Error Analysis Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Error Analysis.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The systematic study of how errors arise in calculations or models, how large they are, and how they propagate through subsequent steps.

Error analysis asks "how wrong could my answer be?" — not just "what is my answer?" — because every measurement and approximation carries uncertainty.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Error analysis studies how errors arise, how big they are, and how they propagate through later steps.

Common stuck point: The procedure for error analysis is the easy part; the trap is reporting an answer without its uncertainty. Asking "Am I quantifying how large my answer's error could be and how input errors grow through the steps?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I quantifying how large my answer's error could be and how input errors grow through the steps?

Worked Examples

Example 1

easy
A student writes: 'a2+b2=a+b\sqrt{a^2+b^2} = a+b.' Identify the error and find a numerical counterexample.

Answer

a2+b2≠a+b in general; counterexample: a=3,b=4 gives 5≠7\sqrt{a^2+b^2} \ne a+b \text{ in general; counterexample: } a=3, b=4\text{ gives } 5 \ne 7

First step

1
Identify the error: the student incorrectly 'distributed' the square root over addition. x+y≠x+y\sqrt{x+y} \ne \sqrt{x}+\sqrt{y} in general.

Full solution

  1. 2
    Counterexample: a=3,b=4a=3, b=4. LHS: 9+16=25=5\sqrt{9+16}=\sqrt{25}=5. RHS: 3+4=73+4=7. 5≠75 \ne 7.
  2. 3
    Correct statement: a2+b2≤∣a∣+∣b∣\sqrt{a^2+b^2} \le |a|+|b| (triangle inequality), with equality only when a=0a=0 or b=0b=0.
This error (distributing roots over sums) is one of the most common algebraic mistakes. Error analysis — identifying what went wrong and why — is essential for building correct mathematical habits.

Example 2

medium
A student 'proves': 'n2>nn^2 > n for all nn' by checking n=2,3,4n = 2, 3, 4. Identify the error in this argument and find a value of nn where the claim fails.

Example 3

medium
A student says: 'If a>ba > b then a2>b2a^2 > b^2.' Find a counterexample.

Example 4

medium
A student concludes from sin⁡θ=12\sin\theta = \frac{1}{2} that θ=30∘\theta = 30^\circ is the only solution. Identify the missing cases for 0≤θ<360∘0 \le \theta < 360^\circ.

Example 5

medium
A student factors x2+5x+6x^2 + 5x + 6 as (x+2)(x+4)(x+2)(x+4). Check by expansion and correct.

Example 6

hard
A student 'proves' 1=21 = 2 by writing a=ba = b, multiplying both sides by aa, subtracting b2b^2, and dividing by a−ba-b. Identify the error.

Example 7

hard
A student integrates ∫x−1dx\int x^{-1} dx as x00\frac{x^0}{0}. Identify the error and give the correct antiderivative.

Example 8

challenge
A student approximates 1+x\sqrt{1 + x} for small xx as 1+x21 + \frac{x}{2}. Estimate the absolute error of this linear approximation at x=0.2x = 0.2 using the next-order term −x28-\frac{x^2}{8}.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find the error: a student solves 2(x+3)=102(x+3) = 10 by writing 2x+3=102x+3=10, then x=3.5x=3.5. What went wrong?

Example 2

medium
A student cancels incorrectly: x2+3xx=x2+3\dfrac{x^2+3x}{x} = x^2+3. Identify and correct the error.

Example 3

easy
A measurement is 48 but the true value is 50. What is the percent error?

Example 4

easy
An estimate is 120; the actual is 100. What is the absolute error?

Example 5

easy
A student repeatedly makes sign errors when distributing -(x - 3). What is the correct expansion?

Example 6

easy
Rounding 3.14159 to two decimal places gives 3.14. What is the rounding error?

Example 7

easy
A student gets 7 * 8 = 54. Diagnose the error type: is it conceptual or a recall slip?

Example 8

easy
Two added measurements each have absolute error 0.2. What is the maximum total absolute error in the sum?

Example 9

easy
A student writes (x + 2)^2 = x^2 + 4. Identify the missing term in this common error.

Example 10

easy
A speedometer reads 60 with a possible error of +/-2. Express the reading as a range.

Example 11

medium
A rectangle has length 10 +/- 0.1 and width 5 +/- 0.1. Estimate the relative error in the area using relative-error addition.

Example 12

medium
A student consistently solves 3x = 12 as x = 36. Diagnose the systematic error and state the correct value.

Example 13

medium
An approximation of pi as 3.14 is used to compute a circle's circumference C = 2*pi*r with r = 10. Find the percent error in C from using 3.14 instead of 3.14159.

Example 14

medium
A measured radius r = 4 has 1% error. The volume of a sphere is V = (4/3)*pi*r^3. Estimate the percent error in V.

Example 15

medium
A student computes 0.1 + 0.2 in a program and gets 0.30000000000000004. Diagnose the error source and whether it is a mistake by the student.

Example 16

medium
Across a test, a student loses points only on problems involving negative exponents. What does this error pattern indicate, and what's the fix?

Example 17

challenge
Two quantities a = 100 +/- 1 and b = 99 +/- 1 are subtracted. Compute the result, its absolute error, and the relative error, and explain why subtraction of near-equal numbers is dangerous.

Example 18

challenge
Using the propagation rule, for f = x^a y^b the relative error is approximately |a|(dx/x) + |b|(dy/y). For f = x^2 / y with dx/x = 2% and dy/y = 3%, compute the percent error in f.

Example 19

challenge
A student's answer to sqrt(2) is 1.41. Estimate the maximum absolute error, then determine the relative error and whether 1.41 is accurate to 3 significant figures (sqrt(2) = 1.41421356...).

Example 20

medium
A student computes the mean of 10, 20, 30 as (10+20+30)/2 = 30. Diagnose the error and give the correct mean.

Example 21

medium
Two independent measurements have relative errors 3% and 4%. For their product, estimate the combined relative error using error addition.

Example 22

medium
A timer reads 9.8 s for an event that truly took 10.0 s. Express both the absolute and percent error.

Example 23

easy
A student writes (a+b)2=a2+b2(a+b)^2 = a^2 + b^2. Give a numerical counterexample.

Example 24

easy
An estimate gives 95 when the true value is 100. Compute the percent error.

Example 25

easy
A student writes log⁡(a+b)=log⁡a+log⁡b\log(a+b) = \log a + \log b. Diagnose the error.

Example 26

easy
A measurement is 510 with possible error ±10\pm 10. Express the value as an interval.

Example 27

easy
A clock loses 2 seconds per hour. What is the percent error per hour?

Example 28

easy
A student claims 16+9=4+3\sqrt{16+9} = 4+3. Compute both sides and identify the error.

Example 29

medium
For f=xyf = xy with relative errors Δxx=2%\frac{\Delta x}{x} = 2\% and Δyy=3%\frac{\Delta y}{y} = 3\%, estimate the relative error in ff.

Example 30

medium
A student computes 5−3×2=45 - 3 \times 2 = 4. Identify the error and give the correct result.

Example 31

medium
A measurement is reported as 4.5±0.14.5 \pm 0.1. Compute the maximum relative error.

Example 32

medium
For f=xyf = \frac{x}{y} with Δxx=1%\frac{\Delta x}{x} = 1\% and Δyy=2%\frac{\Delta y}{y} = 2\%, estimate the relative error in ff.

Example 33

medium
A student writes ∣x−5∣=−3|x - 5| = -3 has solutions x=2x = 2 or x=8x = 8. Identify the error.

Example 34

medium
A student rounds 3.6+2.73.6 + 2.7 to one decimal place as 6.46.4. Identify and correct the error.

Example 35

medium
An approximation of e≈2.72e \approx 2.72 is used in a calculation that needs e≈2.71828e \approx 2.71828. Compute the percent error in this approximation.

Example 36

hard
For f=x3f = x^3 with Δxx=1%\frac{\Delta x}{x} = 1\%, estimate the percent error in ff.

Example 37

hard
A pendulum period is T=2πL/gT = 2\pi\sqrt{L/g}. If LL is known to 2%2\% and gg to 1%1\%, estimate the percent error in TT.

Example 38

hard
Estimate the relative error in f=x2yf = \frac{x^2}{y} if Δxx=2%\frac{\Delta x}{x} = 2\% and Δyy=5%\frac{\Delta y}{y} = 5\%.

Example 39

hard
A student says lim⁡x→0sin⁡xx=0\lim_{x\to 0} \frac{\sin x}{x} = 0. Diagnose the error and give the correct limit.

Example 40

challenge
Two measurements a=5.00±0.01a = 5.00 \pm 0.01 and b=4.99±0.01b = 4.99 \pm 0.01 are subtracted. Compute the absolute and relative errors of a−ba - b and explain the danger of subtracting near-equal values.
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