Same digit. Different value. Why?

A bead count tells us how many. The rod position tells us the value of each bead. Explore whole places, fractions, zero and signs.

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Question

Why is one hundred worth ten tens—and one unit worth ten tenths? Can the same idea explain both?

Place	Value of one bead	Power notation
Hundreds	100	10²
Tens	10	10¹
Units	1	10⁰
Tenths	0.1 = 1/10	10⁻¹
Hundredths	0.01 = 1/100	10⁻²

A cue about direction

Each place to the left is ten times the value of its neighbour. Each place to the right is one tenth of its neighbour.

Six tens and eight units represent 68 — Conceptual diagram from the master: 68 = (6 × 10) + (8 × 1). Open full size

The decimal boundary

The decimal point separates units from tenths. Three active tenths beads represent 0.3.

Zero units and three tenths represent 0.3 — The exponent describes the place value, not a power applied to the digit. Open full size

Zero holds a place

The zero in 206 records no tens. Removing it gives 26, a different value. No active beads represent a zero digit; the rod label still matters.

Try it

Which number has the same value as 0.5?

Choose an answer when you are ready.

Sign applies to the whole number

Negative five units and six negative tenths represent minus 5.6 — −5.6 = −[(5 × 1) + (6 × 0.1)]. The sign applies to both contributions. Open full size

Four regions: positive and negative whole and fractional places — These regions organise one signed number. They are not a Cartesian coordinate plane. Open full size

Reflect

Where would −0.3 appear? Why does having zero whole units not make its fractional part positive?

What a finite frame can show

A frame has a largest place and a smallest place. It can show 0.125 exactly if thousandths are available. It cannot show the infinitely recurring decimal 1/3 exactly. Rounded or truncated displays must be identified. Zero is neither positive nor negative.

Explore this on the interactive frame