One Fraction in Several Bases

Keep the fraction; change its units

Begin with three quarters in base two. The fraction stays at three quarters throughout this comparison. The first illuminated place contributes one half. The next contributes one quarter. Together they make the selected fraction, so its binary expansion is zero point one one.

Move closer to the second place. This one is a quarter, even though its digit is also one. Equal digits do not imply equal contributions. The position tells us which unit is being counted. Here the two contributions have different sizes and add to three quarters.

Now use base four. A first-place unit is one quarter, and the digit three counts three of those units. We have reached the same fraction in just one fractional place. The expansion has become shorter because a quarter is already one of this base’s first-place units.

Return to base two and look at both places together. Nothing has been added to or removed from the fraction. One half plus one quarter and three quarters describe the same quantity with different groupings. A base changes the available place-value units, not the value being described.

Connect familiar decimal notation

Set the base to ten, still keeping three quarters. The first digit is seven. It contributes seven tenths, which is less than three quarters. There is still a difference to account for. That difference belongs to the smaller places that follow.

Select the second place. Its five contributes five hundredths. Seven tenths is seventy hundredths, so the two contributions give seventy-five hundredths. Reducing seventy-five over one hundred gives three quarters. The familiar decimal point seven five is a sum of these two positional contributions.

Compare base four again. The single digit three contributes three quarters immediately. This is why the written digits cannot be compared without knowing the base. A three in this first fractional place is larger than a seven in the first fractional place of base ten.

Return to the decimal expansion and follow the smaller place. The geometry makes the change of scale explicit. The label tells us the exact positional unit. Use those together: a visible smaller piece, a unit of one hundredth, and five copies of that unit.

A fraction can require a continuing expansion

Begin a new comparison with one half in base two. Its first fractional unit is already a half, so the expansion ends after its first digit. The remaining places contribute zero. Ending is a property of this fraction in this base, not an instruction to stop drawing.

Change to base three. No whole count of thirds is exactly one half. One third is too small, and two thirds is too large. The expansion therefore continues into ninths, twenty-sevenths, and smaller units. Here the digit one repeats at every place.

Follow the next place. One third plus one ninth gives four ninths, still below one half. Each additional one contributes a smaller unit. The exact fraction remains one half; the displayed finite set of places is only a beginning of its continuing representation.

Now change to base ten. Five tenths equals one half, so the expansion ends in its first place again. The same fraction has a finite binary expansion, a repeating ternary expansion, and a finite decimal expansion. Its exactness does not depend on which notation is shortest.

Repeating in ten; ending in three

Consider one third in base ten. Three tenths leaves a difference, so the first digit is followed by smaller contributions. The displayed threes recur. Each three counts a different unit: tenths, hundredths, thousandths, and so on. Repeating a digit does not repeat its size.

Move into the hundredths place. Its three hundredths is one tenth of the preceding three tenths. The pieces shrink while the digit remains the same. This distinction is essential when reading any repeating expansion: the pattern is repeated at a new scale.

Change to base three. One third is now exactly one first-place unit. The digit one gives the complete fraction and the later contributions vanish. We did not improve an approximate fraction into an exact one. We changed how an already exact fraction is written.

Return to base ten and view the arrangement as a whole. A longer expansion is not a larger fraction, and a shorter expansion is not a smaller fraction. Read the digits with their positional units. That rule works equally well for terminating and repeating representations.

Group the same fraction into different positional units

Take five eighths in base two. Its digits are one, zero, one: one half, no quarters, and one eighth. These contributions total five eighths exactly. Keep that fraction fixed through the next base changes. We are changing its positional description, so no new quantity should be inferred from a different-looking collection of digits.

Change to base four. The expansion has two fractional digits, both two. The first contributes two quarters, which is one half. The second contributes two sixteenths, which is one eighth. Those are the same two nonzero contributions as before, now counted in base-four units rather than written as three binary places.

Change to base eight. A first-place unit is already one eighth, so the single digit five gives the complete fraction. A one-digit expression can represent the same amount as a three-digit expression when their units differ. Counting digits is not a way to compare the sizes of fractions written in different bases.

Use base ten. Five eighths becomes zero point six two five. Six tenths contributes the largest part, two hundredths supplies the next part, and five thousandths completes it. The contributions are different from the binary grouping, yet their exact sum is still five eighths. The familiar decimal notation is one option among these positional systems.

Try base twelve. The two digits are seven and six. Seven twelfths plus six one-hundred-forty-fourths equals ninety one-hundred-forty-fourths, which reduces to five eighths. The second digit six counts smaller units than the first digit seven. Every base change must be read together with its new powers and units.

A fixed quantity can alternate between short and recurring descriptions

Begin a second comparison with three fifths in base five. The first fractional digit is three because each unit is a fifth. The fraction is represented exactly in one place. This short expression follows directly from the denominator matching the first positional unit of the chosen base.

Switch to base ten. The digit becomes six because each tenth is half as large as a fifth. Six tenths and three fifths cover the same amount. The digit count doubled while the unit size halved, showing the same reciprocal adjustment that appeared in our earlier equivalent-fraction examples.

Switch to base two and inspect the first place. Its digit is one, contributing a half. That leaves one tenth still unresolved. Binary units continue through quarters, eighths and sixteenths; no power of two contains a factor five, so this fraction cannot end after a finite number of binary positions.

Select the fourth binary place. The first four digits are one, zero, zero, one. They contribute one half plus one sixteenth, or nine sixteenths. That prefix is smaller than three fifths by three eightieths. The repeating block continues, so the finite visible prefix must not be mistaken for the complete exact expansion.

Return to base five and its single digit three. The fraction has remained exact throughout the comparison. The recurring binary representation was not a damaged version of the quantity, and the short base-five representation did not change its value. Base choice determines which positional units can finish the description in finitely many steps.

Bases can share useful factors

Take one seventh and choose base seven. One first-place unit equals one seventh, so a single digit one represents the fraction. The connection is direct: the denominator is the base, and the first fractional unit is its reciprocal.

Keep one seventh and change to base fourteen. Each first-place unit is now one fourteenth. Two of them make one seventh, so the first digit becomes two. The fraction has stayed fixed while the unit became half as large and its count doubled.

Change to base twenty-one. Three units of one twenty-first make one seventh. Again the expansion ends at its first place. Seven, fourteen, and twenty-one all contain seven as a factor, which makes sevenths especially direct to express in those bases.

Return to base ten. One seventh now begins with the recurring block one four two eight five seven. Ten does not supply a power whose reciprocal can express a seventh with a finite whole count. The next lesson develops this connection between denominators, powers, and the place where an expansion ends.