Primary & Middle School Mathematics · Grades 1–9

14Numbers up to 10 000

One thousand, two thousand, three thousand … big numbers are built from small ones, using a wonderful idea: the place of a digit tells its value. This chapter teaches how to read, write, compare and order numbers up to $10\,000$ .

14.1 Reading and writing numbers

Definition 14.1 (Place value)

A number is written with the digits $0, 1, 2, \dots, 9$ . Reading from the right, the places are: units, tens, hundreds, thousands. In $3\,254$ :

3\,254 = 3 \text{ thousands} + 2 \text{ hundreds} + 5 \text{ tens} + 4 \text{ units}.

The place-value table of 3\,254, read “three thousand two hundred fifty-four”. — The place-value table of $3\,254$ , read “three thousand two hundred fifty-four”.

Example 14.2 (The zero that holds a place)

“Four thousand and seven” is written $4\,007$ : a $4$ in the thousands, a $7$ in the units, and zeros in between to keep each digit in its right place. Without them, $47$ would be a very different number!

Example 14.3 (Exchanging)

Ten units make one ten; ten tens make one hundred; ten hundreds make one thousand. So $10$ hundreds $= 1\,000$ , and $32$ tens $= 320$ : the number $320$ contains $32$ tens (not just the digit $2$ !).

14.2 Comparing and ordering

Method 14.4 (Comparing two numbers)

Count the digits: more digits means a bigger number ( $1\,002 > 998$ ).
Same number of digits: compare digit by digit starting from the left. The first difference decides: $5\,614 > 5\,589$ because $6 > 5$ at the hundreds.
Write the answer with the signs $<$ (smaller than) or $>$ (bigger than): the open side always faces the bigger number.

Numbers on a line graduated in hundreds: further right means bigger. 240 < 690 < 905. — Numbers on a line graduated in hundreds: further right means bigger. $240 < 690 < 905$ .

Example 14.5

Order from smallest to biggest: $780$ , $8\,700$ , $807$ , $87$ . First by the number of digits: $87$ (two), then $780$ and $807$ (three), then $8\,700$ (four). Between $780$ and $807$ : compare the hundreds digits, $7 < 8$ , so $780 < 807$ . Final order:

87 < 780 < 807 < 8\,700 .

14.3 Even and odd

Definition 14.6 (Even, odd)

A number is even when it can be split into two equal whole parts — its units digit is $0$ , $2$ , $4$ , $6$ or $8$ . Otherwise it is odd (units digit $1$ , $3$ , $5$ , $7$ or $9$ ).

Example 14.7

$46$ is even ( $46 = 23 + 23$ ); $47$ is odd (two equal parts would need half an object). Only the last digit matters: $3\,578$ is even, $2\,341$ is odd.

Even numbers split into two equal rows; odd numbers always leave one alone.

14.4 Exercises

Exercise 14.1 ★

Write in digits: “two thousand three hundred forty-five”; “six thousand and fifty”; “nine thousand and one”.

Solution

Solution of Exercise 14.1.

$2\,345$ ; $6\,050$ ; $9\,001$ .

Exercise 14.2 ★

Write in words: $1\,208$ ; $4\,070$ ; $9\,999$ .

Solution

Solution of Exercise 14.2.

$1\,208$ : one thousand two hundred eight. $4\,070$ : four thousand seventy. $9\,999$ : nine thousand nine hundred ninety-nine.

Exercise 14.3 ★

In $7\,382$ : which digit is in the tens place? In the thousands place? How many hundreds does the number contain in total (careful: not just the hundreds digit — think of Example 14.3)?

Solution

Solution of Exercise 14.3.

Tens digit: $8$ . Thousands digit: $7$ . Hundreds contained in total: $73$ (because $7\,382 = 73$ hundreds $+ 82$ , i.e. $73 \times 100 + 82$ ).

Exercise 14.4 ★

Decompose as in Definition 14.1: $5\,631$ ; $2\,046$ ; $8\,500$ .

Solution

Solution of Exercise 14.4.

$5\,631 = 5$ thousands $+ 6$ hundreds $+ 3$ tens $+ 1$ unit.

$2\,046 = 2$ thousands $+ 0$ hundreds $+ 4$ tens $+ 6$ units.

$8\,500 = 8$ thousands $+ 5$ hundreds.

Exercise 14.5 ★

Copy and complete with $<$ or $>$ :

456 \;?\; 465, \qquad 1\,203 \;?\; 989, \qquad 6\,090 \;?\; 6\,009, \qquad 9\,999 \;?\; 10\,000 .

Solution

Solution of Exercise 14.5.

$456 < 465$ (tens: $5 < 6$ ); $1\,203 > 989$ (four digits beat three); $6\,090 > 6\,009$ (tens: $9 > 0$ ); $9\,999 < 10\,000$ .

Exercise 14.6 ★

Order from smallest to biggest: $530$ ; $3\,500$ ; $353$ ; $5\,033$ ; $503$ .

Solution

Solution of Exercise 14.6.

$353 < 503 < 530 < 3\,500 < 5\,033$ .

Exercise 14.7 ★

On a number line graduated in hundreds from $0$ to $1\,000$ , place (approximately) the numbers $150$ , $480$ , $820$ and $990$ .

Solution

Solution of Exercise 14.7.

$150$ sits halfway between $100$ and $200$ ; $480$ just before $500$ ; $820$ just after $800$ ; $990$ almost at $1\,000$ .

Exercise 14.8 ★

Even or odd? $34$ ; $57$ ; $70$ ; $2\,481$ ; $6\,548$ .

Solution

Solution of Exercise 14.8.

Even: $34$ , $70$ , $6\,548$ (last digit $4$ , $0$ , $8$ ). Odd: $57$ , $2\,481$ (last digit $7$ , $1$ ).

Exercise 14.9 ★

Count by steps: from $2\,970$ , count five steps of $10$ ; from $860$ , count four steps of $100$ ; from $9\,996$ , count four steps of $1$ . What do you notice at the end of the last one?

Solution

Solution of Exercise 14.9.

From $2\,970$ : $2\,980$ , $2\,990$ , $3\,000$ , $3\,010$ , $3\,020$ .

From $860$ : $960$ , $1\,060$ , $1\,160$ , $1\,260$ .

From $9\,996$ : $9\,997$ , $9\,998$ , $9\,999$ , $10\,000$ — the last step crosses into a five-digit number: our whole chapter’s limit!

Exercise 14.10 ★★

Using each of the digits $3$ , $7$ , $0$ , $5$ exactly once, write the biggest possible four-digit number, then the smallest one (a number cannot start with $0$ ).

Solution

Solution of Exercise 14.10.

Biggest: $7\,530$ (digits in decreasing order). Smallest: $3\,057$ (smallest nonzero digit first, then the rest increasing — the $0$ right after the $3$ ).

Exercise 14.11 ★★

I am a three-digit number. My digits are $2$ , $5$ and $8$ , I am even, and I am bigger than $800$ . Who am I?

Solution

Solution of Exercise 14.11.

Bigger than $800$ : the hundreds digit is $8$ . Even: the units digit must be $2$ (using each of $2$ , $5$ , $8$ once). So the number is $852$ .