Order & compare large numbers
Theory
Ordering and comparing large numbers is a Year 5 place value skill: lining numbers up by place value and comparing digits from the left, using the symbols < and >, and writing lists in ascending or descending order. This Year 5 maths page explains how to compare numbers to the millions, read them on a number line, find halfway points and make the largest or smallest number from a set of digits.
To compare two numbers is to decide which is larger. A number with more digits is always larger: any seven-digit number is bigger than any six-digit number, so \(1\,000\,000 > 999\,999\).
If two numbers have the same number of digits, compare them place by place from the left. The first place where the digits differ decides which number is larger.
The symbol < means “is less than” and > means “is greater than”. The open side always faces the larger number: \(3\,190\,000 < 3\,910\,000\).
Ascending order goes from smallest to largest. Descending order goes from largest to smallest.
Comparing large numbers needs no formulas, just place value and the comparison symbols.
How to compare two large numbers
- Count the digits. More digits means a larger number.
- Line up the places and compare from the left: millions, then hundred thousands, then ten thousands, and so on.
- Stop at the first difference. The number with the larger digit there is larger. Write < or > with the open side towards it.
How to order a list
- Compare the first digits of all the numbers, then the next place for any that tie.
- Write the list smallest to largest for ascending order, or largest to smallest for descending order.
- Check each neighbour pair: every number should be smaller (or larger) than the next.
(i) \(5\,602\,000\) ___ \(5\,620\,000\)
(ii) \(1\,000\,010\) ___ \(998\,500\)
(i) The millions and hundred thousands match, so compare the ten thousands.
(ii) Count the digits.
| \(\text{(i) ten thousands}\) | \(:\) | \(0 < 2\) |
| \(\text{(ii) digits}\) | \(:\) | \(7 > 6\) |
(i) \(5\,602\,000 < 5\,620\,000\).
(ii) \(1\,000\,010 > 998\,500\).
All four have 7 millions. Compare the hundred thousands: 3, 5, 3, 5. For the two with 5, compare the ten thousands.
| \(7\,530\,214\) | \(>\) | \(7\,503\,142\) |
\(7\,530\,214\) is the largest.
The hundred thousands digits are 1, 0, 6 and 1. For the two with 1, compare the ten thousands.
| \(4\,016\,000\) | \(<\) | \(4\,106\,000\) |
| \(4\,106\,000\) | \(<\) | \(4\,160\,000\) |
| \(4\,160\,000\) | \(<\) | \(4\,610\,000\) |
\(4\,016\,000\), \(4\,106\,000\), \(4\,160\,000\), \(4\,610\,000\).
\(8\,070\,000\) has no hundred thousands, so it is smallest. The other two differ in the thousands.
| \(8\,707\,000\) | \(>\) | \(8\,700\,000\) |
| \(8\,700\,000\) | \(>\) | \(8\,070\,000\) |
\(8\,707\,000\), \(8\,700\,000\), \(8\,070\,000\).
| \(\text{each gap}\) | \(=\) | \(100\,000\) |
| \(8 \text{ gaps}\) | \(=\) | \(800\,000\) |
| \(6\,000\,000 + 800\,000\) | \(=\) | \(6\,800\,000\) |
The arrow points to \(6\,800\,000\).
(i) What is the largest number you can make?
(ii) What is the smallest seven-digit number?
| \(\text{largest}\) | \(:\) | \(8\,754\,310\) |
| \(\text{smallest}\) | \(:\) | \(1\,034\,578\) |
(i) \(8\,754\,310\).
(ii) \(1\,034\,578\). The zero cannot go first, so the \(1\) leads and the \(0\) goes next.
| \(\text{starts}\) | \(:\) | \(3\,62\ldots\) |
| \(\text{thousands digit}\) | \(=\) | \(4\) |
The number is \(3\,624\,000\).
Common pitfalls
Frequently asked questions
How do you compare two large numbers?
First count the digits: more digits means larger. If they have the same number of digits, compare place by place from the left. The first place that is different decides, so \(3\,190\,000 < 3\,910\,000\).
What does ascending order mean?
Ascending order means from smallest to largest. Descending order is the reverse, from largest to smallest.
What do the symbols < and > mean?
The symbol < means is less than and > means is greater than. The open side faces the larger number, as in \(1\,000\,000 > 999\,999\).
What is the smallest seven-digit number?
The smallest seven-digit number is \(1\,000\,000\) (one million). The largest is \(9\,999\,999\).
How do you find the number halfway between two millions?
Halfway between two whole millions is the smaller million plus \(500\,000\). Halfway between \(8\,000\,000\) and \(9\,000\,000\) is \(8\,500\,000\).
How do you make the largest number from some digits?
Write the digits from largest to smallest. For the smallest number, write them from smallest to largest, but never start with zero.