Significant Digits: How Many Digits Do You Keep? | Units Matter

Why does a calculator give you ten digits when your ruler only earned three—and why does writing 9.80 m/s² instead of 9.8 m/s² actually change your physics answer? Learn how significant digits (also called significant figures, sig digs, or sig figs) come straight from your measuring tool, how to count them, and how to round every calculation cleanly. In this video: • The opening mystery: why g = 9.8 m/s² vs. g = 9.80 m/s² gives a different weight • Measuring a pencil with two rulers: 14.3 cm (cm marks) vs. 14.32 cm (mm marks) • What significant digits are: all certain digits + 1 estimated digit • Counting rules: non-zero digits (14.3 cm), captive zeros (1.05 m), leading zeros (0.187 kg = 187 g), and trailing zeros after a decimal (9.80 m/s², 0.500 kg) • Whole-number trailing zeros (5,000 m) and how scientific notation makes precision clear (5 × 10³ m vs. 5.000 × 10³ m) • Exact numbers (7 gym mats, 1 kg = 1,000 g) vs. approximate conversion factors (1,609 m = 1 mi) • Multiplying & dividing rule (fewest sig digs): 60.0 mi/h = 26.8 m/s, rug area 3.25 m × 2.1 m = 6.8 m², skateboard speed 12.0 m / 4.5 s = 2.7 m/s • Adding & subtracting rule (fewest decimal places): 2.3 m + 15.27 m + 0.415 m = 18.0 m • Why rounding early hurts: 7 square gym mats (1.15 m × 1.15 m → 9.26 m², not 9.24 m²) + calculator STO memory tip • Solving the opening question: Fg = mg for a 75.2 kg person (740 N with 9.8 m/s² vs. 737 N with 9.80 m/s²) • Go further: writing ± uncertainty in lab reports (14.3 cm ± 0.1 cm vs. 14.32 cm ± 0.01 cm) Chapters 00:00 Why 9.8 m/s² vs. 9.80 m/s² matters 00:18 Measuring a pencil: how the tool sets your digits 01:14 What significant digits are (certain + 1 estimated) 01:37 Counting significant digits: non-zeros & zeros 02:23 Trailing zeros & scientific notation 03:23 Exact numbers vs. approximate conversion factors 03:46 Multiplying & dividing: keep the fewest sig digs 04:58 Adding & subtracting: keep the fewest decimal places 05:39 Multi-step calculations: round only at the very end 06:14 Calculator tip: storing intermediate values 06:29 Solving the opening mystery: Fg = mg with 9.8 vs. 9.80 m/s² 07:38 Go further: writing ± uncertainty in lab reports 08:01 The 5 rules of significant digits, every time Answers: 14.3 cm · 14.32 cm · 26.8 m/s · 6.8 m² · 2.7 m/s · 18.0 m · 9.26 m² · 737 N Narrated by Mr. Johnson · Units Matter · Animated with Manim (Community Edition) #physics #significantfigures #significantdigits #measurement