1% better every day: the exact arithmetic, with a calculator

The formula, step by step

If something grows by a fixed percentage each day, the factor after n days is (1 + r) to the power of n, where r is the daily rate written as a decimal. For 1 percent a day, r is 0.01 and the factor is 1.01 multiplied by itself n times. After one day it is 1.01. After two days it is 1.01 x 1.01 = 1.0201. After 100 days it is 1.01 to the power of 100, which equals 2.7048 to four decimals. After 365 days it is 37.7834.

The mirror calculation uses 0.99. A 1 percent daily decline gives 0.99 to the power of 365, which equals 0.0255, so almost nothing is left. The calculator above computes these with exact integer arithmetic, so no rounding error builds up across hundreds of multiplications.

For comparison, the third line shows adding 1 percent of the starting amount each day with no compounding. After 365 days that gives 1 + 0.01 x 365 = 4.65. The gap between 4.65 and 37.7834 is what compounding adds.

Is it true that you get 37 times better in a year?

No. The arithmetic is exact, but people do not improve by a fixed percentage every day for a year. Skills level off, gains get harder, and quality is not a number that multiplies. The figure is a way to see why small, repeated improvements matter, not a measurement of what you will achieve.

The honest use of the number is as an illustration of direction. Tiny changes repeated daily add up, and tiny losses repeated daily add up the other way.

Exact values for common time spans

Factors at 1 percent per day, rounded to four decimals.

Why real progress does not follow the curve

The arithmetic is a model. Four reasons the real world differs:

• Habits level off. In the 2010 Lally study, automaticity rose quickly at first and then flattened towards a plateau rather than growing without limit.
• Gains shrink. Beginners improve fast, experts improve slowly, so a fixed daily percentage is unrealistic for any skill.
• Most outcomes are not numbers. Fitness, writing and calm do not multiply the way money does.
• Consistency is the real lever. The useful message is to show up daily with a small, repeatable action, not to hit a daily percentage.

How to use the idea without the myth

1. Pick one small daily action. Choose something you can repeat tomorrow: one page, five squats, ten minutes of study.
2. Mark each day on a chain. A visible record turns the idea of small gains into something you can see.
3. Watch the direction, not the number. After a month, ask whether you are doing more of what you want than before. That is the useful question.
4. Allow for plateaus. A flat week is normal. Keep the routine and change one thing if it stays flat.

Common questions

What is 1.01 to the power of 365?

It is 37.7834 to four decimals, calculated with exact integer arithmetic. It means a quantity that grows 1 percent each day ends the year about 37.8 times its starting size.

What is 0.99 to the power of 365?

It is 0.0255 to four decimals. A quantity that falls 1 percent each day keeps about 2.6 percent of its starting size after a year.

Does 1% better every day really work for habits?

The arithmetic is correct, but improvement in people is not a fixed daily percentage. Use the figure as an illustration of why small repeated actions add up, not as a target.

Can I try a different percentage?

Yes. Enter any daily rate from 0.01 to 10 percent and any number of days up to 3,650. The calculator shows the better, worse and no-compounding results together.

Sources

Where the facts on this page come from:

• How are habits formed: Modelling habit formation in the real world, European Journal of Social Psychology, Lally, van Jaarsveld, Potts and Wardle, 2010
• Making health habitual: the psychology of habit-formation and general practice, British Journal of General Practice, Gardner, Lally and Wardle, 2012