doing math

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Orders of magnitude

level 1 course

Get a feel for thousands, millions, and billions.

Best in your head · no pen, no calculator why?

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the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

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

Big named numbers, or writing a number as a × 10ⁿ, where a is at least 1 and under 10.

  1. Turn each name into zeros: ten 1, a hundred 2, thousand 3, million 6, billion 9, trillion 12.
  2. Multiplying adds the zero counts. Dividing subtracts them.
  3. For a × 10ⁿ, move the decimal point until one nonzero digit is in front. n counts the moves.
  4. Below 1, the point moves right, so n is negative.
worked example

Example: Write 45,000,000 as a × 10ⁿ with 1 ≤ a < 10. What is n?

  1. Move the point left until one digit is in front: 4.5.
  2. That takes 7 moves, so n is 7.

Answer: 7

Daily or weekly to yearly

An amount per day or per week, over a whole year.

  1. Per day: multiply by 365, done as × 300 plus × 65.
  2. Per week: multiply by 52, done as × 50 plus × 2.
  3. Add the two parts.
worked example

Example: You spend $4 a day on coffee. How much is that in a year of 365 days?

  1. $4 × 300 = $1,200.
  2. $4 × 65 = $260.
  3. $1,200 + $260 = $1,460.

Answer: $1,460

Turn a count into time

Turning seconds, minutes or hours into days or years.

  1. A day has 86,400 seconds, 1,440 minutes or 24 hours.
  2. A year has about 31.5 million seconds, 525,600 minutes or 8,760 hours.
  3. Divide the count by the right one. Cancel matching zeros first.
worked example

Example: A year has about 31.5 million seconds. About how many years is a billion seconds? Round to the nearest year.

  1. Divide 1,000,000,000 by 31,500,000.
  2. Divide both by a million: 1,000 ÷ 31.5.
  3. 1,000 ÷ 31.5 ≈ 32, so about 32 years.

Answer: 32

Tips by skill

  • TipBig-number names: Turn each name into its count of zeros. Add them to multiply, subtract them to divide, then name the result.
  • TipDaily to yearly: Per day, multiply by 365 (× 300 plus × 65). Per week, multiply by 52 (× 50 plus × 2).
  • TipHow long is that?: Divide by a day (86,400 seconds, 1,440 minutes, 24 hours) or a year (about 31.5 million seconds, 525,600 minutes, 8,760 hours).
  • TipPowers of ten: Count the places the decimal point moves to leave one nonzero digit in front, not the zeros. Moving right makes n negative.

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.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

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