Convert fractions and recurring decimals

Decimal

\(\displaystyle 0.35\)

Which fraction is exactly equal to the finite decimal shown?

Terminating and recurring describe the decimal expansion

A terminating decimal has a finite representation: after some place, all later digits may be taken as zero

A recurring decimal repeats a fixed block of digits forever, possibly after an initial non-repeating prefix

Recurring decimals are nonterminating too. Nonterminating does not mean nonrecurring or irrational

A finite string of displayed digits alone cannot establish what all later digits do

Show exactly which digits repeat

A bar covers just the recurring block. Digits before the bar do not belong to that repeating block

For keyboard answers in this topic, enclose the recurring block in parentheses: 0.12(3) and 0.(123) represent the two different values above

These parentheses mean repeat the enclosed digits forever, not multiplication

Use the earliest repeating start and shortest repeating block. Include the whole part and keep any leading zeros in the prefix or block

Type 0.(03), not 0.(3), for recurring zero three

Exact decimal

\(\displaystyle 0.2\overline{45}\)

Which keyboard entry represents the exact decimal shown?

Retain the exact place-value method for terminating decimals

A denominator that can be made into a power of ten gives an exact terminating decimal

In the reverse direction, write the decimal over the power of ten determined by its places, then simplify

Appending zeros changes the written precision, not the exact value

Scale a reduced denominator to a power of ten

Convert the fraction to an exact terminating decimal

Sixteen is two to the fourth power

Four decimal places express ten-thousandths exactly

Construct an exact decimal without rounding

Convert a finite decimal to a simplified fraction

Write the decimal as a fraction in simplest form

Three decimal places mean thousandths

Divide numerator and denominator by one hundred and twenty-five

Keep every place value before simplifying

Reduce first, then inspect the denominator factors

For a fraction in lowest terms, a terminating decimal is possible exactly when the denominator has no prime factors other than two and five

Enough factors of two and five can make such a denominator into a power of ten

A different prime factor in a reduced denominator cannot divide a power of ten, so the ordinary decimal expansion does not terminate

Inspect the reduced denominator, not the original one. Common factors may remove the other primes

Fraction

\(\displaystyle \frac{18}{24}\)

Reduce the fraction fully and enter its denominator

Cancel the greatest common factor before testing the denominator primes

Fraction

\(\displaystyle \frac{18}{24}=\frac{3}{4}\)

Why does the fraction have a terminating decimal representation?

Fraction

\(\displaystyle \frac{7}{24}\)

The fraction is already reduced. Factor its denominator and enter the prime factor other than two or five

Break twenty-four into prime factors

A repeated remainder forces the digits to repeat

Long division carries a remainder to the next decimal place. Multiply that remainder by ten, then divide again

The first decimal digit is one and the new remainder is four

The same remainder four now returns, so the digit six repeats forever

A remainder of zero would end the expansion. Otherwise there are only finitely many possible remainders, so a remainder must eventually repeat

A cycle that begins immediately

Convert the fraction to an exact recurring decimal. Type the whole part and decimal point, then any non-repeating digits, and enclose just the infinitely repeating block in parentheses

Use the shortest non-repeating prefix and shortest recurring block. Keep leading zeros, include the whole part, and do not put spaces inside the number

Find the whole part and starting remainder

Multiply remainder 1 by 10. The next digit is 3, leaving remainder 1

Remainder 1 has appeared before. The digits from its first occurrence repeat

The repeating cycle begins immediately after the decimal point

The shortest repeating block is 3. Keep any zero at the start of the block

In keyboard notation, put just the recurring digits inside parentheses

0.(3)

The parentheses mark infinitely repeated digits, not multiplication or rounding

Construct a two-digit recurring block

Separate the non-repeating prefix from the cycle

Convert the fraction to an exact recurring decimal. Type the whole part and decimal point, then any non-repeating digits, and enclose just the infinitely repeating block in parentheses

Use the shortest non-repeating prefix and shortest recurring block. Keep leading zeros, include the whole part, and do not put spaces inside the number

Find the whole part and starting remainder

Multiply remainder 7 by 10. The next digit is 5, leaving remainder 10

Multiply remainder 10 by 10. The next digit is 8, leaving remainder 4

Multiply remainder 4 by 10. The next digit is 3, leaving remainder 4

Remainder 4 has appeared before. The digits from its first occurrence repeat

The first 2 decimal digits do not belong to the repeating cycle

The shortest repeating block is 3. Keep any zero at the start of the block

In keyboard notation, put just the recurring digits inside parentheses

0.58(3)

The parentheses mark infinitely repeated digits, not multiplication or rounding

Find the earliest repeating start

A zero can be part of the repeating block

Convert the fraction to an exact recurring decimal. Type the whole part and decimal point, then any non-repeating digits, and enclose just the infinitely repeating block in parentheses

Use the shortest non-repeating prefix and shortest recurring block. Keep leading zeros, include the whole part, and do not put spaces inside the number

Find the whole part and starting remainder

Multiply remainder 1 by 10. The next digit is 0, leaving remainder 10

Multiply remainder 10 by 10. The next digit is 3, leaving remainder 1

Remainder 1 has appeared before. The digits from its first occurrence repeat

The repeating cycle begins immediately after the decimal point

The shortest repeating block is 03. Keep any zero at the start of the block

In keyboard notation, put just the recurring digits inside parentheses

0.(03)

The parentheses mark infinitely repeated digits, not multiplication or rounding

Keep both zeros before the repeating digit

Divide the magnitude and restore the whole-value sign

Convert the fraction to an exact recurring decimal. Type the whole part and decimal point, then any non-repeating digits, and enclose just the infinitely repeating block in parentheses

Use the shortest non-repeating prefix and shortest recurring block. Keep leading zeros, include the whole part, and do not put spaces inside the number

Work with the positive magnitude first and restore the negative sign at the end

Find the whole part and starting remainder

Multiply remainder 5 by 10. The next digit is 4, leaving remainder 2

Multiply remainder 2 by 10. The next digit is 1, leaving remainder 8

Multiply remainder 8 by 10. The next digit is 6, leaving remainder 8

Remainder 8 has appeared before. The digits from its first occurrence repeat

The first 2 decimal digits do not belong to the repeating cycle

The shortest repeating block is 6. Keep any zero at the start of the block

In keyboard notation, put just the recurring digits inside parentheses

-2.41(6)

The parentheses mark infinitely repeated digits, not multiplication or rounding

Retain the negative sign and whole part

Use the display as a finite observation

A calculator or spreadsheet has room for only finitely many displayed digits. If one divided by three is displayed as 0.333333, that string is a finite approximation

Exact long division supplies the missing evidence: the remainder one returns after every digit, so the digit three repeats forever

Use a digital display to inspect digits, then use a zero or repeated remainder to justify whether division terminates or recurs

In the next table the readout is explicitly truncated, not rounded. Increasing its length does not turn it into an exact infinite expansion

Increase n from one through six. Each row divides ten to the power n by seven and shows its integer quotient and remainder. The starting remainder for one divided by seven is one

What establishes the recurring block for one seventh?

Write the complete shortest cycle

An equation solution can be a fraction

Earlier equation work used positive whole-number solutions. The same reversible operations also find a fractional solution

Divide both sides by nine rather than rounding seven divided by nine

Check by substituting the exact fraction

This short bridge is enough for the one-multiplier equations below; it does not assume later general rational-equation methods

Solve the equation and give an exact fraction. Equivalent fractions are accepted

Undo multiplication by nine on both sides

Shift equal recurring tails into alignment

Let x be the exact decimal. Multiplying by ten shifts a one-digit recurring block by one place

Subtract the first equality from the second. The infinite fractional tails are identical and cancel exactly

For a two-digit recurring block, multiply by one hundred so one whole block moves before the decimal point

Convert one repeating digit to an exact fraction

Only the digits under the bar repeat forever

Write the exact recurring decimal as a fraction in simplest form, not a rounded approximation

First work with the positive magnitude, \(x\)

The fractional tails are identical; subtract the second equation from the first

The equality is exact because the full recurring tails cancelled

Align one-digit tails and simplify

Preserve a leading zero in a recurring block

Only the digits under the bar repeat forever

Write the exact recurring decimal as a fraction in simplest form, not a rounded approximation

First work with the positive magnitude, \(x\)

The fractional tails are identical; subtract the second equation from the first

The equality is exact because the full recurring tails cancelled

Use a shift equal to one complete block

Move the prefix before aligning the recurring tail

Multiply by ten to move the prefix before the decimal point, and by one hundred to move the prefix and one recurring digit

Subtract the two expressions with identical fractional tails

Do not use a denominator of ninety-nine here: the required two shifts differ by ninety, not ninety-nine

Align tails after a one-digit prefix

Only the digits under the bar repeat forever

Write the exact recurring decimal as a fraction in simplest form, not a rounded approximation

First work with the positive magnitude, \(x\)

The fractional tails are identical; subtract the second equation from the first

The equality is exact because the full recurring tails cancelled

Keep the prefix out of the repeating block

Shift a prefix and a two-digit recurring block

Only the digits under the bar repeat forever

Write the exact recurring decimal as a fraction in simplest form, not a rounded approximation

First work with the positive magnitude, \(x\)

The fractional tails are identical; subtract the second equation from the first

The equality is exact because the full recurring tails cancelled

Retain the zero at the start of the prefix

Restore the sign after converting the positive magnitude

Check zero and the recurring-nine boundary

Zero and integers have terminating decimal representations

Every non-zero terminating decimal has an alternative representation ending in recurring nines. The nines are not a tiny amount below that exact value

This is an exact equality, not rounding. When constructing a decimal from a fraction, use its ordinary terminating form if division reaches remainder zero

Recurring decimal

\(\displaystyle 1.2\overline{9}\)

Write the exact value as a terminating decimal

Align 100x and 10x so the recurring-nine tails cancel

Which statement is exact?

Use a digital display to investigate, not to prove

This independent investigation is not automatically marked. Predict the decimal behaviour of one eighth, one sixth and one seventh from their reduced denominators

Use a calculator or spreadsheet to divide each numerator by its denominator. Record the finite display and the number of places it shows

Use exact long division to determine whether a remainder becomes zero or repeats. Explain any difference between the displayed digits and the exact recurring notation

Do not infer irrationality or non-recurrence from a finite display. The next topic considers numbers without terminating or recurring decimal expansions

Explain why keeping the exact fraction is often safer than copying a rounded decimal into a later calculation

Review predictions with exact denominator and remainder evidence

The denominator has only a factor two, and the remainder eventually reaches zero

A factor three remains and the remainder four repeats after the prefix digit

The starting remainder one returns after six digits

The displayed six-digit truncation is not the exact fraction, even though it is close

Keep the exact value and the representation conditions

A reduced denominator with only prime factors two and five gives a terminating decimal representation

A different prime factor prevents termination of the ordinary long-division expansion; a repeated non-zero remainder gives a recurring block

The prefix and recurring block have different roles. Preserve their zeros and show exactly which digits repeat

To convert a recurring decimal to a fraction, align identical infinite tails with powers of ten, subtract, solve the remaining equation exactly and simplify

A recurring decimal is both nonterminating and rational. A finite calculator display cannot decide all later digits

Fixed notation checks and the ungraded digital investigation are guided activities, not proof of independent mastery in every case