Integers, Place Value and Ordering
Numbers tell us how much, where and in what order. We will begin with whole numbers and zero, then use place value and number lines to compare positive numbers, negative numbers and decimals.
Foundation skills for both tiers: start here to build the number sense needed for later calculations. Work through one example at a time.
Start with integers
Integers are whole numbers, including negative whole numbers and zero:
… −3, −2, −1, 0, 1, 2, 3 …
The dots mean the list continues in both directions. Integers have no fractional part.
Positive numbers are greater than zero. Negative numbers are less than zero. The minus sign in −3 tells us the number is negative three.
Zero is an integer. It is neither positive nor negative.
A horizontal number line puts numbers in order. Equal distances represent equal changes in value.
Zero separates negative numbers on the left from positive numbers on the right. Moving right increases the value; moving left decreases it.
−2 is greater than −5 because it is further right. We write −5 < −2. The symbol < means “less than”.
The size of the digits alone does not decide the order of negative numbers.
Example Question:
Which number is greater: 0.5 or 0.45?
Think in tenths and hundredths. Example 7 shows why 0.5 is the greater value.
Exam Tip
Look for minus signs before comparing the digits.
Read whether the question wants smallest first or largest first.
For decimals, align the decimal points. A longer string of decimal digits does not automatically mean a larger number.
Method — Compare, then order
1. Check the signs. Negative numbers are below zero; positive numbers are above zero.
2. Compare place values for positive numbers. Start with the largest place. If those digits match, move one place to the right.
3. Use the number line for negatives. The number further right is greater. Among two negative numbers, the one closer to zero is greater.
4. Put the numbers in the requested order. Increasing order means smallest to largest. Decreasing order means largest to smallest.
Worked Examples
Example 1 — Identify integers
Which of these are integers: −3, 0, 4, 2.5 and −0.5?
−3 is a negative whole number. Zero and 4 are whole numbers too.
The integers are −3, 0 and 4.
2.5 lies between 2 and 3. −0.5 lies between −1 and 0. Neither is a whole number.
Place value — a digit’s position matters
The digits are the symbols 0 to 9. A digit’s place value tells you how much it represents.
In 47,362, the 4 represents 40,000. It does not represent just 4.
Example 2 — Read and write whole numbers
Read 47,362 using its place values.
| Place | Digit and value |
|---|---|
| Ten-thousands | 4 → 40,000 |
| Thousands | 7 → 7,000 |
| Hundreds | 3 → 300 |
| Tens | 6 → 60 |
| Units (ones) | 2 → 2 |
47,362 = 40,000 + 7,000 + 300 + 60 + 2
Read it as forty-seven thousand, three hundred and sixty-two. A comma groups the thousands; it is not a decimal point.
Now write forty thousand, five hundred and seven in digits.
There are 4 ten-thousands, 0 thousands, 5 hundreds, 0 tens and 7 units.
40,507
The zeros hold empty places. Removing them to write 457 would change the positions of the other digits and the value of the number.
Place value — continue through the decimal point
To the right of units come tenths, hundredths and thousandths. Each place is worth one tenth of the place immediately to its left.
Ten tenths make one whole. Ten hundredths make one tenth. Ten thousandths make one hundredth.
Example 3 — Read a decimal carefully
Read 12.305 and explain its digits.
| Place | Digit and value |
|---|---|
| Tens | 1 → 10 |
| Units | 2 → 2 |
| Tenths | 3 → 0.3 |
| Hundredths | 0 → 0 |
| Thousandths | 5 → 0.005 |
12.305 = 10 + 2 + 0.3 + 0.005
Read it as twelve point three zero five. Say the digits after the point separately.
The 5 represents five thousandths. The zero keeps the hundredths place empty. 12.305 is different from 12.35.
To write six point zero four, keep the empty tenths place: 6.04.
Example 4 — Compare positive whole numbers
Which is greater: 47,362 or 47,326?
Both have 4 ten-thousands, 7 thousands and 3 hundreds. Those places match.
Compare the tens: 6 tens is more than 2 tens. We can decide now.
47,362 > 47,326
The symbol > means “greater than”. Read the statement from left to right.
Example 5 — Compare negative numbers
Which is greater: −5 or −2?
On the opening number line, −5 is five units left of zero. −2 is only two units left of zero.
−2 lies further right, so it is greater:
−2 > −5
Although 5 is greater than 2, the minus signs put these numbers on the other side of zero.
Distance and value are different ideas. −4 and 4 are both four units from zero, but 4 is greater because it lies to the right.
Example 6 — Order negatives, zero and positives
Put 3, −1, 0, −4 and 2 in increasing order.
Increasing means smallest first. Begin with the negative number furthest left: −4.
Next comes −1, then zero, then the positive numbers 2 and 3.
Increasing order: −4, −1, 0, 2, 3.
For decreasing order, start at the other end:
Decreasing order: 3, 2, 0, −1, −4.
Example 7 — Align decimal place values
Which is greater: 0.5 or 0.45?
Five tenths is fifty hundredths, so 0.5 = 0.50. The equals sign means the values are the same.
Compare 50 hundredths with 45 hundredths:
0.50 > 0.45, so 0.5 > 0.45.
Now put 0.65, 0.7 and 0.705 in increasing order. Write matching decimal places to compare:
0.65 = 0.650
0.7 = 0.700
Compare their place values from left to right:
| Value written in thousandths | What decides its position |
|---|---|
| 0.650 | 6 tenths: the smallest. |
| 0.700 | 7 tenths and 0 thousandths. |
| 0.705 | 7 tenths and 5 thousandths: greater than 0.700. |
Increasing order: 0.65, 0.7, 0.705.
Trailing zeros at the end of a decimal do not change its value. Zeros inside a number can hold important places.
Negative decimals follow the same number-line rule. −0.7 is further left than −0.65, so −0.7 < −0.65.
Comparison symbols — read their meaning
| Symbol | Meaning and example |
|---|---|
| = | Equal to: 0.5 = 0.50 |
| ≠ | Not equal to: 0.5 ≠ 0.05 |
| < | Less than: −4 < 1 |
| > | Greater than: 7 > 3 |
| ≤ | Less than or equal to: 3 ≤ 3 |
| ≥ | Greater than or equal to: 5 ≥ 2 |
The open side of < or > faces the greater value. First decide which number is greater, then choose the symbol.
The extra line in ≤ or ≥ allows equality. For example, 3 ≤ 3 is true, but 3 < 3 is false.
Example 8 — Use “or equal to”
A lift can carry at most 5 people. Which counts are allowed: 4, 5 or 6?
“At most 5” means 5 or fewer. Both 4 and 5 are allowed; 6 is not.
4 ≤ 5 and 5 ≤ 5 are true.
“At least 2 people” means 2 or more. Both 2 ≥ 2 and 5 ≥ 2 are true.
These symbols include the stated limit. They do not mean the value must equal it.
Example 9 — Compare temperatures between integers
Three places have temperatures of −2°C, 1°C and −0.5°C. Put them in order from coldest to warmest.
The symbol °C means degrees Celsius. A negative temperature is below 0°C.
−0.5 is halfway between −1 and 0. It is greater than −2 but less than 1.
Coldest to warmest: −2°C, −0.5°C, 1°C.
−2 < −0.5 < 1
This last line combines two comparisons: −2 is less than −0.5, and −0.5 is less than 1.
Key ideas to remember
Integer: a whole-number value, including negatives and zero.
Place value: what a digit is worth because of its position.
Increasing: smallest to largest. Decreasing: largest to smallest.
Number line: read from left to right to move through increasing values.
Check your answer
Check that increasing order moves right along a number line. Decreasing order moves left.
Keep every minus sign. A negative number is always less than zero and less than any positive number.
Align decimal points before comparing digits. You can write 0.7 as 0.70 or 0.700 without changing its value.
Read each inequality aloud. For ≤ and ≥, remember that equality is allowed.
When copying an ordered list, check that every original number appears once.
Think Like a Mathematician
Why is −2 greater than −7?
Look back at the opening number line. −7 is seven units left of zero. −2 is two units left of zero.
Both are negative. Moving from −7 towards −2 goes right, so the value increases.
−7 is further from zero, but that does not make it greater. Distance tells us how far; the sign tells us which side.
The numbers −4 and 4 are equally far from zero. Their distances match, but their positions and values do not.
Common mistakes to avoid
Comparing negative digits as if they were positive. −5 is left of −2, so −5 is smaller.
Leaving out zero when listing integers. Zero is a whole-number value and belongs in the list.
Calling zero positive. Positive numbers are greater than zero; zero itself is neither positive nor negative.
Dropping placeholder zeros. Read each place before copying the number. 40,507 is not 457.
Thinking more decimal digits means more value. Compare 0.50 with 0.45: fifty hundredths is greater than forty-five hundredths.
Reversing < and >. Decide the order first, then read your completed statement aloud.
Mixing up increasing and decreasing. Write “smallest first” or “largest first” beside the question before ordering.
Try it yourself
Start with integers and place value, then compare and order. Explain what decides each answer before opening the worked solution.
1. Which are integers: −6, 0, 8, 1.4 and −2.5?
−6, 0 and 8 are whole-number values.
1.4 lies between 1 and 2. −2.5 lies between −3 and −2.
The integers are −6, 0 and 8.
2. State the value of the digits 7 and 6 in 47,362.
The 7 is in the thousands place: it represents 7000.
The 6 is in the tens place: it represents 60.
3. Write fifty thousand, two hundred and nine in digits.
Use 5 ten-thousands, 0 thousands, 2 hundreds, 0 tens and 9 units.
50,209
The zeros hold the empty thousands and tens places.
4. In 8.407, what are the values of the 4 and the 7? Read the number aloud.
The 4 is in the tenths place: 0.4.
The 7 is in the thousandths place: 0.007.
The zero holds the hundredths place.
Read 8.407 as eight point four zero seven.
5. Choose < or > to compare 36,482 and 36,428.
The ten-thousands, thousands and hundreds match.
Compare the tens: 8 tens is greater than 2 tens.
36,482 > 36,428
6. Choose < or > for each pair: (a) −2 and −6; (b) −4 and 1.
(a) −2 is further right than −6.
−2 > −6
(b) Every negative number lies left of every positive number.
−4 < 1
7. Put 4, −3, 0, −7 and 2 in increasing order.
Increasing means smallest first.
Order the negatives first: −7 lies left of −3. Then comes zero, then 2 and 4.
−7, −3, 0, 2, 4
8. Put −1, 5, −4 and 0 in decreasing order.
Decreasing means largest first.
Start with 5, then zero. Of the negative numbers, −1 is greater than −4.
5, 0, −1, −4
9. Put 0.8, 0.78 and 0.805 in increasing order.
Align the decimal places:
0.8 = 0.800
0.78 = 0.780
0.780 has 7 tenths, so it comes first. Compare 0.800 with 0.805: only the thousandths differ.
0.78, 0.8, 0.805
10. Which statements are true: 3 ≤ 3, 3 < 3, 5 ≥ 2 and 2 ≥ 5?
3 ≤ 3 is true: equality is allowed.
3 < 3 is false: 3 is equal to 3, not less.
5 ≥ 2 is true: 5 is greater than 2.
2 ≥ 5 is false: 2 is below 5.
11. Temperatures are −1.5°C, 0°C and −3°C. Order them from coldest to warmest. Between which consecutive integers does −1.5 lie?
Negative temperatures lie below zero. −3 is further left than −1.5.
Coldest to warmest: −3°C, −1.5°C, 0°C.
−1.5 is halfway between −2 and −1 on a number line.
12. Which is greater, −0.8 or −0.75? Then state the distance of −4 and 4 from zero.
Write −0.8 as −0.80. It is eighty hundredths left of zero; −0.75 is only seventy-five hundredths left.
−0.75 is further right, so −0.75 > −0.8.
−4 and 4 are each 4 units from zero. Equal distance does not mean equal value.
Frequently Asked Questions
Is zero an integer?
Yes. It is a whole-number value. It is neither positive nor negative.
Can an integer be written with a decimal point?
Yes. For example, 3.0 equals 3, so it is an integer. Decide from the value, not just how it is written.
Why is −2 greater than −5?
−2 lies further right on a number line. For two negative numbers, the one closer to zero is greater.
Why does 0.5 equal 0.50?
Five tenths is the same amount as fifty hundredths. Adding a trailing zero changes the way the decimal is written, not its value.
Do zeros always leave a number unchanged?
No. Adding a trailing decimal zero is different from removing a placeholder. For example, 6.04 and 6.4 have different values.
What does the extra line in ≤ or ≥ mean?
It allows equality as well as the comparison. “At most 5” allows 5, while “less than 5” does not.
For ordering positive and negative fractions alongside decimals, continue with Fractions, Decimals & Percentages.
Continue Learning
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