[{"data":1,"prerenderedAt":173},["ShallowReactive",2],{"cheatsheet-significant-figures":3},{"sheet":4},{"sheetSlug":5,"topicSlugs":6,"primaryTopic":8,"title":9,"subtitle":10,"sections":11},"significant-figures",[7],"measurement-and-significant-figures",1,"Significant Figures & Measurement","Counting sig figs, rounding, scientific notation, SI units and prefixes, and dimensional analysis.",[12,24,32,40,48,62,75,84,92,148,156,164],{"heading":13,"kind":14,"items":15},"Exact vs measured","rules",[16],{"ref":17,"mode":20,"provenance":21,"text":22,"source_hash":23},{"type":18,"topic":8,"id":19,"field":18},"rule","NUM-R019","transform","original","Exact numbers (counts, definitions) have infinite sig figs and never limit precision. Measured values are inexact and do limit sig figs.","4ad02151e84dcfcbf14bca73b0221926460fe93097fc8b1c8c19201bc0b67b8c",{"heading":25,"kind":14,"items":26},"Counting significant figures",[27],{"ref":28,"mode":30,"provenance":21,"text":31},{"type":18,"topic":8,"id":29,"field":18},"NUM-R002","verbatim","Rules for counting significant digits: (1) All nonzero digits are significant. (2) Zeros between nonzero digits are always significant. (3) Leading zeros (before the first nonzero digit) are never significant. (4) Trailing zeros after a decimal point are significant. (5) Trailing zeros without a decimal point are NOT significant.",{"heading":33,"kind":14,"items":34},"Scientific notation",[35],{"ref":36,"mode":20,"provenance":21,"text":38,"source_hash":39},{"type":18,"topic":8,"id":37,"field":18},"NUM-R003","Write as a.bcd × 10ⁿ with exactly one nonzero digit before the decimal (1.23 × 10⁴, not 12.3 × 10³). Every digit shown in the coefficient is significant.","fd7a227d8bbbafb112c72e9db28f0cd6b3833a29b31ef1b54dfc35d78ebfb6ac",{"heading":41,"kind":14,"items":42},"Rounding",[43],{"ref":44,"mode":20,"provenance":21,"text":46,"source_hash":47},{"type":18,"topic":8,"id":45,"field":18},"NUM-R006","If the first dropped digit is ≥ 5, round the last kept digit up; cascade the carry (999.6 to 3 sf → 1.00 × 10³).","ea2f4ab0562c1f2d975e850e1341974eeef4a92f92153bdbb1a457b06f7c8a51",{"heading":49,"kind":50,"items":51},"Arithmetic with sig figs","steps",[52,57],{"ref":53,"mode":20,"provenance":21,"text":55,"source_hash":56},{"type":18,"topic":8,"id":54,"field":18},"NUM-R007","Add/subtract: round to the fewest decimal places among inputs. Multiply/divide: round to the fewest sig figs. Exact numbers never limit sig figs.","544f560a1b8790caf2aed52551fc314851c09729dfe453b414063e0f9fb1af24",{"ref":58,"mode":20,"provenance":21,"text":60,"source_hash":61},{"type":18,"topic":8,"id":59,"field":18},"NUM-R016","Multi-step: carry all digits through every step and round only the final result. Early rounding propagates error.","7d5f97b94e7c68cda301e72d3779bbd7ee0fd9868e91bd23046814905ac06692",{"heading":63,"kind":50,"items":64},"Dimensional analysis",[65,70],{"ref":66,"mode":20,"provenance":21,"text":68,"source_hash":69},{"type":18,"topic":8,"id":67,"field":18},"NUM-R010","Multiply by conversion factors equal to 1 with the old unit in the denominator so units cancel. Track every unit through every step.","9c57d0e231a16b8f3f655eb1d0cf58b929cf1f05a554986641f25037792daab2",{"ref":71,"mode":20,"provenance":21,"text":73,"source_hash":74},{"type":18,"topic":8,"id":72,"field":18},"NUM-R012","Squared/cubic units: raise the factor to that power (1 m = 10² cm → 1 m² = 10⁴ cm²). Compound units (g/cm³ → kg/m³): convert top and bottom separately.","d43c9b95895800922bd006a1a3e208b1d58bee21219a3d9088fe29d8d45b4a22",{"heading":76,"kind":77,"items":78},"Common SI base units","data",[79],{"ref":80,"mode":20,"provenance":21,"text":82,"source_hash":83},{"type":18,"topic":8,"id":81,"field":18},"NUM-R020","length → m · mass → kg · time → s · temperature → K · amount → mol. Liter, m/s, and kg/m³ are derived units.","0c1b1ba0719e2ec68f08421cf58ed1637e8f97d0c69fe5cbda23308f9362072a",{"heading":85,"kind":14,"items":86},"Temperature",[87],{"ref":88,"mode":20,"provenance":21,"text":90,"source_hash":91},{"type":18,"topic":8,"id":89,"field":18},"NUM-R013","K = °C + 273.15 (273.15 is exact). °C = (°F − 32) × 5/9. °F = °C × 9/5 + 32.","ae9235832f1a416f1b6756aebde9e17e113d96779695fcfc4843712eccb6a655",{"heading":93,"kind":77,"columns":94,"items":98},"Common SI prefixes",[95,96,97],"Prefix","Symbol","Value",[99],{"ref":100,"mode":20,"provenance":21,"rows":102,"source_hash":147},{"type":18,"topic":8,"id":101,"field":18},"NUM-R009",[103,107,111,115,119,123,127,131,135,139,143],[104,105,106],"tera","T","10¹²",[108,109,110],"giga","G","10⁹",[112,113,114],"mega","M","10⁶",[116,117,118],"kilo","k","10³",[120,121,122],"deci","d","10⁻¹",[124,125,126],"centi","c","10⁻²",[128,129,130],"milli","m","10⁻³",[132,133,134],"micro","μ","10⁻⁶",[136,137,138],"nano","n","10⁻⁹",[140,141,142],"pico","p","10⁻¹²",[144,145,146],"femto","f","10⁻¹⁵","832d6943dd049c880c867b7d90797bf817e85ee4f70c1b3bc0e302f80770ade3",{"heading":149,"kind":77,"items":150},"Common conversions",[151],{"ref":152,"mode":20,"provenance":21,"text":154,"source_hash":155},{"type":18,"topic":8,"id":153,"field":18},"NUM-R014","Exact (defined): 1 in = 2.54 cm · 1 lb = 0.45359237 kg · 1 mi = 1609.344 m · 1 gal = 3.785411784 L · 1 mL = 1 cm³ · 1 Å = 10⁻¹⁰ m · 1 atm = 101,325 Pa = 760 torr · 1 cal = 4.184 J (thermochemical). Approximate: 1 atm ≈ 760 mmHg.","d8c80682ce0abccb939b3e9c1d9d5661e799026e8d30df9cd03edbf135e5e1a9",{"heading":157,"kind":14,"items":158},"Density as a conversion",[159],{"ref":160,"mode":20,"provenance":21,"text":162,"source_hash":163},{"type":18,"topic":8,"id":161,"field":18},"NUM-R021","Density links mass and volume; write it in the orientation that cancels the known unit and leaves the unknown.","672c4f34fa0e73aa62487b7c90c77e3ac572cdfcf566a9167cdf2468df48796e",{"heading":165,"kind":166,"items":167},"Watch out","pitfalls",[168],{"ref":169,"mode":20,"provenance":21,"text":171,"source_hash":172},{"type":18,"topic":8,"id":170,"field":18},"NUM-R005","You cannot add sig figs a measurement does not have. To show a whole-number result's trailing zeros are significant, use a trailing decimal (300.) or scientific notation.","7f47e065dd21e6fb0fb58783e794296385d3dd09320b50420f6796b7e229491e",1787246033543]