The Calendar That Took Four Tries to Get Right Another Peek Beyond the Point

Jonali buys cinnamon (50g), cumin (100g), cardamom (25g), and pepper (250g), and working with these weights in kilograms (0.05kg, 0.1kg, 0.025kg, 0.25kg) sets up the real question this chapter answers: how does multiplication actually work once decimals are involved? Arshad buying 5 pens at Rs 9.5 each computes 9.5 x 5 by rewriting 9.5 as the fraction 95/10, giving (95x5)/10 = 475/10 = 47.5 -- the fraction form makes the decimal multiplication provably correct, not just a memorised shortcut.

Multiplying 5.8 x 1.24 by first multiplying the whole numbers 58 x 124 = 7192, then counting a total of 3 digits after the decimal points in both original numbers (1 from 5.8, 2 from 1.24), places the decimal point 3 places from the right in the result: 7.192. A genuinely important pattern emerges when checking whether a product is always bigger than the numbers being multiplied: 2.25 x 8 = 18 (bigger, as expected), but 0.25 x 8 = 2 (SMALLER than 8), and 0.25 x 0.8 = 0.2 (smaller than both factors) -- multiplying by a number less than 1 always shrinks the result, exactly the same surprising fact seen already with fractions.

Dividing 237 by 8 using place-value long division, the exact same method used for whole numbers, simply continues past the ones place: after reaching a remainder of 5 ones, regrouping into 50 tenths continues the division, placing a decimal point in the quotient at exactly that moment, and continuing through hundredths and thousandths gives 237 / 8 = 29.625 precisely. A shopkeeper's 9.5kg of sugar split evenly into 4 bags uses the identical method, reaching 2.375kg per bag.

Ravi's scooter trip, 126km in 2.5 hours, needs 126 divided by 2.5 to find average speed -- and dividing by a decimal is handled by converting the divisor into a fraction first (2.5 = 25/10), then multiplying by its reciprocal: 126 x 10/25 = 1260/25 = 50.4 km/hr. The same technique, multiplying both dividend and divisor by the same power of 10, works for any decimal divisor: 4.68 / 0.13 becomes 468/13 once both are scaled up by 100, turning an awkward decimal division into an ordinary whole-number one.

Dividing 10 by 3 using long division never terminates, producing 3.333... forever, the remainder always cycling back to the same value at every step. Dividing 1 by 7 produces 0.142857142857..., and the six-digit block 142857 carries a genuinely remarkable property: multiplying it by 1, 2, 3, 4, 5, or 6 just rotates its own six digits into a new starting position (142857 x 3 = 428571, still the same six digits, just shifted). In 1927, the Austrian mathematician Emil Artin conjectured that infinitely many such cyclic numbers exist among the reciprocals of primes -- and nearly a century later, this conjecture is still not fully proven, remaining a genuinely open question in mathematics today.

Earth actually orbits the Sun in 365.2422 days, not a clean 365 -- ignored, this shortfall of 0.2422 days per year builds up to over 24 days after just a century. The first fix, a leap year every 4 years (366 days instead of 365), overcorrects slightly: over exactly 4 years, it adds 1461 days against an actual 1460.9688, and over 100 years, 36,525 days against an actual 36,524.22 -- now over-compensating by more than a third of a day per century. The second fix removes the leap day in century years not divisible by 400 (so 1900 wasn't a leap year, but 2000 was), bringing 1000 years to 3,65,240 days against an actual 3,65,242.2, still short by 2.2 days. The current rule -- divisible by 4, except century years, unless also divisible by 400 -- brings 1000 years to within just 0.2 days of the true value, a gap the calendar's own designers judged small enough not to bother correcting any further, at least for now.