Fractions, decimals & series

Write 1/19 or 1/29 as its repeating decimal — no long division

Getting the full repeating block of 1/19 by long division is 18 grinding steps.

Spot it when

You need the decimal expansion of 1/n where n ends in 9 (19, 29, 49…).

e.g.

1/19

Worked examples

The method builds the answer directly — worked results:

1/19

0.‾052631578947368421 (multiplier 2, period 18)

1/29

0.‾0344827586206896551724137931 (multiplier 3, period 28)

1/49

0.‾020408163265306122448979… (multiplier 5)

Real moments — exams, homework, competitions — where this shortcut earns its keep.

Homework

A number-theory exercise asks for the full period of 1/19 and you do not want to do 18 divisions.

becomes

1/19

⚡

Use multiplier 2 (one more than 1): build digits right-to-left → 0.052631578947368421.

Competition

A competition asks for the 10th digit of 1/29 — you only need the cycle, not the whole division.

becomes

1/29

⚡

Multiplier 3 generates the 28-digit cycle digit by digit; just count to the 10th.

In class

Converting 1/49 for a recurring-decimal proof in class.

becomes

1/49

⚡

Multiplier 5 (one more than 4): each step multiplies the running digit, carrying as you go.

एकाधिकेन पूर्वेण

Ekadhikena Pūrveṇa

“By one more than the previous one”

When the last digit is 5, the prefix does all the work — multiply it by itself plus one.

Conventional

$85^2 = 85\times 85$ via long multiplication (6+ steps, carrying).

Vedic

$8\times 9 = 72$ , append $25 \to 7225$ (1 mental step).

⚡ 1 step vs 6+ steps for squaring numbers ending in 5.

Use when

Avoid when