Place value to millions
Every digit has a value that depends on where it sits. Once you can name those places, big numbers stop feeling big.
Reading the columns
Digits sit in columns, and the columns have names: ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions.
Reading a long number is just reading the columns from the left. The chart below holds 3 472 016.
The same digit, four different values
A 4 is not always worth four. It is worth four only in the ones column.
Move it one column left and it is worth forty. Another column and it is four hundred. In the ten thousands column that same little 4 is worth forty thousand.
The digit never changes. Its address does.
Partitioning makes arithmetic easier
Breaking a number into its parts turns one hard sum into a few easy ones.
3 472 is 3 000 and 400 and 70 and 2. To add 200 to it, you only have to touch the 400. To take away 70, you only touch the 70.
This is also why the written methods work — they are partitioning with the working tidied away.
Zero is holding a place
In 3 472 016 there is a 0 in the hundreds column. It is not nothing — it is the thing stopping every other digit sliding one place right.
Take it out and you get 347 216, a completely different number. A zero in the middle of a number is doing real work.
Read it in groups of three
A seven-digit number looks frightening until you notice it is three small numbers wearing a coat.
Split the digits into groups of three from the right. Read each group as if it were on its own, then say the name of the group: million, thousand, and nothing at all for the last one.
So 4 205 037 is four — million — two hundred and five — thousand — and thirty-seven. The spaces in the written number are there to mark exactly these groups.
Comparing big numbers without counting digits twice
To decide which of two numbers is larger, line them up from the right and count how many digits each has. More digits wins, every time, for whole numbers.
Only when the digit counts match do you compare from the left, one column at a time, stopping at the first column where they differ.
That first difference decides it. Nothing to the right of it can change the answer, however large those digits look — which is why 89 999 is smaller than 100 002 despite all those nines.
Worked examples
In the number 254 608, what is the value of the 5?
Name the columns from the right: 8 ones, 0 tens, 6 hundreds, 4 thousands, 5 ten thousands, 2 hundred thousands. The 5 is in the ten thousands column, so it is worth 50 000 — not 5.
Answer: 50 000
Write 6 000 + 300 + 9 as one number.
There are 6 thousands, 3 hundreds, no tens and 9 ones. The tens column still needs filling, so a zero goes there to hold the place: 6 309.
Answer: 6 309
Which is larger, 89 999 or 100 002?
Do not compare digit by digit from the left — count the digits first. 89 999 has five digits and 100 002 has six, so 100 002 is larger. A six-digit number always beats a five-digit one.
Answer: 100 002
What is 1 000 more than 47 300?
Only the thousands column is affected. 47 300 has 7 thousands, so it becomes 8 thousands and the rest stays put: 48 300.
Answer: 48 300
Write 3 040 600 in words.
Split into groups of three from the right: 3 | 040 | 600. Read each with its name: three million, forty thousand, six hundred. The zero in the thousands group is not read out, but it holds the place.
Answer: Three million, forty thousand, six hundred
What is the value of the 7 in 2 706 145?
Group from the right: 2 | 706 | 145. The 7 sits in the hundreds column of the thousands group, so it means seven hundred thousand.
Answer: 700 000
Which is larger, 908 000 or 1 010 000?
Count the digits: six against seven. More digits wins for whole numbers, so there is no need to compare any columns.
Answer: 1 010 000
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