Place Value

Understanding Place Value

Every digit in a number has a value depending on its position, called its place value. The same digit can represent different amounts in different places.

For whole numbers, place values increase by powers of ten from right to left:

• Units (ones) 6 the rightmost digit
• Tens 6 ten times the units
• Hundreds 6 ten times the tens
• Thousands 6 ten times the hundreds
• And so on...

For example, in the number 3,482:

• The digit 2 is in the units place and represents 2
• The digit 8 is in the tens place and represents 80
• The digit 4 is in the hundreds place and represents 400
• The digit 3 is in the thousands place and represents 3,000

Decimals extend place value to the right of the decimal point, decreasing by powers of ten:

• First decimal place: tenths ($\frac{1}{10}$)
• Second decimal place: hundredths ($\frac{1}{100}$)
• Third decimal place: thousandths ($\frac{1}{1000}$)
• And so on...

For example, in 56.374:

• 3 is in the tenths place and represents $\frac{3}{10}$
• 7 is in the hundredths place and represents $\frac{7}{100}$
• 4 is in the thousandths place and represents $\frac{4}{1000}$

The value of a digit is calculated by multiplying the digit by the place value it occupies.

For instance, in 5,206:

$5 \times 1000 = 5000$, $2 \times 100 = 200$, $0 \times 10 = 0$, $6 \times 1 = 6$

Total value = $5000 + 200 + 0 + 6 = 5206$

Reading Numbers

To read whole numbers correctly, say the digit followed by its place value group, starting from the left.

Use commas to separate thousands in large numbers for clarity (e.g. 12,345).

For example, 7,345 is read as "seven thousand three hundred and forty-five".

When reading decimals, say the whole number part, then say "point", followed by each digit individually.

For example, 12.406 is read as "twelve point four zero six".

It is important to distinguish between a digit and its place value:

• A digit is a single number from 0 to 9.
• Place value is the value of that digit depending on its position.

For example, in the number 5,082:

• The digit 8 is in the tens place, so its place value is 80.
• The digit 0 is in the hundreds place, so its place value is 0 (zero hundreds).

Understanding this helps avoid confusion when reading or writing numbers.

For instance, the number 4,305 is read as "four thousand three hundred and five" 6 note the zero means no tens.

Writing Numbers in Words and Figures

To write numbers in words, break the number into place value groups and write each part in words.

For example, 2,417 is written as "two thousand four hundred and seventeen".

For decimals, write the whole number part in words, then say "point", and write each digit after the decimal point as a separate word.

For example, 3.205 is written as "three point two zero five".

When converting words to figures, identify the place values mentioned and write the corresponding digits in the correct positions.

For example, "five thousand and sixty-two" becomes 5,062.

Use correct terminology for place values:

• Thousands, hundreds, tens, units for whole numbers
• Tenths, hundredths, thousandths for decimals

This helps avoid mistakes when writing numbers from words or vice versa.

For example, "seven hundred and four thousand, three hundred and twenty-one" is written as 704,321.

Example: Write 9,482.36 in words.

Answer: "Nine thousand four hundred and eighty-two point three six".

Comparing and Ordering Numbers

To compare numbers, use place value to determine which is greater or smaller.

For example, compare 3.4 and 3.45 by adding zeros to make decimal places equal: 3.40 and 3.45.

Start by comparing the digits in the highest place value:

• If digits differ, the number with the larger digit in that place is greater.
• If digits are the same, move to the next place value to the right.

This applies to both whole numbers and decimals.

For decimals, compare digits place by place after the decimal point, adding zeros if necessary to make the decimal places equal length.

For example, to compare 4.56 and 4.6:

Rewrite 4.6 as 4.60 to compare tenths and hundredths:

• 4.56 has 5 tenths and 6 hundredths
• 4.60 has 6 tenths and 0 hundredths

Since 6 tenths > 5 tenths, 4.6 is greater than 4.56.

Ordering numbers means arranging them from smallest to largest (ascending) or largest to smallest (descending).

Always compare place values carefully to order correctly.

Example: Order these numbers in ascending order: 3.2, 3.15, 3.205

Compare tenths:

• 3.2 = 3.20
• 3.15
• 3.205

Compare hundredths:

• 3.15 has 1 tenth and 5 hundredths
• 3.20 has 2 tenths and 0 hundredths
• 3.205 has 2 tenths, 0 hundredths, and 5 thousandths

So order is: 3.15, 3.2 (3.20), 3.205

Example: Compare 5,432 and 5,423.

Look at thousands: both 5,000.

Look at hundreds: both 400.

Look at tens: 3 tens (30) vs 2 tens (20). Since 30 > 20, 5,432 > 5,423.