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· 7 min read
The phase cycle takes 29.53 days but the orbit takes only 27.32, and the 2.21-day gap is the entire explanation. With the eight phases, their illumination, when each rises and sets, and how they flip south of the equator.
· 7 min read
Every problem in the chapter is the same four moves. The mole ratio is the hinge, and it is the step students lose — worked here end to end, limiting reactant and percent yield included.
· 7 min read
Four routes for four kinds of solution, each worked with real numbers. Including the part most pages leave out: the square-root formula for a weak acid is an approximation with a validity limit, and it fails quietly.
· 6 min read
Multiply each atomic weight by its subscript and add. The only three places the arithmetic goes wrong are a subscript after a closing bracket, the hydrate dot, and a capital letter typed as a lowercase one.
· 14 min read
A four-year public in-state budget is 30,990 dollars today. Eighteen years out at 4 percent cost inflation the same four years cost 266,593 — 2.15 times the sticker you can see now. And the sticker was never the price: the average net cost of attendance is 21,340.
· 4 min read
A fair draw needs more than a random number: equal probability, no reuse of a biased method, and a result someone else can check. Here is how to run one.
· 4 min read
The Fibonacci sequence adds each pair of terms to make the next; the ratio of neighboring terms homes in on the golden ratio φ ≈ 1.618. See how, and where the pattern turns up.
· 17 min read
We measured the coverage curve of French on 313 million words of subtitles: the top 1,000 forms account for 80.77% of running text. Then we took a normal French sentence in which 80% of the tokens are inside that top 1,000, and showed that a reader who knows every one of them learns nothing from it.
· 4 min read
A z-score is z = (x − μ) / σ — how many standard deviations a value sits from the mean. Learn how to compute it, read the standard normal curve, and turn it into a percentile.
· 3 min read
Convert between degrees and radians with radians = degrees × π/180 (and back with × 180/π), understand why 180° = π, and learn the common angles by heart.
· 5 min read
A grade average is the weighted mean of your grades: each grade times its weight, summed, divided by the total weight. Learn the method, a worked example, and how GPA differs from a percentage.
· 4 min read
Use the distance formula d = √((x₂−x₁)² + (y₂−y₁)²) to measure the straight-line distance between two points on a plane, with a worked example and the midpoint too.
· 4 min read
Daylight saving time shifts clocks an hour in spring and back in autumn. Learn spring-forward and fall-back, why one day is 23 hours and another 25, and how US and EU dates differ.
· 4 min read
Find the slope of a line from two points with m = (y₂ − y₁) / (x₂ − x₁), and read what a positive, negative, zero, or undefined slope tells you.
· 5 min read
Converting between time zones is adding or subtracting the difference in UTC offsets, then handling any day rollover. Learn the method, a worked example, and the DST caveat.
· 5 min read
Solve any quadratic equation with x = (−b ± √(b² − 4ac)) / 2a, and read the discriminant to know how many real solutions to expect.
· 4 min read
Voltage drop is the voltage lost along a wire due to its resistance. Learn how length, gauge, and current affect it, and why it matters for your circuits.
· 5 min read
Your age in days is the whole number of days between your birth date and today. Learn how to count it by hand, convert it to weeks and months, and why leap years matter.
· 4 min read
Round to any decimal place or number of significant figures, and learn when to use round half up versus round half to even to avoid bias.
· 4 min read
Building a countdown to a target date is just counting whole days from today to the target, then splitting that total into weeks. Learn the method with a worked example.
· 5 min read
Size the series resistor for any LED with one formula: R = (Vs − Vf) / I. Follow the steps, see a worked 5 V example, and pick the right resistor.
· 4 min read
Learn what counts as a significant figure, the rules for leading, trailing, and captive zeros, and how sig figs keep your calculated answers honest.
· 4 min read
Horsepower measures the rate of doing work. Learn where the unit came from, how mechanical hp differs from metric PS/CV, and how both relate to kilowatts.
· 4 min read
Break any whole number into its prime building blocks with a factor tree or repeated division — and write the result as a clean prime factorization with exponents.