Chapter 9 · Section 4Textbook pages 174–176

Midpoint & Density മധ്യബിന്ദു

Which number is exactly halfway between two others? For 2 and 4 the answer is obviously 3. For −2½ and 4¾ it is less obvious – until we find a one-line rule: the midpoint is half the sum. The same rule shows something surprising: between any two numbers, however close, there are infinitely many more.

📖 What you already know (pages 6 and 8 of this guide)

The distance between two numbers is the bigger minus the smaller: |x − y|.
Moving d to the right of x lands on x + d.
½(a + b) = ½a + ½b.

1. Halfway between two numbers

🎬 Animation: midpoints of 2, 4 and of −2, 4

2. A rule for every pair

Let x be the smaller number and y the bigger one. Follow the textbook's three pictures.

🎬 Animation: x + ½(y − x) = ½(x + y)

⭐ Key box (textbook page 175)

സംഖ്യാരേഖയിൽ ഏതു രണ്ടു ബിന്ദുക്കളുടെയും മധ്യബിന്ദു, അവയെ സൂചിപ്പിക്കുന്ന സംഖ്യകളുടെ തുകയുടെ പകുതിയെ സൂചിപ്പിക്കുന്ന ബിന്ദുവാണ്.

The midpoint of any two points on the number line is the point of half the sum of their numbers:

midpoint of x and y = ½(x + y)

Two worked examples (page 175)

−2½ and 4¾: ½ × (−2½ + 4¾) = ½ × 2¼ = 1⅛.
−4.8 and 1.2: ½ × (−4.8 + 1.2) = ½ × (−3.6) = −1.8.

🎚️ Find any midpoint

x y

The rule works in either order and with any signs: ½(x + y) = ½(y + x).

3. Numbers never run out സംഖ്യാസാന്ദ്രത

Take two points, however close. Their midpoint lies between them. Then the midpoints of that midpoint with each end lie between them too. And so on, for ever.

🎬 Animation: keep halving

📜 Side box: density of numbers (page 176)

However close two points of a line are, there are infinitely many other points between them. With numbers: for any x and y, z₁ = ½(x + y) is between them; then z₂ = ½(x + z₁) and z₃ = ½(y + z₁) are between x and y too – and we can go on as long as we like.

🔍 Zoom into a gap

rounds of halving

Between ⅓ and ½. Each round puts a new midpoint into every gap, so the number of points doubles every time and never stops.

4. Equal distances: |x − 1| = |x − 4|

🎬 Animation: the point equally far from 1 and 4

💡 The idea

|x − 1| is the distance of x from 1 and |x − 4| its distance from 4. They are equal exactly when x is halfway between 1 and 4, so x = ½(1 + 4) = 2½.

🌍 Real life

Two friends living at the 3 km and 11 km marks of a road meet halfway, at the 7 km mark. The average of two test marks, 62 and 78, is their midpoint, 70. A carpenter finds the centre of a plank from its ends' ruler readings the same way.

5. Check yourself

🧭 Summary

Midpoint of x and y = x + ½(y − x) = ½(x + y).
Between any two real numbers there are infinitely many others (keep taking midpoints).
|x − a| = |x − b| ⇒ x is the midpoint of a and b: x = ½(a + b).

Next: Practice sets 3 and 4 – solutions ▶