Chapter 4: Another Peek Beyond the Point
Decimal Multiplication, Place Value Long Division, Repeating Cycles & The Gregorian Leap Year Algorithm
✖️ Decimal Multiplication
Multiply numbers ignoring decimals, then place the decimal point by adding the decimal digits of both factors.
Multiplying by 10, 100, 1000 shifts the decimal point right by 1, 2, 3 places.
Multiplying decimals between 0 and 1 results in a product smaller than both factors (e.g. 0.25 × 0.8 = 0.2).
➗ Decimal Division
1. Place Value Long Division:
Regroup remaining Ones into Tenths, Tenths into Hundredths, and so on. Place a decimal point in the quotient the moment you cross the Ones place.
2. Decimal Divisor Conversion:
Multiply both dividend and divisor by 10, 100, or 1000 to convert the divisor into a whole number before dividing (e.g. 4.68 ÷ 0.13 = 468 ÷ 13 = 36).
3. Powers of 10 Division Shift:
Dividing by 10, 100, 1000 shifts the decimal point left by as many places as there are zeroes.
🔁 Infinite Cycles & Patterns
Fractions like 10/3 = 3.333... never end because a remainder of 1 keeps recurring infinitely.
1/7 = 0.142857... Multiplying 142857 by 2, 3, 4, 5, 6 cycles the exact same digits! Emil Artin conjectured infinitely many such cyclic numbers exist.
Fractions terminate into exact decimals if and only if the denominator factors solely into powers of 2 and 5 (since 2 × 5 = 10).
🌍 Gregorian Leap Year Algorithm (Earth Year = 365.2422 Days)
• The Astronomical Gap: Earth revolves around the Sun in 365.2422 days. Calendar years (365 days) leave a deficit of 0.2422 days/year (24.22 days per century).
• Rule 1 (+1 day every 4 years): Gives 25 leap days/century (36,525 days) → Overcompensated by 0.78 days!
• Rule 2 (No leap day on century years): Divisible by 100 = 365 days (gives 36,524 days/century) → Undercompensated by 2.2 days per 1000 years!
• Rule 3 (Leap day back on 400th year): Divisible by 400 = 366 days. In 1000 years, error is just 0.2 days (3,65,242 calendar days vs 3,65,242.2 actual solar days).
⚠️ Common Decimal Traps
Trap 1: Trailing Zeros in Multiplication Count
When calculating 0.25 × 0.8, first multiply 25 × 8 = 200, then count 3 decimal places to get 0.200 = 0.2. Do not drop the zero before placing the decimal point!
Trap 2: Dividing by Numbers < 1
Division does NOT always make a number smaller! When the divisor is between 0 and 1, the quotient is strictly greater than the dividend (e.g. 128 ÷ 0.4 = 320 > 128).