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Numbers to a million and beyond

Numbers · level 3, lesson 1 of 7

A question to start

If you counted out loud, one number every second, without stopping to eat or sleep, how long would it take you to count to a million?

Most people guess a few hours, or maybe a day. The real answer is about 11 and a half days. And to count to a billion the same way would take about 32 years. A million and a billion sound like close cousins, but one is a thousand times bigger than the other. This lesson is about reading, writing and feeling numbers this big.

The idea

Every place is ten times the one before

You already know the places up to thousands: ones, tens, hundreds, thousands. Each place is worth ten times the place to its right. Ten ones make a ten, ten tens make a hundred, ten hundreds make a thousand. The rule never stops:

  • ten thousands make ten thousand (10 000)
  • ten ten-thousands make a hundred thousand (100 000)
  • ten hundred-thousands make a million (1 000 000)

A million is a 1 followed by six zeros. It is a thousand thousands.

Here is a way to picture it. A small cube with sides of 1 cm is one. A row of ten cubes is a ten. A flat square of ten rows is a hundred. A stack of ten flats is a cube 10 cm on each side: a thousand. Now make a row of ten of those big cubes, a flat of ten rows, and a stack of ten flats. You get a cube 1 metre on each side, big enough to sit inside, and it holds a million little 1 cm cubes.

1 10 100 1000 (ten flats stacked)

Periods: reading big numbers in groups of three

Long numbers are hard to read: try 4 852 017 306 without help. The trick is to split the digits into groups of three, starting from the right. Each group is called a period, and each period has a name.

Billions Millions Thousands Ones
4 852 017 306

Inside each period the places are always hundreds, tens, ones. So you read each group as an ordinary number from 0 to 999, then say the period's name:

4 852 017 306 is "four billion, eight hundred fifty-two million, seventeen thousand, three hundred six".

Notice that we never say "ones" at the end; the last group is just read as a number. And notice the zero in 017: it holds the hundreds place open. Without it, the number would shrink.

Different countries mark the groups differently. In Canada you will often see a thin space (4 852 017 306) because a comma is used as the decimal point in French. In the United States people usually write commas (4,852,017,306). Both mean exactly the same number. On this site we use spaces.

How many zeros?

Name Written out Zeros How many thousands
one thousand 1 000 3 1
one million 1 000 000 6 a thousand thousands
one billion 1 000 000 000 9 a thousand millions
one trillion 1 000 000 000 000 12 a thousand billions

Each period name is a thousand times the one before, because each period is three places, and three places is \(10 \times 10 \times 10 = 1000\).

Comparing and ordering big numbers

Which is bigger, 2 989 999 or 3 000 001? To compare two whole numbers:

  1. Count the digits. A number with more digits is bigger (as long as neither starts with zero). Both of these have 7 digits, so look further.
  2. Compare from the left, place by place. The first place where they differ decides. Millions: 2 against 3. So 3 000 001 is bigger, even though the other number has lots of 9s.

Why from the left? Because the leftmost place is worth the most. One extra million is more than everything all the smaller places can add up to: the biggest the smaller places can ever make is 999 999, which is still less than a million. So once the millions differ, nothing to the right can change the result.

We write \(3\,000\,001 > 2\,989\,999\). The sign \(>\) means "is greater than", and \(<\) means "is less than". The open end always faces the bigger number.

Rounding to any place

Big numbers are often more than we need. A newspaper says "about 3.5 million people", not "3 456 789 people". Rounding means swapping a number for a nearby simpler one.

To round to a place (say, to the nearest hundred thousand):

  1. Find the two "round" numbers the number sits between. 3 456 789 sits between 3 400 000 and 3 500 000.
  2. Find the halfway point: 3 450 000.
  3. If the number is at or past halfway, round up; if it is before halfway, round down.
3 400 000 3 450 000 3 500 000 3 456 789 this half rounds up

The quick way: look at the digit just to the right of the place you are rounding to. If it is 5 or more, round up. If it is 4 or less, round down. For 3 456 789 to the nearest hundred thousand, the hundred-thousands digit is 4 and the digit to its right is 5, so round up to 3 500 000.

Why does "5 or more" mean "at or past halfway"? The digit to the right tells you how far along you are in tenths of the gap. A 5 means you have gone at least 5 tenths of the way, which is at least halfway.

The same number rounded to different places:

Round 3 456 789 to the nearest Answer
ten 3 456 790
thousand 3 457 000
hundred thousand 3 500 000
million 3 000 000

How big is a million, really?

Big numbers are easy to say and hard to feel. Here are three ways to feel them.

  • Time. A day has \(24 \times 60 \times 60 = 86\,400\) seconds. A million seconds is \(1\,000\,000 \div 86\,400\), which is about 11.6 days. A billion seconds is about 31.7 years.
  • Steps. A child's step is roughly half a metre. A million steps is about 500 km: a long, long walk.
  • Space. Mars goes round the Sun at an average distance of about 228 million km. Earth is about 150 million km from the Sun. Written out, 228 million is 228 000 000: the "228" sits in the millions period.

Worked examples

1. Read and write. Write 60 040 500 in words.

Split into periods from the right: 60 | 040 | 500. That is "sixty million, forty thousand, five hundred".

2. Words to digits. Write "seven billion, three million, twenty" in digits.

Make four periods and fill each one: billions 7, millions 003, thousands 000, ones 020. So 7 003 000 020. Every period except the first must have exactly three digits; the zeros keep each digit in its proper place.

3. Order. Put in order from smallest to biggest: 1 090 000, 1 009 000, 1 900 000, 190 000.

190 000 has only 6 digits, so it is smallest. The other three have 7 digits and the same millions digit (1), so look at the hundred thousands: 0, 0, 9. 1 900 000 is biggest. For the two with 0, compare ten thousands: 9 against 0. So 1 090 000 > 1 009 000. Order: 190 000, 1 009 000, 1 090 000, 1 900 000.

4. Round. Round 7 649 501 to the nearest thousand.

The thousands digit is 9 (in 7 649 501). The digit to its right is 5, so round up. But 9 + 1 = 10, which does not fit in one place, so it carries: 649 thousand becomes 650 thousand. Answer: 7 650 000. Check: it sits between 7 649 000 and 7 650 000, and 501 is past the halfway mark of 500.

5. A million seconds. Show that a million seconds is between 11 and 12 days.

11 days is \(11 \times 86\,400 = 950\,400\) seconds, which is less than a million. 12 days is \(12 \times 86\,400 = 1\,036\,800\) seconds, which is more. So a million seconds is between 11 and 12 days.

Try it

  1. Write 5 300 070 in words.
  2. Write "twelve million, four hundred thousand, nine" in digits.
  3. What is the 8 worth in each number? (a) 8 000 000 (b) 48 000 (c) 2 800 000 000
  4. Put in order, smallest first: 3 333 000, 3 033 300, 3 330 300, 333 330.
  5. Round 4 718 260 to the nearest (a) thousand, (b) hundred thousand, (c) million.
  6. A number rounds to 600 000 when rounded to the nearest hundred thousand. What is the smallest whole number it can be? The biggest?
  7. How many thousands are in a million? How many millions in a billion? Explain with places.
  8. Roughly how many days is 10 million seconds? (Use: a million seconds is about 11.6 days.)
Answers
  1. Periods: 5 | 300 | 070. "Five million, three hundred thousand, seventy."
  2. Millions 12, thousands 400, ones 009: 12 400 009.
  3. (a) 8 million. (b) 8 thousand (8 000). (c) 8 hundred million (800 000 000).
  4. 333 330 has 6 digits, so it is smallest. The others have 7 digits and start with 3; compare the hundred thousands: 3, 0, 3. So 3 033 300 is next. Then 3 333 000 and 3 330 300: compare ten thousands, 3 and 3, then thousands, 3 and 0. Order: 333 330, 3 033 300, 3 330 300, 3 333 000.
  5. (a) Thousands digit 8, next digit 2: round down to 4 718 000. (b) Hundred-thousands digit 7, next digit 1: round down to 4 700 000. (c) Millions digit 4, next digit 7: round up to 5 000 000.
  6. The halfway points around 600 000 are 550 000 and 650 000. 550 000 rounds up to 600 000, so the smallest is 550 000. 650 000 would round up to 700 000, so the biggest is 649 999.
  7. A million is 1 000 000 and a thousand is 1 000. Moving three places left multiplies by \(10 \times 10 \times 10 = 1000\), so there are 1000 thousands in a million. In the same way, a billion is three places further left than a million, so there are 1000 millions in a billion.
  8. 10 million is ten times a million, so about \(10 \times 11.6 = 116\) days: nearly four months. (Exactly: \(10\,000\,000 \div 86\,400\) is about 115.7 days.)

Common mistake

Dropping the zeros inside a period. "Three million, five thousand, twenty" is 3 005 020, not 3 5 20 or 3 520. Every period after the first needs exactly three digits, so fill empty places with zeros.

Comparing by the first digit only. 980 000 is not bigger than 1 200 000 just because 9 is bigger than 1. Count the digits first: the 7-digit number wins.

Where it leads