Reference and Synthesis
Corrections and Judgement Calls on the Source Notes
Every place where the teaching notes are arithmetically wrong, internally inconsistent or incomplete, what was done about it, and how each error could have been caught.
What this page covers
- See each source error alongside its correction
- Understand how each was detected
- Learn the checks that would have caught them
- Know where the source is incomplete rather than wrong
How this list was produced
Every worked example in the source was recomputed. Where the recomputation disagreed with the notes, the discrepancy was traced to a specific line, and the corrected version is what appears on the topic pages — always with the correction shown rather than applied silently.
Three of the errors below were caught by the source contradicting itself: the same quantity computed correctly on one page and incorrectly on another. Those are the most certain of the corrections, since the source itself supplies the right answer.
Arithmetic errors
Solving 2x2 - 5x + 3 = 0, the line reads 25 ± √14 = 264, 244. The 25 has been carried down from under the radical; outside it sits -b = 5. Correct: 5 ± 14 = 32 or 1. Verified by substitution: 2(94) - 152 + 3 = 0 and 2 - 5 + 3 = 0.
The roots must sum to -ba = 52. The printed roots sum to 12.5. A five-second check.
x2 - 13x + 36 is factored as (x - 10)(x - 3). Expanding gives x2 - 13x + 30. Correct: (x - 9)(x - 4), since (-9)(-4) = 36 and -9 - 4 = -13. Alternatively the intended trinomial may have been x2 - 13x + 30, which the printed factors do give.
Multiplying the factors back. The constant term comes out 30, not 36.
For x2 - 5x + 6x3 + 8 the restriction is given as x ≠ 2. The denominator vanishes when x3 = -8, that is at x = -2. At x = 2 the denominator is 16. Correct: x ≠ -2.
An intermediate line reads -11 = √74cos θ, dropping the factor |u| = √13. The final answer -0.3547 is correct and the check at the foot of the same page restores the factor, writing cos θ = u · v|u||v|. Only the intermediate line is wrong.
The product of [[1, -3], [2, 1]] and [[-2, -1], [4, 1]] is given as [[-10, -4], [8, -1]]. Recomputing the (2,1) entry from row 2 and column 1 gives (2)(-2) + (1)(4) = 0, not 8. Correct: [[-10, -4], [0, -1]]. The value 8 appears in a different product on the same page, so the two have most likely been transcribed across.
Internal contradictions
These are the strongest corrections, because the source itself gives the right answer elsewhere.
Page 4 gives θ = ±2pi3 on -π ≤ θ ≤ π. But cos2pi3 = -12. Page 8 solves the same equation and writes reference angle π3, because cosπ3 = 12, therefore θ = ±π3 — which is correct. Four pages apart, the notes disagree. Correct: ±π3.
Cosine is positive in quadrants I and IV. A positive cosine cannot have a solution in quadrant II, where 2pi3 lies.
The radian column gives tanπ4 = 1√2, while the degree column on the same page correctly gives tan 45° = 1. Since π4 and 45° are the same angle, the radian entry is wrong. Correct: 1, since tangent is opposite over adjacent, here 11.
The coordinate definitions are given as sec t = 1y and csc t = 1x. These are swapped. With x = cos t and sec t = 1cos t, secant must be 1x. The right-hand column of the same page has it correct, so the error is confined to the coordinate form.
Wording errors, where the working is right
The instruction reads add half of b2 to both sides. That would add 12b2; the correct quantity is (b2)2 = b24. For b = -8 the correct addition is 16, whereas half of b2 is 32. The worked examples on the same sheet use the correct quantity, so only the wording is wrong.
For y = -2cos 2t the source writes amplitude = -2. Amplitude is a magnitude and cannot be negative. The correct description is amplitude 2 together with a reflection in the horizontal axis. The sketch the source draws is right; only the label is loose.
The ranges are written (-1, 1) with round brackets. The endpoints are attained — sinπ2 = 1 exactly — so square brackets are correct: [-1, 1].
The quotient identity is written once as cot θ = cos θsin θ and once as sin θsin θ. The first is correct; the second would equal 1.
Ambiguities resolved by internal consistency
A handwritten source cannot always be read with certainty. Where a reading was ambiguous, the arithmetic decided it.
The second polynomial in (x3 - 3x2 + 8x + 7) + (…) is hard to read at scan resolution. The stated answer -4x3 - 3x2 - 4x + 10 is consistent only with (-5x3 - 12x + 3): any other reading of the x coefficient gives a different result. That reading is used.
A line reads 23√x = 22√x, which is false. Read as x3√x = x2√x it is correct, since x3/x1/2 = x5/2. The source's handwritten x closely resembles a 2 throughout, and the correct reading is used.
Gaps left as gaps
| Location | What is missing | Action taken |
|---|---|---|
| Week 2, page 6 | A line reading y3 - 5y2 = 2/5 - breaks off | Not reconstructed. Nothing invented |
| Week 9, page 9 | A cosine-rule check is begun and left as = … | The check is completed on the page, with the working shown |
| Week 6, page 4 | A 'prove 2 = 1' fallacy is presented without resolution | Presented as a fallacy, with the division-by-zero step identified |
The 'prove 2 = 1' argument is not an error in the notes. It is a deliberate teaching device: from a = b it derives a + b = b, and the step that fails is dividing by a - b, which is zero. Presented as a puzzle, it is a good one.
The checks that caught them
| Check | Cost | Errors it caught here |
|---|---|---|
| Sum and product of quadratic roots | Seconds | The Week 6 formula error |
| Multiplying factors back | Seconds | The Week 1 factoring error |
| Substituting into the original | Seconds | Would have caught most of the list |
| Quadrant sign check (ASTC) | Seconds | The Week 8 cosine error |
| Comparing two pages of the same source | Minutes | Three of the corrections above |
| Recomputing a matrix entry | Seconds | The Week 11 product error |
Every one of these checks costs a few seconds and uses information independent of the calculation being checked. That independence is what makes a check worth doing — repeating the same arithmetic tends to repeat the same mistake.
Frequently asked questions
Why publish the errors rather than quietly fixing them?
Because a reader working from the same notes will meet them. Recording the correction alongside the original saves that reader from concluding their own working is wrong.
How were they found?
By recomputing every worked example. Several were caught by sum-and-product checks on quadratic roots, and two by the source contradicting itself on different pages.
Does an error mean the notes are unreliable?
No. These are handwritten lecture notes, and a handful of arithmetic slips across 130 pages is unremarkable. The mathematics throughout is sound; the slips are transcription-level.
What about the incomplete parts?
Where the source breaks off or is illegible, nothing has been invented. The gap is recorded as a gap.
Source. Compiled while authoring this collection, by recomputing every worked example in the source.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.