Mathematics · Foundation article

Place value and number systems

Connect digits with tens and hundreds, explain zero and expanded form, and represent the same quantity in binary.

Useful background article: Counting, zero and natural numbers

After reading, you can

Distinguish a number, its numeral and its digits.
Explain decimal place values, zero placeholders and expanded form using groups.
Read a short binary numeral with named place values and check the unchanged quantity.

Separate the number from its written form

Twenty-five can describe a stone count. The number is the value; a numeral is its written representation, usually 25. The stones do not change when you describe their count in words.

A digit is one symbol used in a numeral. In base ten, the digits are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. The numeral 25 contains two digits: 2 and 5.

Build tens and hundreds from ones

Treat one stone as one unit. Gather ten into one group called a ten. Ten such groups contain one hundred units: ten tens equal one hundred. Grouping reorganises the collection without changing its count.

Base ten uses place values 1, 10, 100 and so on from the right. Each place is worth ten times its right neighbour. A 5 contributes five units in the ones place and fifty in the tens place. Its position changes its contribution.

Keep zero where a place has no contribution

205 records two hundreds, zero additional tens and five ones. Its middle zero keeps the 2 in the hundreds place. Removing that zero gives 25, a different quantity. In 250, the 5 instead contributes five tens and the last zero records no additional ones.

Zeros before the first nonzero digit do not change a whole number’s value: 025 represents 25. Ordinary writing uses 25. A teaching display may retain 025 to show all three columns. Similarly, 000 shows empty hundreds, tens and ones; its usual numeral is 0.

Read each place and show its contribution

Anchor the places at the right: ones, tens, then hundreds. Read 205 as two hundred and five. Its expanded form is 2 × 100 + 0 × 10 + 5 × 1 = 200 + 0 + 5 = 205. Every term shows a digit multiplied by its place value.

You may shorten the expanded sum to 200 + 5 because adding zero changes nothing. That does not allow you to remove the middle zero from the numeral 205. To check an expansion, connect each contribution back to the correct place rather than adding the digits 2 + 0 + 5.

Change the base without changing the quantity

Binary is base two. It uses only the digits 0 and 1, with place values 1, 2, 4, 8, 16 and so on from right to left. Each place is twice the next one to its right. A digit tells whether that place’s contribution is absent or included once.

Read binary 11001 using places 16, 8, 4, 2, 1 from left to right: 16 + 8 + 0 + 0 + 1 = 25 in base ten. It does not mean eleven thousand and one. Label the base: the same digit string can mean different values in different systems. Twenty-five stones remain twenty-five stones.

Build a numeral and check the model

Choose a digit from 0–9 for each of the explorer’s hundreds, tens and ones places. Predict the total and its expansion, then compare the quantity model. A hundreds group represents one hundred units, not one unit merely because it appears as one group.

Compare 205 with 250 and try a display beginning with zero. Explain which zero can be omitted from ordinary writing. The optional binary details describe the same quantity. Next is “Number lines, order and comparison,” using positions on a line to compare values.

Build a number from its places

Choose a digit for each decimal place. Compare the chart, blocks and expanded form. Try 205 and 250: the same digits represent different quantities when their positions change.

Try a starting example

Hundreds × 100

Tens × 10

Ones × 1

Three-place chart

Usual decimal numeral: 205

The chart keeps all three places visible. Leading zeros can be left out of the usual whole-number numeral. A zero between other digits, as in 205, keeps those digits in their correct places.

One small square represents 1. A strip contains 10 such squares. A large square contains 10 strips, or 100 small squares.

2 × 100 + 0 × 10 + 5 × 1 = 205
Total in ones: 205

Compare the same quantity in binary

Binary uses digits 0 and 1. From right to left, its place values are 1, 2, 4, 8 and so on: each is twice the next place to its right. The quantity stays the same.

11001101 (base 2) = 205 (base 10)

Place value: 128Digit: 1Contribution: 1 × 128 = 128
Place value: 64Digit: 1Contribution: 1 × 64 = 64
Place value: 32Digit: 0Contribution: 0 × 32 = 0
Place value: 16Digit: 0Contribution: 0 × 16 = 0
Place value: 8Digit: 1Contribution: 1 × 8 = 8
Place value: 4Digit: 1Contribution: 1 × 4 = 4
Place value: 2Digit: 0Contribution: 0 × 2 = 0
Place value: 1Digit: 1Contribution: 1 × 1 = 1

128 + 64 + 0 + 0 + 8 + 4 + 0 + 1 = 205

Worked examples

Keep an empty tens place visible

Two groups contain one hundred stones each, with five extra stones and no separate tens groups. Someone writes 250. Does that numeral describe the collection?

The collection has two hundreds, zero separate tens and five ones. Put the digits 2, 0 and 5 in those places: 205.
Check the contributions: 2 × 100 + 0 × 10 + 5 × 1 = 205 stones. The middle zero contributes nothing but preserves position.
250 would describe two hundreds and five tens, with no additional ones: 250 stones. It moves the 5 into a different place.

The numeral is 205. Five separate stones contribute five units rather than fifty.

Check two representations of twenty-five

A collection contains twenty-five seeds. A label records its count as binary 11001. Explain why this agrees with decimal 25.

Name the binary places from left to right: 16, 8, 4, 2, 1. Match the digits 1, 1, 0, 0, 1 to them.
Include sixteen, eight and one; the two zero digits contribute nothing. The total is 16 + 8 + 1 = 25 seeds.
Decimal 25 gives two tens and five ones, also twenty-five. The grouping and written form differ; the seed count agrees.

Binary 11001 and decimal 25 represent the same number. State the base when the notation could be ambiguous.

Try it yourself

In decimal 606, what does each 6 contribute? Give an expanded form that also shows the tens place.
Show the answer and reasoning
The left 6 contributes 600 and the right 6 contributes 6: 6 × 100 + 0 × 10 + 6 × 1 = 606.
The same digit has different values because its places differ. The zero preserves an empty tens contribution between hundreds and ones.
The three-place teaching display shows 040. What is its usual numeral? Would deleting the last zero preserve its value too?
Show the answer and reasoning
The usual numeral is 40. Deleting the last zero from 040 gives 04, which represents 4.
The leading zero contributes no hundreds. The last zero holds the ones place; removing it moves the 4 from tens to ones.
Read binary 10010 using places 16, 8, 4, 2, 1. What decimal number does it represent? Explain the contributions.
Show the answer and reasoning
Eighteen: 16 + 0 + 0 + 2 + 0 = 18.
Only the sixteen and two places contribute. Decimal 18 represents the same quantity as one ten and eight ones.

What to carry forward

Digits form a numeral, which represents a number’s value.
In base ten, position determines each digit’s contribution, and zero can preserve a place.
Expanded forms and named bases help you check that different representations describe the same quantity.

References

Sources for the concepts and teaching approaches in this article.