Figures that Count

solved Uncertainty

Q. Which of these are written to four significant figures?

How it's solved

Solved with Uncertainty

Working

0.007 m²: 1 significant figure
2.64 × 10²⁴ kg: 3 significant figures
0.2370 g cm⁻³: 4 significant figures
6.320 J: 4 significant figures
6.032 N m⁻²: 4 significant figures
0.0006032 m²: 4 significant figures

Conclusion

Zeros in front of a number only place the decimal point, so 0.007 has one figure and 0.0006032 has four. Zeros after the point at the end were measured, so 0.2370 and 6.320 have four. 6.032 has four, and in 2.64 × 10²⁴ the power of ten counts for nothing: three.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :figures_that_count do4  kind :mcq5  describes "Which of these are written to four significant figures?"67  option "0.007 m²"8  option "2.64 × 10²⁴ kg"9  option "0.2370 g cm⁻³"10  option "6.320 J"11  option "6.032 N m⁻²"12  option "0.0006032 m²"13end

solution.rb

1require_relative "../questions/figures_that_count"2require_relative "../science/units_and_measurement/uncertainty"34solve :figures_that_count, using: Uncertainty do5  answer options.select { |o| o.content.to_measurement.significant_figures == 4 }67  conclusion "Zeros in front of a number only place the decimal point, so 0.007 has one figure " \8             "and 0.0006032 has four. Zeros after the point at the end were measured, so 0.2370 " \9             "and 6.320 have four. 6.032 has four, and in 2.64 × 10²⁴ the power of ten counts for " \10             "nothing: three."11end

playground.rb

1require_relative "../solutions/figures_that_count"23play_with :figures_that_count do4  title "Figures that Count"5end

The laws it drew on

quantity	written	units	within	from
first measured	$a$	—	—	Uncertainty
error in first	$Δ a$	—	—	Uncertainty
relative error in first	$δ_{a}$	—	—	Uncertainty
second measured	$b$	—	—	Uncertainty
error in second	$Δ b$	—	—	Uncertainty
relative error in second	$δ_{b}$	—	—	Uncertainty
product	$p$	—	—	Uncertainty
error in product	$Δ p$	—	—	Uncertainty
relative error in product	$δ_{p}$	—	—	Uncertainty
total	$t$	—	—	Uncertainty
difference	$d$	—	—	Uncertainty
least count	$l_{c}$	m	—	Uncertainty
scale division	$d_{m}$	m	—	Uncertainty
divisions read past	$N_{d}$	—	—	Uncertainty

δ_{a} = \frac{Δ a}{a}

first error over first

δ_{b} = \frac{Δ b}{b}

second error over second

p = a \cdot b

product of measurements

δ_{p} = δ_{a} + δ_{b}

relative errors add

Δ p = δ_{p} \cdot p

error from relative error

t = a + b

sum of measurements

d = a - b

difference of measurements

l_{c} = \frac{d_{m}}{N_{d}}

division shared out