Classify real numbers as rational or irrational using their decimal expansions

Learning goal: Know that every real number is either rational or irrational, understand informally that every number has a decimal expansion that is terminating, repeating, or non-repeating and non-terminating, and explain informally why a repeating decimal must be a rational number.

A rational number is any number that can be written as a fraction p/q where p and q are integers and q is not zero. Every integer is rational, because 7 equals 7/1 and −3 equals −3/1. Every terminating decimal is rational: 0.375 means 375/1000, which reduces to 3/8. Mixed numbers, negative fractions and everyday percentages such as 35% are rational, because each can be rewritten as one integer over another nonzero integer (35% = 35/100 = 7/20); a percent sign does not do this on its own, since π% would mean π/100, which is not a fraction of integers. The definition is about whether such a fraction exists, not about how the number happens to be written on the page. The number 0.5 looks like a decimal, but it is rational because 1/2 is available.

An irrational number is a real number that is not rational: no fraction of integers equals it exactly. The real numbers are the rational numbers and the irrational numbers together, and every real number belongs to exactly one of those two sets. There is no third category and no overlap. So the sentence "this number is either rational or irrational" is always true for a real number, and the job in this lesson is to decide which one applies and to say why.

Every number has a decimal expansion. Every real number, rational or not, has a decimal expansion, and that expansion can have only three shapes. It can be terminating, meaning the digits stop, as with 3/8 = 0.375. It can be repeating, meaning that from some point on a block of digits repeats forever, as with 1/3 = 0.333… or 5/12 = 0.41666…, where the block is a single digit, or 1/7 = 0.142857142857…, where the block is six digits long. Or it can be non-repeating and non-terminating: the digits never stop and never settle into a repeating block. The bar notation records a repeating block, so 0.41666… is written 0.416 with a bar over the final 6, and 0.142857… is written with a bar over all six digits. A fraction of integers, as the next paragraph shows, can produce only the first two shapes; the third shape belongs to the irrational numbers.

Why a fraction always terminates or repeats. Consider the long division for 1/7. At each step the remainder is one of the numbers 0, 1, 2, 3, 4, 5 or 6, because a remainder is always smaller than the divisor. If the remainder is ever 0, the expansion terminates. Otherwise, after at most six steps some remainder must show up a second time, and from that step on the digits repeat exactly, because the same remainder produces the same next digit and the same next remainder. For 1/7 the remainders run 3, 2, 6, 4, 5, 1 and then 3 again, which is why the block has six digits. The same argument works for any fraction p/q: there are only q possible remainders, so the expansion must terminate or repeat with a block no longer than q − 1 digits. This is the informal reason that rational numbers are exactly the numbers whose decimal expansions terminate or repeat.

Which fractions terminate. Write the fraction in lowest terms and factor the denominator. If the only prime factors are 2 and 5, the expansion terminates, because the fraction can be rescaled to have a denominator that is a power of 10: 3/8 = 375/1000, and 7/20 = 35/100. If any other prime appears in the denominator of the lowest-terms fraction, the expansion repeats: 5/12 has denominator 12 = 2² × 3, and the factor 3 forces 0.41666…. The condition "in lowest terms" matters: 3/6 has a 3 in the denominator, but 3/6 = 1/2 = 0.5 terminates.

Irrational numbers have the third kind of expansion. A number whose decimal expansion never terminates and never repeats cannot be a fraction of integers, because we just saw that every fraction terminates or repeats. So √2 = 1.41421356…, π = 3.14159265… and the number 0.101001000100001…, where the runs of zeros grow by one each time, are all irrational. The last example is useful because its digits follow a clear rule, yet no block ever repeats, so a pattern is not the same thing as a repeating block. A common misconception is that "irrational" means "has a decimal that goes on forever." That is wrong: 1/3 goes on forever and is rational. The test is whether a repeating block exists, not whether the digits stop.

Why a repeating decimal is rational — a worked model inside the lesson. Ohio's Grade 8 standard asks you to understand this informally; the multiply-and-subtract method here is the reason behind it and is an extension, not a required skill. Let x = 0.363636…. The repeating block has two digits, so multiply by 100 to move one full block past the decimal point: 100x = 36.363636…. Now subtract the original equation: 100x − x = 36.363636… − 0.363636…, so 99x = 36, and x = 36/99 = 4/11. The subtraction works because both numbers have the same infinite tail, and equal tails cancel exactly. Check by dividing: 4 ÷ 11 = 0.3636…, which confirms it. When the repeating block starts after some non-repeating digits, use two multiplications. For x = 0.8333…, 10x = 8.333… and 100x = 83.333…; subtracting gives 90x = 75, so x = 75/90 = 5/6. The multiplier for the first equation moves the point to the start of the block, and the multiplier for the second moves it one full block further.

Deciding on a classification. To show a number is rational, produce a fraction of integers, or show its expansion terminates or repeats. To show a number is irrational, you need an argument that no such fraction exists; a calculator display showing many digits with no visible repetition is not proof, because the block might be longer than the screen. In this course you may take √2 and π as known irrationals, and you can also reason that the sum of a rational number and an irrational number is irrational, so 2 + √2 is irrational. A square root of a whole number is rational only when that whole number is a perfect square: √49 = 7 is rational, while √50 is not.

Worked classification: Decide whether 7/40, 2/9, √64 and 0.12112111211112… (runs of 1s growing by one each time, forever) are rational or irrational. 7/40 is a fraction of integers, so it is rational; 40 = 2³ × 5 has only the primes 2 and 5, so its expansion terminates: 7/40 = 175/1000 = 0.175. 2/9 is rational, and because 9 = 3², the expansion repeats: 2/9 = 0.222…. √64 = 8, which equals 8/1, so it is rational even though it is written with a root symbol. 0.12112111211112… has runs of ones that grow by one each time, so no block repeats and the digits never stop; it is irrational.

Worked conversion (extension), block of three: Let x = 0.351351351…. The block 351 has three digits, so 1000x = 351.351351…. Subtracting, 1000x − x = 999x = 351, so x = 351/999. Both are divisible by 27: 351 = 27 × 13 and 999 = 27 × 37, so x = 13/37. Check: 13 ÷ 37 begins 0.351351…, as required.

Worked conversion (extension), delayed block: Let x = 2.0454545…. The block 45 begins after one non-repeating digit. 10x = 20.4545… and 1000x = 2045.4545…. Subtracting the first from the second, 990x = 2025, so x = 2025/990. Dividing numerator and denominator by 45 gives 45/22. Check: 45 ÷ 22 = 2.04545…, which matches.

Worked misconception: A student says 0.777… is irrational because it never ends. Test the definition: the digit 7 repeats forever, so the expansion is repeating, and repeating means rational. Convert it: x = 0.777…, 10x = 7.777…, 9x = 7, so x = 7/9. A number with an endless expansion is irrational only when the expansion is also non-repeating.

Expansion sort: Make a three-column record headed terminating, repeating, and non-repeating non-terminating. Place 5/8, 4/11, 1/6, √9, √7, π, 0.25 and 0.202002000… (runs of zeros that grow by one each time, forever) in the columns by computing or reasoning about each expansion, and beside each rational number write it as a fraction of integers in lowest terms. For each denominator, state its prime factorization and say why that factorization predicts terminating or repeating.

Independent performance: Write nine numbers of your own: three fractions whose denominators have only the primes 2 and 5, three fractions whose denominators have another prime factor, and three numbers that are not fractions of integers (a square root of a non-square whole number, an expression with π, and a decimal you construct whose digits follow a rule but never settle into a repeating block). For each fraction, compute its decimal expansion by long division to at least six digits and say whether it terminates or repeats, naming the repeating block where there is one. For each non-fraction, write the first six digits of its expansion and explain why no repeating block can appear. Then write a short explanation of why a decimal that never ends is not automatically irrational, using one of your repeating examples. Extension, if you wish: convert one of your repeating decimals back to a fraction with the multiply-and-subtract method and check it. Success criteria: every fraction is in lowest terms before its denominator is factored, every classification names the shape of the expansion (terminating, repeating, or non-repeating and non-terminating), and the explanation names the repeating block as the deciding feature.