 The idea

Choose a unit (metre, centimetre…). Then every position except O can be named by a number: positions to the right are positive (അധിസംഖ്യ), positions to the left are negative (ന്യൂനസംഖ്യ), and O itself is 0. One number now tells both the distance and the side.

3. Moves to the right: start + change = finish

A point moves from position 5 to position 8. The change of position (സ്ഥാനമാറ്റം) is 8 − 5 = 3. But knowing only "it moved 3" does not fix the end: 3 to the right of 5 is 8, while 3 to the left is 2. So a change, too, needs a direction.

🎬 Animation: the four right moves of the textbook table

Starting position	Change	Final position
5	3 (right)	8 = 5 + 3
3	5 (right)	8 = 3 + 5
−5	3 (right)	−2 = −5 + 3
−3	5 (right)	2 = −3 + 5

In every row, final = start + change. The last two rows use the Class 8 rule −x + y = y − x: −5 + 3 = 3 − 5 = −2 and −3 + 5 = 5 − 3 = 2.

4. Moves to the left – and what 5 + (−3) must mean

🎬 Animation: left moves as negative changes

Starting position x	Change y	Final position x + y
5	−3 (3 left)	2 = 5 + (−3)
3	−5 (5 left)	−2 = 3 + (−5)
−5	−3 (3 left)	−8 = (−5) + (−3)
−3	−5 (5 left)	−8 = (−3) + (−5)

💡 How the meaning is chosen (pages 123–124)

We have never added a negative number before, so 5 + (−3) has no meaning yet. We choose one that keeps an old rule: the order of adding never matters (5 + 3 = 3 + 5, ½ + ¼ = ¼ + ½). So let 5 + (−3) = −3 + 5, and from Class 8, −3 + 5 = 5 − 3 = 2. That is exactly the final position! In the same way 3 + (−5) = 3 − 5 = −2, (−5) + (−3) = −5 − 3 = −(5 + 3) = −8 and (−3) + (−5) = −8.

⭐ Key box (textbook page 125)

x ഏതു സംഖ്യയായാലും, y ഏതു അധിസംഖ്യയായാലും

x + (−y) = x − y

For any number x and any positive number y, adding a negative number means subtracting. With this meaning, whatever the signs of the starting position x and the change y, the final position z is given by the single equation z = x + y.

🧪 Fill in the textbook table (page 125)

start move

The textbook's table uses starts 7, 3, −7, −3 with moves of 3 or 7 to the right or left. Try them all: the final position is always start + move.

📜 Side box: one fact, many sentences ഒരു കാര്യം പല വാക്യം (page 126)

Without negative numbers we would need a rule for every case. Let x and z be distances of the starting and final positions from O, and y the size of the move:

x and y in the same direction: z = x + y, in that direction;
opposite directions and x > y: z = x − y, on the side of x;
opposite directions and x < y: z = y − x, on the side of y.

With negative numbers all three collapse into z = x + y. That is the whole point of the chapter: new numbers and new meanings so that one equation does every job.

🌍 Real life

A lift: ground floor 0, basements −1, −2. Starting at floor 3 and going 5 floors down is 3 + (−5) = −2: the second basement. A bank account at ₹500 that pays a bill of ₹800 ends at 500 + (−800) = −300, an overdraft. A thermometer at −2 °C that warms by 5 °C reads −2 + 5 = 3 °C.

5. Check yourself

🧭 Summary

Positions right of O are positive, left of O negative; O is 0. Moves right are positive changes, moves left negative changes.
x + (−y) = x − y: adding a negative number is subtracting. So 5 + (−3) = 2, 3 + (−5) = −2, (−5) + (−3) = −8.
With this meaning, final position = starting position + change in every case: z = x + y.
The meaning was chosen to keep an old rule (order of adding does not matter). The whole chapter works this way.

Next: Warm-up sums and Practice set 1 – animated solutions ▶