Significant Figures in Physics: Complete Guide with Rules & Real Examples | 9th Class PTCB 2025

Significant Figures in Physics: Complete Guide with Rules & Real Examples | 9th Class PTCB 2025

1.11 Significant Figures 🔢 Introduction: Why Can't We Measure Everything Exactly? 🤔

Imagine you're at a candy store 🍬. You can count the exact number of candies in a jar - 1, 2, 3, 4... all the way to 50 candies. That's an exact count!

But now, try to measure the height of that candy jar with a ruler 📏. Can you say it's exactly 15 cm? Or is it 15.2 cm? Or maybe 15.23 cm?

Here's the truth: We can count things exactly, but we cannot measure physical quantities exactly! ⚠️

All measurements include uncertainties depending upon:

• The instrument we use 🔬
• How carefully we measure 👁️
• Environmental conditions 🌡️

This is why we need to understand Significant Figures - they help us express how reliable our measurements are! ✨

What Are Significant Figures? 📊

Definition: The significant figures (or significant digits) are the digits in a measurement that are reliably known plus one estimated (doubtful) digit.

Think of them as the "honest digits" in your measurement - they tell you which numbers you can trust! 💯

Real-World Example: Measuring a Pencil ✏️

Let's say you want to measure the length of a pencil using a ruler.

Scenario 1: Simple Ruler (Markings Every 1 cm)

You place the pencil on the ruler and see:

• The pencil clearly goes beyond 7 cm ✅
• But it doesn't quite reach 8 cm ❌
• You estimate it's about 7.4 cm 📐

Your measurement: 7.4 cm Significant figures: 2 (the 7 is certain, the 4 is estimated)

Scenario 2: Better Ruler (Markings Every 1 mm)

With a more precise ruler, you can see:

• The pencil goes beyond 7.4 cm ✅
• It's closer to 7.43 cm 📏
• You estimate it as 7.43 cm

Your measurement: 7.43 cm Significant figures: 3 (7 and 4 are certain, 3 is estimated)

💡 Key Point: The more refined your instrument, the more significant figures you can report!

The Story of Two Students 👨‍🎓👩‍🎓

Figure 1.14 shows an interesting situation. Two students are asked to measure the length of a rod using the same ruler.

The Measurement Challenge 🎯

The rod's length falls between 4.6 cm and 4.7 cm. What happens?

Student A says: "The length is 4.6 cm" ✏️

• She thinks the edge is closer to the 6 mm mark

Student B says: "The length is 4.7 cm" ✏️

• He thinks the edge is closer to the 7 mm mark

Who is Right? 🤷

Both students agree on the digit 4 - this is the certain digit ✅

But they disagree on the next digit (6 or 7) - this is the doubtful digit ❓

This doubtful digit was determined by estimation and has a probability of error.

Important: In any measurement:

• The accurately known digits (like 4) ✅
• PLUS the first doubtful digit (like 6 or 7) ❓
• Together form the SIGNIFICANT FIGURES 🔢

Rules for Determining Significant Figures 📋

Let's learn the rules with real-world examples!

Rule 1: All Non-Zero Digits Are Significant ✨

All digits from 1 to 9 are ALWAYS significant.

Examples:

• 234 has 3 significant figures 🎯
• 5.67 has 3 significant figures 🎯
• 8 has 1 significant figure 🎯

Real-Life: If your height is 172 cm, all three digits (1, 7, 2) are significant! 📏

Rule 2: The Tricky Zeros! 0️⃣

Zeros are tricky - sometimes they count, sometimes they don't! Let's break it down:

(a) Zeros BETWEEN Digits → SIGNIFICANT ✅

A zero between two non-zero digits is ALWAYS significant.

Examples:

• 5.06 m → 3 significant figures (5, 0, 6 all count!)
• 1002 kg → 4 significant figures
• 3.0005 L → 5 significant figures

Real-Life: If you weigh 60.5 kg, the zero counts! That's 3 significant figures. ⚖️

(b) Leading Zeros → NOT SIGNIFICANT ❌

Zeros on the LEFT side (before the first non-zero digit) are NOT significant.

These zeros are just placeholders to show where the decimal point is!

Examples:

• 0.0034 m → 2 significant figures (only 3 and 4 count)
• 0.5 cm → 1 significant figure (only 5 counts)
• 0.00078 kg → 2 significant figures (only 7 and 8 count)

Real-Life: The thickness of a paper is 0.0005 m - only 1 significant figure (the 5)! 📄

💡 Tip: Leading zeros are just saying "the decimal point is here" - they don't add precision!

(c) Trailing Zeros AFTER Decimal → SIGNIFICANT ✅

Zeros on the RIGHT side AFTER a decimal point ARE significant.

These zeros mean you actually measured to that precision!

Examples:

• 2.40 mm → 3 significant figures (2, 4, 0 all count!)
• 5.00 cm → 3 significant figures
• 3.200 m → 4 significant figures

Real-Life: If you measure your desk as 1.20 m, that's 3 significant figures. The zero shows you measured precisely! 📐

Why? If the length was just 1.2 m, you'd write 1.2 not 1.20. Writing 1.20 means you're confident it's not 1.19 or 1.21!

(d) Scientific Notation - Easy Mode! 🚀

In scientific notation, ALL digits before the exponent are significant.

Examples:

• 3.50 × 10⁷ m → 3 significant figures (3, 5, 0)
• 1.2 × 10³ kg → 2 significant figures (1, 2)
• 4.000 × 10⁵ L → 4 significant figures (4, 0, 0, 0)

Real-Life: The distance from Earth to the Sun is 1.50 × 10⁸ km - that's 3 significant figures! ☀️

💡 Pro Tip: Scientific notation makes it SUPER EASY to count significant figures - just count all the digits you see before the ×10!

Real-World Examples from Daily Life 🌍 Example 1: Cooking 🍳

Recipe says: "Add 250 mL of water"

• 3 significant figures (2, 5, 0)
• This means you should measure pretty carefully!

But if it says: "Add about 2 cups of water"

• 1 significant figure
• Approximate measurement is okay!

Example 2: Your Phone Screen 📱

Screen size: "6.5 inches"

• 2 significant figures (6, 5)
• Manufacturers measure this carefully!

Example 3: Speed Limit 🚗

Sign says: "50 km/h"

• 1 or 2 significant figures (depends on context)
• This is a rounded, approximate limit

Example 4: Medicine Dosage 💊

Prescription: "Take 5.00 mg"

• 3 significant figures
• Very precise! Medical measurements need accuracy!

Quick Quiz 1 ⏰

Question: Name some repetitive processes occurring in nature which could serve as a reasonable time standard.

Possible Answers:

• ⏰ The rotation of the Earth (24 hours)
• 🌙 The revolution of the Moon around Earth (≈27 days)
• ⏱️ The oscillation of a pendulum
• 💓 The beating of your heart
• 🌊 Ocean tides

Quick Quiz 2 🧮

How many significant figures are there in each of the following?

(a) 1.25 × 10⁷ m

Answer: 3 significant figures ✅

• All digits before the exponent count: 1, 2, 5

(b) 12.5 cm

Answer: 3 significant figures ✅

• All non-zero digits count: 1, 2, 5

(c) 0.125 m

Answer: 3 significant figures ✅

• Leading zero doesn't count
• Only 1, 2, 5 are significant

(d) 0.000125 km

Answer: 3 significant figures ✅

• All leading zeros don't count
• Only 1, 2, 5 are significant

💡 Notice: All four measurements have the same number of significant figures even though they look different! They're just written in different units! 🎯

Practice Problems for You! 📝

Try these yourself:

1. How many significant figures? 450 m

   • Is the zero significant? (Tricky!) 🤔

2. How many significant figures? 0.00600 L

   • Watch out for leading AND trailing zeros!

3. How many significant figures? 100.0 kg

   • The decimal point makes a difference!

4. How many significant figures? 2.0 × 10⁴ cm

   • Easy one with scientific notation!

Why Do Significant Figures Matter? 🎯 In Science 🔬

• They show the precision of your measurements
• They prevent you from claiming false accuracy
• They help communicate uncertainty honestly

In Engineering 🏗️

• Building tolerances need precise measurements
• Safety depends on knowing measurement limits

In Medicine 💉

• Drug dosages must be accurate
• Lives depend on precise measurements!

In Everyday Life 🏠

• Helps you understand product specifications
• Makes you a smarter consumer!

Key Takeaways 💡

✅ Significant figures = Reliable digits + One doubtful digit

✅ Non-zero digits (1-9) are ALWAYS significant

✅ Zeros between digits are significant

✅ Leading zeros (left side) are NOT significant

✅ Trailing zeros after decimal are significant

✅ Scientific notation makes counting easy!

✅ More precise instruments → More significant figures

✅ Never claim more precision than your instrument allows! ⚠️

Remember This! 🧠

"Significant figures are not just numbers - they're a promise of honesty in science. They tell the world: 'This is what I know for sure, and this is where my measurement gets fuzzy.'" 💭

The more refined your measuring instrument, the more significant figures you can confidently report! 📏✨