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Sums

In a paragraph

A box of 12 × 7 + 3 apples is a little more than 80. Everybody knows 2 + 2 = 4, but fewer people know that 2^10 is exactly 1024, which is why a kilobyte was once 1024 bytes. What is 144 ÷ 12 = ? Splitting a $120 + $45 bill three ways?

The classic right triangle: 3² + 4² = 5². A circle of radius 5 has an area of π × 5² and a circumference of 2π × 5. The diagonal of a unit square is √2, and a tip of 15% of 80 is twelve.

Light travels about 3 × 10⁸ m/s, so in a minute it covers 3e8 × 60 metres. In trigonometry, sin(30°) is one half, and sin(π/6) is the same in radians. Avogadro's number times two is 6.022e23 × 2.

Long and nested should work

1. Nested brackets: (((2 + 3) × (4 − 1))^2 − 17) ÷ 4
2. Twenty terms: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20
3. Square brackets: [3 + (4 × [2 − 7])] × 2
4. Tower of powers: 2^2^3
5. Negative and fractional powers: 2^-3 + 4^(1/2) − 8^(−1/3)
6. Roots of roots: √(√(√256))
7. Implicit times: 3(2 + 5)(4 − 1)
8. Factorials: 10! ÷ (5! × 5!)
9. Squared minus: −(−3)^2
10. Floating point: 0.1 + 0.2 = 0.30000000000000004
11. Phi: (1 + √5) ÷ 2
12. Big numbers: 99999999 × 99999999
13. Scientific notation: 1.5e-7 × 3.2e12
14. A power in a tag: 6.02214076 × 10²³ ÷ 1000, and 5 × 10^-3
15. Superscripts: 2¹⁰ − 3⁴ + 10⁻²
16. Modulo: 7 mod 3 + 15 mod 4
17. Percent: 25% of 360
18. Money with thousands: $1,234.56 × 12

Written in fractions: the answer is a fraction, and a/b opens on should work

1. 1/2 + 3/4 − 5/8
2. Fractions of one character: ½ + ¼, and 2 × 1½
3. (2/3) ÷ (4/9)

Functions should work

1. Identity: sin(45°)^2 + cos(45°)^2
2. Inverse trig: arctan(1) × 4
3. Hyperbolic: tanh(0.5) + sinh(1)
4. Logs: log(1000) − ln(1) + log(0.01)
5. Nested functions: sqrt(abs(−16)) + round(2.567) + floor(−2.5)
6. Trig without brackets: sin 30 + cos 60
7. Radians: cos(π) + sin(π/2)

Long and mixed: fractions, roots, logs, trig and π should work

1. All of them: (3/4 + √(16) × log(100)) ÷ (2π − sin(π/6))
2. Phi squared: ((1 + √5)/2)^2 − ((1 + √5)/2) − 1
3. An identity and more: sin(π/4)^2 + cos(π/4)^2 + tan(π/4) × log(10)
4. Roots of powers: 2^(1/2) × 3^(1/3) + √(√81) − 8^(2/3)
5. Square of a sum of roots: (sqrt(2) + sqrt(3))^2 − 2sqrt(6)
6. Logs and fractions: log(2)/log(8) + ln(1) − (5/6 − 1/3) × π
7. Compound interest: 1000 × (1 + 0.05/12)^(12 × 10)
8. Inverse trig: arcsin(1/2) + arccos(1/2) + arctan(1)
9. In degrees: ((2 + 3)^2 − √(144) ÷ (4 − 1)) × sin(30°) + ln(exp(2))
10. Nested functions: sqrt(abs(log(0.001))) × cos(π/3) + floor(π^2) − round(exp(0) + 1/3)
11. A long equation: sin(x) + √(x) = log(x + 10) + π/4
12. A long inequality: x^2/4 − 3x + ln(2) < √(x + 9)
13. Drawn: $\frac{\sqrt{16} + \log(1000)}{2\pi} - \sin\left(\frac{\pi}{6}\right)$
14. Drawn: $\sqrt{\frac{\pi^2}{6}} \cdot \cos\left(\frac{\pi}{3}\right) + \ln\left(e^{2}\right)$
15. A continued fraction: $\frac{1}{1 + \frac{1}{1 + \frac{1}{2}}} + \sqrt[3]{8} \cdot \cos(\pi)$
16. Powers of fractions: $\left(\frac{3}{4}\right)^{2} \div \frac{9}{16} + \log(100) - \tan\left(\frac{\pi}{4}\right)$
17. An integral of a mix: $\int_{0}^{\pi} \left(\sin x + \frac{x}{\pi}\right) \, dx$
18. A sum of a mix: $\sum_{k=1}^{20} \frac{\sqrt{k}}{k^2 + 1}$
19. A drawn equation: $\frac{x}{2} + \sqrt{x + 1} = \ln(x + 5) + 3$

Equations and inequalities

Of x should work

1. Linear: Solve 2x + 3 = 7 first.
2. Quadratic: x² − 5x + 6 = 0
3. A cubic: x^3 = 2x + 1
4. Quintic: x^5 − 3x^3 + x − 1 = 0
5. Biquadratic: x⁴ − 10x² + 9 = 0
6. Factored: (x − 1)(x − 2)(x − 3) = 0
7. With a tag power: x² − 2x + 1 = 0
8. Exponential: 2^x = 1024
9. Logarithmic: log(x) = 3
10. Mixed: ln(x) + x = 5
11. Trig: sin(x) = cos(x)
12. Radical: √(x + 3) = x − 3
13. Absolute value: abs(x − 2) = 5
14. Rational: 1/x + 1/(x + 1) = 1
15. Both sides: 3x + 7 = 2x − 5
16. Decimals: 0.5x − 1.25 = 3.75
17. No real root: x^2 + 1 = 0
18. Cubic inequality: x^3 − x > 0
19. Linear inequality: 2x + 1 ≥ 3x − 4
20. Square inequality: x² ≤ 9, and x^2 − 4 > 0
21. Double inequality: −2 < 3x + 1 < 7

Together: two or three equations should work

1. Two lines: Solve 2x + y = 5, x − y = 1 together.
2. Three planes: x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2.
3. A circle and a line: x² + y² = 25 and x + y = 7.
4. One line twice: x + y = 1, 2x + 2y = 2

Functions to draw should work

1. A parabola: y = x² − 4x + 3
2. A named function: f(x) = sin(x)/x
3. Growth: g(x) = 2^x
4. A cubic: h(x) = x³ − 3x
5. With asymptotes: y = tan(x)
6. A line: y = 2x + 1
7. Not one: y = 5, y = x, y = mx + b

Calculus

Derivatives, series and differential equations should work

1. At a point: Find d/dx x^3 at x = 2.
2. In words: the derivative of sin(x) at x = pi
3. To draw: d/dx (x^3 − 3x)
4. Taylor: the Taylor series of sin(x) about 0 to order 7
5. Maclaurin: the Maclaurin series of e^x
6. A differential equation: y' = x + y, y(0) = 1
7. Another: dy/dx = x*y with y(0) = 1

The cards of the calculator

Conversions should work

1. A run of 5 km in miles.
2. Set the oven from 350 °F to °C.
3. How many cups in 2 liters?
4. A 2 TB in GB disk.
5. Convert 100 pounds in kilograms.

Dates should work

1. The project ran from March 3, 2024 to June 1, 2026.
2. Open between 2024-01-01 and 2024-12-31.
3. A range: 1 May 2026 – 4 July 2026
4. From today to 25 December 2026.
5. The warranty ends 90 days after 1 May 2026.
6. Three weeks back: 3 weeks before March 1, 2027
7. A month on: 2026-01-31 plus 1 month

Money should work

1. A $250,000 mortgage at 6.5% for 30 years.
2. Or a 5-year car loan of $20k at 4%.
3. Prices rose from $120 to $150 last year.
4. Rates increased from 3% to 5%.
5. Savings at 5% compounded monthly.
6. 4.5% APR, compounded daily

Choices and complex numbers should work

1. There are 52 choose 5 poker hands.
2. Orders: 10 nPr 3
3. A function: nCr(10, 3)
4. With its answer: C(52, 5) = 2,598,960
5. Multiply (3 + 4i)(1 − 2i) by hand.
6. Its length: |3 + 4i| is 5
7. A sum: 2 + 3i

IP networks should work

1. Home network: 192.168.0.1/24
2. A big private one: 10.0.0.0/8
3. A small office: 172.16.5.4/28
4. With a mask: 192.168.1.10 255.255.255.0
5. Two hosts linked: 203.0.113.8/31
6. IPv6 documentation: 2001:db8::/32
7. IPv6 link-local: fe80::1/64
8. IPv6 in full: 2001:0db8:85a3:0000:0000:8a2e:0370:7334/64
9. Not an address: 300.1.1.1/24
10. Not a mask: 10.0.0.1 255.0.255.0

Logic should work

1. To simplify: F = AB + A'C + BC
2. Of its minterms: F(A, B, C) = Σm(1, 3, 5, 7)
3. With rows that may be either: F(A, B, C, D) = Σm(0, 2, 5, 7, 8, 10, 13, 15) + d(1)
4. Of its maxterms: ΠM(0, 2, 4)
5. In words: A AND NOT B OR C
6. With signs: Y = A ⊕ B, and P ∧ ¬Q
7. As a textbook prints it: F = AB′ + C′D
8. Not logic: R AND D, O'Brien, ROCK AND ROLL, the letters 'A' and 'B'

Formulas and constants should work

1. Einstein's E = mc² and Newton's F = ma
2. Ohm's law V = IR, and the ideal gas law PV = nRT
3. Pythagoras: a² + b² = c², and kinetic energy KE = ½mv²
4. The speed of light, about 300,000 km/s
5. Planck's constant and the reduced Planck constant
6. Avogadro's number, and the Boltzmann constant
7. The gravitational constant, and the acceleration due to gravity
8. The mass of the electron, and the elementary charge
9. Not: a Raspberry Pi, a bar of soap, e = mc, F = mass

Chemistry

A reaction is balanced; a formula gets its molar mass, name and oxidation numbers; a composition its empirical formula; a mixture of gases the pressure of each; a compound named its formula; an element its place in the table. A click opens it in the Chemistry panel.

Reactions should work

1. Water: 2H2 + O2 → 2H2O
2. Rust, to balance: Fe + O2 → Fe2O3
3. Burning propane: C3H8 + O2 = CO2 + H2O
4. With an arrow typed: CH4 + 2O2 -> CO2 + 2H2O
5. The Haber process: N2 + 3H2 ⇌ 2NH3
6. Photosynthesis: 6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂
7. As Wikipedia writes it: 2H₂ + O₂ → 2H₂O
8. Ions and states: Ag⁺(aq) + Cl⁻(aq) → AgCl(s)
9. Ions in plain text: Na+ + Cl- → NaCl
10. Neutralization: HCl + NaOH → NaCl + H2O
11. Redox, to balance: Cl2 + NaOH → NaCl + NaClO3 + H2O
12. A half-reaction: Fe^3+ + e- → Fe^2+

Formulas should work

1. Battery acid is H2SO4.
2. Table salt, NaCl, and baking soda, NaHCO3
3. Breathe out CO₂, drink H₂O
4. Blue crystals of CuSO4·5H2O
5. With brackets: Ca(OH)2, (NH4)2SO4 and K4[Fe(CN)6]
6. As Wikipedia writes it: Fe₂O₃, rust
7. An ion: SO₄²⁻, and typed, MnO4^-
8. Caffeine, C8H10N4O2
9. Silver chloride does not dissolve: AgCl
10. Chalk, CaCO3, and quicklime, CaO

Compositions should work

1. Glucose is 40% C, 6.7% H and 53.3% O by mass.
2. By name: carbon 40%, hydrogen 6.7%, oxygen 53.3%
3. An oxide of iron: Fe 69.9%, O 30.1%
4. With colons: C: 85.6%, H: 14.4%
5. Of elements that are gases: water is hydrogen 11.2%, oxygen 88.8%

Mixtures of gases should work

1. Air is 78% nitrogen, 21% oxygen and 1% argon.
2. Dry air: N2 78.08%, O2 20.95%, Ar 0.93%, CO2 0.04%
3. As Wikipedia writes it: N₂ 78%, O₂ 21%, Ar 1%
4. Natural gas: methane 95%, ethane 3%, propane 2%
5. A mix for diving, heliox: 79% helium and 21% oxygen
6. By mass, air is 75.5% N₂, 23.1% O₂ and 1.3% Ar

Names of compounds should work

1. A pinch of sodium chloride
2. Concentrated sulfuric acid
3. Rust is iron(III) oxide.
4. Fertilizer: ammonium nitrate
5. Blue copper(II) sulfate pentahydrate
6. A brown gas: dinitrogen tetroxide
7. Carbon monoxide and carbon dioxide
8. Hydrochloric acid in the stomach
9. For cakes: sodium hydrogen carbonate

Elements should work

1. Iron (Fe) rusts.
2. Na (sodium) reacts with water.
3. The element carbon
4. Gold is element Au.
5. Copper (Cu) and zinc (Zn) make brass.

Chemistry traps should not work

They look like chemistry but are not. A red one here is a false icon.

1. Codes: a PS5, the W3C, Y2K, H1N1, an SN2 reaction, HeLa cells, NaN, a PoS terminal, the CoP26
2. Capitals: the CNN, the BBC, the NHS, KOH alone, COVID-19
3. One element: vitamin B12, an N95 mask, a V8 engine, O2 alone
4. Letters and arrows: Home → About, the chords C → G, S → N
5. Words: He is here, In 2024, As we know, No way, the US and UK, a CO2e figure, H2O.com
6. Not compounds: citric acid, the climate, copper sulfate (which copper?), 40% of people and 60% of cats
7. Not elements: Ohm (Ω), Pi (π), the element of surprise
8. Gases not near 100%: 50% nitrogen and 20% argon

Physics

Parts in series and parallel get their value; a resistor's colours or SMD code its resistance; a level in dBm or dBV its watts and volts. A click opens it in the Physics panel.

Series and parallel should work

1. Two in parallel: 4.7 kΩ ∥ 10 kΩ
2. In series and parallel: 100 + 220 || 330
3. With brackets: (1k + 2k2) || 4k7
4. Capacitors add in parallel: 10 µF || 22 µF
5. Coils: 10 mH ∥ 10 mH

Resistor colour codes should work

1. A resistor of brown, black, red and gold bands
2. Yellow-violet-red-gold
3. Read from its other end, a resistor: gold, red, violet, yellow
4. Five bands, 1%: a resistor of brown black black red brown

SMD resistor codes should work

1. An SMD resistor marked 473 is 47 kΩ.
2. R for the point: a 4R7 resistor, and a 0R22 shunt resistor
3. SMD resistor codes 103, 472 and 01C
4. Of 4 digits: the resistor's code is 4702.

Levels in decibels should work

1. The output is −30 dBm.
2. A transmitter of 30 dBW
3. Consumer audio, −10 dBV
4. A weak signal of 20 dBµV, or 0 dBmV

Physics traps should not work

They look like physics but are not. A red one here is a false icon.

1. Logic: a || b, x || 0
2. Two kinds of part: 10 µF || 10 kΩ
3. Colours of other things: the flag is red, white and blue; red, orange, yellow, green, blue, violet
4. Decibels alone: a 3 dB gain, 85 dB(A)
5. Names and cases of parts: resistors R1, R2 and R47, a 1206 resistor of 100 Ω
6. Codes of other parts: a capacitor marked 104

Biology

A sequence of DNA or RNA gets its length, GC and mass, a primer its melting point; a cross its ratios, and a cross of the X its sons and daughters. A click opens it in the Biology panel.

Sequences should work

1. A primer: 5′-AGCGGATAACAATTTCACACAGGA-3′
2. M13 forward, GTAAAACGACGGCCAGT
3. As GenBank writes it: atggctagca aaggagaaga acttttcact ggagttgtcc
4. GFP's start: ATGGCTAGCAAAGGAGAAGAACTTTTCACTGGAGTTGTCCCAATTCTTGTTGAATTAGATGGTGATGTTAATGGGCACAAATTTTCTG
5. An mRNA: AUGGCCAUUGUAAUGGGCCGCUGAAAGGGUGCCCGAUAG

Crosses should work

1. A dihybrid cross: AaBb × AaBb
2. A test cross: Tt x tt
3. True-breeding parents, AA × aa
4. Colour blindness, as Wikipedia writes it: X^BX^b × X^bY
5. In plain text: XAXa × XaY
6. Raised as text: XᴴXʰ × XʰY

Biology traps should not work

They look like biology but are not. A red one here is a false icon.

1. Genes that do not pair: AA x BB, Ab × Ab, XX x YY, AA x AA
2. A size: a 2 x 4
3. Short or of one base: GATTACA, ACCTGGTTGGCCAA, GGGGGGGGGGGGGGGGGGGG
4. Words: WHAT A CATCHY HEADLINE, a cat

Maths drawn as maths

KaTeX, as ChatGPT and Notion draw it should work

1. Fractions: $\frac{1}{2} + \frac{1}{3}$
2. Roots and powers: $\sqrt{144} + 2^{3}$
3. A cube root: $\sqrt[3]{27} \cdot 2$
4. Absolute values: $\left|-5\right| + |3|$
5. The constant e: $\ln(e^3) + e^0$
6. Trig in radians: $\cos(\pi) + \sin\left(\frac{\pi}{2}\right)$
7. A factorial: $10!$
8. A quadratic: $x^2 - 5x + 6 = 0$
9. Another letter: ${t^2} + 3t - 10 = 4 + 8t$
10. A double inequality: $-2 < 3x + 1 < 7$
11. Two equations: $\begin{cases} 2x + y = 5 \\ x - y = 1 \end{cases}$
12. An integral: $\int_0^1 x^2 \, dx$
13. An integral of sine: $\int_{0}^{\pi} \sin x \,\mathrm{d}x$
14. To infinity: $\int_{0}^{\infty} e^{-x^2} \, dx$
15. A sum: $\sum_{k=1}^{10} k^2$
16. The Basel sum: $\sum_{n=1}^{\infty} \frac{1}{n^2}$
17. A product: $\prod_{n=2}^{\infty} \left(1 - \frac{1}{n^2}\right)$
18. A limit: $\lim_{x \to 0} \frac{\sin x}{x}$
19. The limit of e: $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x}$
20. A derivative to draw: $\frac{d}{dx} x^3$
21. A differential equation: $\frac{dy}{dx} = x + y,\quad y(0) = 1$
22. A choice: $\binom{52}{5}$
23. A complex number: $(3+4i)(1-2i)$
24. A display formula, on its own line: $$\frac{-3 + \sqrt{9 + 16}}{2}$$
25. A function to draw: $y = \frac{1}{x}$
26. Another: $f(x) = e^{-x^2}$
27. Formula: $E = mc^2$
28. Formula: $F = \frac{G m_1 m_2}{r^2}$
29. Formula: $x = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$
30. Formula: $T = 2\pi\sqrt{\frac{L}{g}}$
31. Not ours: $e^{i\pi} + 1 = 0$ and $x_1 + x_2$
32. A function in pieces: $\begin{cases} x & x > 0 \\ -x & x \le 0 \end{cases}$

Matrices should work

1. 2×2: $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$
2. 3×3 named: $A = \begin{pmatrix} 4 & 7 & 2 \\ 3 & 6 & 1 \\ 2 & 5 & 3 \end{pmatrix}$
3. A determinant: $\begin{vmatrix} 2 & -1 \\ 5 & 3 \end{vmatrix}$
4. det of a fraction: $\det\begin{bmatrix} 2 & 0 \\ 0 & \frac{1}{2} \end{bmatrix}$
5. Singular: $\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$
6. Not square: $\begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \end{bmatrix}$
7. An old array: $\left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right]$
8. MathML: $\left[\begin{array}{cc}5& 1\\ 2& 3\end{array}\right]$
9. Of letters, not ours: $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$

MathJax 3, as Stack Exchange draws it should work

1. A fraction sum: $\frac{3}{4}-\frac{1}{8}$
2. A root: $\sqrt{2}\times \sqrt{8}$
3. An equation: $2^x=1024$
4. An integral: ${\int }_1^e\frac{1}{x}dx$
5. A sum: $\sum _{i=1}^{100}i$
6. Formula: $A=\pi r^2$
7. Formula: $PV=nRT$

MathJax 2, its TeX in a script after it should work

1. A sum: ½ + ¼$\frac{1}{2} + \frac{1}{4}$
2. An equation: 3x − 4 = 11$3x - 4 = 11$

Wikipedia and MathML should work

1. Wikipedia's markup: $3^2 + 4^2$3² + 4²
2. Wikipedia, a formula: $E=mc^{2}$E = mc²
3. MathML: $\sqrt{144}+2^3$
4. MathML, a fraction: $\frac{22}{7}-\pi$
5. MathML, sine applied: $\sin x=0.5$
6. MathML, an integral: ${\int }_0^{2}xdx$
7. MathML, a choice: $\left(\genfrac{}{}{0ex}{}{10}{3}\right)$

Select it yourself

These get no icon in the text. Select one with the mouse: an icon comes at the end of what you select, with the same tip and click. Or right-click a selection for Calculate “…”.

1. A whole number, its factors and bases: 360, or 97
2. A decimal, as a fraction: 0.375
3. A list, its statistics: 4 8 15 16 23 42
4. With commas: 1, 2, 3, 4
5. Codes: 0xFF, 0b1010, f81d4fae-7dec-11d0-a765-00a0c91e6bf6
6. A Unix time: 1700000000, and Base64: SGVsbG8gd29ybGQ=
7. A quantity, in every unit of its group: 5 km, or 72°F
8. A date, from today: March 3, 2024; a time on: in 90 days
9. A sum in a sentence: Twelve dozen is 12 × 12, a gross.
10. A letter of its own: 3t + 2 = 11
11. Chemistry alone: Fe, iron, KOH, O2, water
12. Colours alone: red red brown
13. An SMD code alone: 4R7, or R47
14. Logic of + alone: AB + BC
15. A protein: MQIFVKTLTGKTITLEVEPSDTIENVKAKIQDKEGIPPDQQRLIFAGKQLEDGRTLSDYNIQKESTLHLVLRLRGG

A column of a table, its statistics:

Not yet, and traps

The calculator can, the detector cannot yet not yet

1. Other letters in the text: 3t + 2 = 11, and 5n − 4 = 16
2. The constant e in the text: e^3 + ln(e^2)
3. Absolute bars in the text: |−5| + |3|
4. An integral in the text: ∫₀¹ x² dx
5. A limit in the text: lim x→0 sin(x)/x
6. A sum in words: Σ k² for k = 1 to 10
7. TeX source not drawn: \frac{1}{2} + \frac{1}{3}

Traps should not work

They look like maths but are not. A red one here is a false icon.

1. An ISBN: 978-3-16-148410-0
2. An address alone, no network: 192.168.0.1
3. A ratio: 3:2
4. A version: 2.0.1, and Windows 10 to 11
5. A score: 21-14
6. Coordinates: 40.7128° N, 74.0060° W, and point C(3, 4)
7. Speed: x2 playback
8. A deal: buy 2 + get 1 free
9. A size: a 1920 x 1080 screen, I am 5 ft tall, He is 6 ft 2 in
10. A temperature range: −5 to 10 °C
11. Times: from 9 to 5, from 9:00 to 17:00, open 24/7
12. A phone: +44 20 7946 0958, or 555-123-4567
13. An age rating: 18+
14. Solved already: x = 5, y = 3, and x = - 3
15. Code in text: i = i + 1
16. Roman numerals: chapter (i) 2, and i/2
17. Dates alone: 2024-10-18, on 3/4/2024 and 5/6/2024, 3 years later
18. Numbers of other things: see pages 10-20, version 1.2.3, COVID-19, the score was 3 - 1
19. Words: a loan to a friend, the 30 years war at 5%

let total = 2 + 3; // code is not read

A page that changes

Inside a box that scrolls

Maths that comes later

Pages change after they load. New maths gets its icon too: