Comparing and Ordering Natural Numbers

Use the symbols <, >, and = to compare natural numbers, and arrange them in ascending or descending order using place value and the number line.

Comparison Symbols

We use three symbols to compare numbers:

Symbol	Meaning	Example
$<$	less than	$345 < 872$
$>$	greater than	$5\,000 > 999$
$=$	equal to	$200 + 50 = 250$

Tip

A helpful trick: the pointy end of $<$ and $>$ always points to the smaller number. Think of it as a hungry crocodile that opens its mouth towards the bigger number!

How to Compare Two Numbers

When comparing two natural numbers, follow these steps:

1
Count the digits
The number with more digits is always larger. Example: 8 945 (4 digits) > 673 (3 digits).
2
If digit count is equal, compare from the left
Look at the leftmost (highest place value) digit first. The number with the larger digit in that position is greater.
3
Move right if digits are equal
If the leftmost digits are the same, compare the next digit to the right, and so on.
4
If all digits match, the numbers are equal
Write = between them.

Compare 34 512 and 34 891

Both numbers have 5 digits, so we compare digit by digit from the left:

Ten-thousands: 3 vs 3 — equal, move right
Thousands: 4 vs 4 — equal, move right
Hundreds: 5 vs 8 — 5 < 8, so the first number is smaller

$34\,512 < 34\,891$

Comparing Rule

a > b \iff \text{a has a larger digit at the first differing position from the left}

Start comparing from the highest place value and work right until you find a difference.

The Number Line

A number line is a straight line where numbers are placed in order from left to right. Numbers further to the right are always larger.

$\longleftarrow \quad 0 \quad 100 \quad 200 \quad 300 \quad 400 \quad 500 \quad \longrightarrow$

On a number line:

Numbers increase as you move right
Numbers decrease as you move left
The number line helps you visualise which number is larger

Note

The number line extends infinitely in both directions. For natural numbers, we start at 0 and go right: 0, 1, 2, 3, ...

Ordering Multiple Numbers

To arrange a group of numbers in order:

Ascending order (from smallest to largest): $12 < 45 < 103 < 2\,500$
Descending order (from largest to smallest): $2\,500 > 103 > 45 > 12$

1
Find the smallest (or largest) number first
Look for the number with the fewest digits, or compare leading digits.
2
Remove it from the list and repeat
Each time, pick the next smallest (or largest) from the remaining numbers.
3
Write them in a sequence
Separate with commas or use the correct inequality symbols.

Order these numbers from smallest to largest: 8 400, 840, 84 000, 8 040

Count digits first:

840 — 3 digits (smallest)
8 040 — 4 digits
8 400 — 4 digits
84 000 — 5 digits (largest)

For the two 4-digit numbers, compare: 8 040 vs 8 400

Thousands: 8 = 8 — equal
Hundreds: 0 < 4, so 8 040 < 8 400

Ascending order: $840 < 8\,040 < 8\,400 < 84\,000$

Real-World Examples

Real world: comparing distances

Three Slovak cities have these distances from Bratislava by road:

Košice: 400 km
Banská Bystrica: 213 km
Žilina: 201 km

Ordering from shortest to longest distance: $201 < 213 < 400$ , so Žilina is closest, then Banská Bystrica, then Košice.

Real world: comparing populations

Three towns have these populations:

Town A: 24 850 people
Town B: 24 580 people
Town C: 42 850 people

Compare: Town C (42 850) has 5 digits starting with 4, while Towns A and B start with 2. So Town C is largest. Comparing A and B: both start with 24, hundreds digit 8 = 8, tens: 5 > 8... wait, 850 vs 580: hundreds 8 vs 5, so 24 850 > 24 580.

Order (descending): $42\,850 > 24\,850 > 24\,580$

Real world: comparing prices

A bicycle costs 389 €, a scooter costs 1 250 €, and a skateboard costs 98 €.

The scooter (1 250 €, 4 digits) is most expensive. Comparing bicycle (389 €) and skateboard (98 €): bicycle has 3 digits, skateboard has 2, so bicycle is more expensive.

Order (ascending): $98 < 389 < 1\,250$

Common Mistakes

Thinking more digits always means a larger number when comparing across different groups (e.g. forgetting to count digits first)
Putting the open end of > or < towards the wrong number
Confusing ascending (smallest first) with descending (largest first)
Stopping the comparison too early — always check all positions until you find a difference
Forgetting that 0 acts as a placeholder — 5 020 is NOT the same as 520

Practice Problems

Compare using <, > or =: 73 410 and 73 041

Show answer

73 410 > 73 041. The hundreds digits differ: 4 > 0.

Order from largest to smallest: 9 999, 10 000, 9 099, 10 100

Show answer

10 100 > 10 000 > 9 999 > 9 099

Three friends ran these distances: Ana 4 215 m, Bohu 4 512 m, Cyril 4 125 m. Who ran the most? Who ran the least?

Show answer

Most: Bohu (4 512 m). Least: Cyril (4 125 m). Order: 4 125 < 4 215 < 4 512.