1.1 Place value and ordering numbers — Learn

Prerequisites: reading and writing whole numbers; the four operations with whole numbers.

By the end of this section you can: state the value of any digit in a number; multiply and divide by powers of ten; and compare and order integers, decimals and negative numbers using $=, \neq, <, >, \leq$ and $\geq$.

1. The value of a digit

Every digit in a number has a place value — what it is worth because of where it stands. In the whole-number part the columns are ones, tens, hundreds, and so on. After the decimal point they are tenths, hundredths, thousandths.

In $4\,357.286$ the digit $8$ is in the hundredths column, so it is worth $0.08$.

Check. What is the digit $2$ worth in $4\,357.286$? What about the $7$?

2. Multiplying and dividing by 10, 100, 1000

Multiplying by 10 makes every digit worth ten times as much, so the digits move one place to the left. Dividing by 10 moves them one place to the right. Nothing is added or removed — we are only changing place value.

Watch out. "Add a zero." Adding a zero changes the number: $4.35 \times 10 = 43.5$, not $4.350$. Move the digits; do not decorate the number with zeros.

3. Comparing and ordering

These symbols always point their open end at the larger number:

To order decimals, compare the same column in each number, working left to right.

Example 1. Which is larger, $1.7$ or $0.95$?

The ones digit decides it: $1.7$ has $1$ one; $0.95$ has $0$ ones. So $1.7 > 0.95$.

Example 2. Write $0.4,\ 0.09,\ 0.35$ in order.

Give each number the same number of decimal places: $0.40,\ 0.09,\ 0.35$. Now compare tenths: $0.09$ has $0$ tenths, so it is smallest. Between $0.40$ and $0.35$, the tenths are equal, so compare hundredths: $0 < 5$, so $0.40 > 0.35$. The order is

Watch out. A longer decimal is not always larger. $0.4 = 0.40 > 0.35$, even though "0.35" has more digits. Compare place value, not length of writing.

4. Negative numbers

Negative numbers sit to the left of zero. On a number line, the number further left is smaller: $-7 < -2$, even though $7 > 2$.

Example 3. Put $-3,\ 2,\ -5,\ 0,\ -0.5$ in order.

Reading left to right on the number line: $-5 < -3 < -0.5 < 0 < 2$.

Try it: put two numbers in order

Predict first. Do not move the sliders yet. If $a = -2.5$ and $b = 1.8$, which is greater? Then move the sliders and check. Finish by answering the question in the panel.

Explain: why is $0.3$ greater than $0.29$, even though $0.29$ has more digits?

5. Progress check

The drink prices were 65p and the selling price was £1.20. Use ordering to answer without calculating: is £1.20 greater than 65p? Which of the two stall costs, £1.85 or £3.00, is larger?

You should now be able to

• state the value of any digit in an integer or decimal;
• multiply and divide by 10, 100 and 1000 by moving digits;
• compare and order integers, decimals and negative numbers;
• use the symbols $=, \neq, <, >, \leq, \geq$ correctly.