Orders of magnitude
Get a feel for thousands, millions, and billions.
Best in your head · no pen, no calculator why?
the idea
Big numbers climb a ladder. A thousand has 3 zeros, a million 6, a billion 9 and a trillion 12, so each step up is × 1,000. To multiply two of them, add their zeros: a million times a thousand has 6 + 3 = 9 zeros, a billion.
To feel how big a number is, turn it into time or money you know. A million seconds is under 12 days. A billion seconds is about 32 years.
techniques
Count zeros, then moves
- Turn each name into zeros: ten 1, a hundred 2, thousand 3, million 6, billion 9, trillion 12.
- Multiplying adds the zero counts. Dividing subtracts them.
- For a × 10ⁿ, move the decimal point until one nonzero digit is in front. n counts the moves.
- Below 1, the point moves right, so n is negative.
worked example
Write 45,000,000 as a × 10ⁿ with 1 ≤ a < 10. What is n?
- Move the point left until one digit is in front: 4.5.
- That takes 7 moves, so n is 7.
Answer: 7
Daily or weekly to yearly
- Per day: multiply by 365, done as × 300 plus × 65.
- Per week: multiply by 52, done as × 50 plus × 2.
- Add the two parts.
worked example
You spend $4 a day on coffee. How much is that in a year of 365 days?
- $4 × 300 = $1,200.
- $4 × 65 = $260.
- $1,200 + $260 = $1,460.
Answer: $1,460
Turn a count into time
- A day has 86,400 seconds, 1,440 minutes or 24 hours.
- A year has about 31.5 million seconds, 525,600 minutes or 8,760 hours.
- Divide the count by the right one. Cancel matching zeros first.
worked example
A year has about 31.5 million seconds. About how many years is a billion seconds? Round to the nearest year.
- Divide 1,000,000,000 by 31,500,000.
- Divide both by a million: 1,000 ÷ 31.5.
- 1,000 ÷ 31.5 ≈ 32, so about 32 years.
Answer: 32
watch out for
- Jumping two steps up the ladder: a million times a thousand is a billion, not a trillion. Each step is × 1,000.
- Counting zeros instead of moves. 45,000,000 has 6 zeros, but the point moves 7 places, so n is 7.
- Dropping a factor of ten: a year is about 31.5 million seconds, not 315 million.
practice
Big-number names
worked example
A billion times 100 is:
- a hundred million
- a hundred billion
- a hundred trillion
- a trillion
Answer: a hundred billion
- Count zeros: a billion has 9 and 100 has 2. 9 + 2 = 11 zeros: a hundred billion.
Daily to yearly
worked example
A dog walker costs $135 a week. How much is that in a year of 52 weeks?
Answer: $7,020.00
- $135 × 52 = $135 × 50 + $135 × 2 = $6,750 + $270 = $7,020.
How long is that?
worked example
About how long is 2 million hours?
- 230,000 years
- 2,300 years
- 23,000 years
- 230 years
Answer: 230 years
- A year has 8,760 hours (365 × 24).
- 2,000,000 ÷ 8,760 ≈ 228, so about 230 years.
Powers of ten
worked example
Write 4 as a × 10ⁿ with 1 ≤ a < 10. What is n? (Use a minus sign if n is negative.)
Answer: 0
- 4 is already between 1 and 10, so the point doesn't move.
- So 4 = 4 × 10⁰, and n = 0.