PERIODIC AND NONPERIODIC DECIMAL NUMBERS

The summing and differencing of RATIONAL NUMBERS in "fractional form" (a/b) requires conversion to common denominators. TEEEDJUS! So a SHORTCUT was developed: the "decimal number" with denominator a power of 10, as in 1/8 = 0.125. This representation is DEFAULT for "irrational real numbers". Herein, however, we study "decimals" in the RATIONAL NUMBER SYSTEM.

Given 1/8 = 0.125, the "decimal number" (on the right side) has three "parts":

  1. The decimal point ("."), separating the "decimal string" into a "left side" and "right side".
  2. The "left side" is the characteristic of the decimal number. (In 0.125, the characteristic is 0.)
  3. The "right side" is the mantissa of the decimal number. (In 0.125, the mantissa is 125).

We can now define a rational number in terms of the decimal number epresentation:

  1. A number in decimal form is RATIONAL if, starting at some point in its MANTISSA, it BECOMES PERIODIC WITH a DIGIT-STRING that REPEATS INFINITELY. Example: 1/3 = 0.33333.... (Let's write that as 0.|3|, meaning that digits between the verticals form THE PERIOD and are repeated.) Another example: 1/7 = 0.|142857|.

  2. If the decimal form is NONPERIODIC (without a 'repeater'), then the number is IRRATIONAL, as in the square root of 2.
THEOREM: Any RATIONAL NUMBER has a PERIODIC DECIMAL EXPANSION; conversely, ANY PERIOD DECIMAL REPRESENTS A RATIONAL NUMBER. (Proof.)

Fraction → decimal representation is easy: divide denominator into numerator. But we need also to understand decimal representation → fraction.

Let's practice the method by a case we already understand 1/3 = .|3|.

  1. Start by pretending we don't know 1/3, but have n = .|3|.
  2. We use n to generating another with the same "repeater" (and same mantissa), so that we can difference two different numbers with the same mantissas. That's easy. We form 10n = 3.|3|. If we form the difference 10n - n, then the equal mantissas "cancel out" and we have 10n — n = 9n = 3.
  3. Given 9n = 3, we divide both sides by 3, and reduce "the fraction" to obtain: n = 3/9 = 1/3 -- our desired answer.

Let's try that with a problem wherein we don't know the answer at the outset:

  1. Let n = 0.1695|737|. But, unlike the .|3| case, the period doesn't begin immediately after the decimal point. So we need to bring the PERIOD up to the decimal point.

  2. With four digits in between decimal point and "repeater", we multiply by 10 · 10 · 10 · 10 = 10000. Then, we have 10000n = 1695.|737|.

  3. Next, we need to bring "one whole PERIOD across the decimal point". The PERIOD has three places so we multiply by 10 · 10 · 10 = 1000. We obtain 1000 · 10000n = 10000000n = 1695737.|737|.

  4. We now have two numbers, 10000000n and 1000n have equal mantissas. So their difference ELIMINATES THE MANTISSA. 10000000n - 10000n = 999000n = 1695737.|737| — 1695.|737| = 1694042.

  5. We now have 999000n = 1694042, or n = 1694042/9990000, the FRACTIONAL FORM of this number, which may be reducible.

We've seen, above, TWO KINDS OF REPEATNG DECIMALS REDUCIBLE TO "FRACTIONS":

  1. The "pure kind", such as 0.|3| = 1/3, wherein the "period" begins immediately after the DECIMAL POINT.
  2. The "mixed kind", such as 0.1695|737| = 1694042/9990000, wherein the "period" does not begin with DECIMAL POINT. The MIXED type occurs whenever the FRACTIONAL DENOMINATOR contains 2 or 5, the FACTORS of THE BASE, 10.

A RATIONAL NUMBER such as 1/3, when expressed decimally displays itself as THE SUM OF A GEOMETRIC PROGRESSION (GP): a MANTISSA of 3's repeated infinitely.

As another case:

We learn from these two cases of fraction → GP that 1/3 = 0.|3|, with a 'repeater' of a SINGLE DIGIT, while 1/7 = 0.|142857|, with a PERIOD of SIX DIGITS. A PRINCIPLE is involved here.

Let 1/p denote a FRACTION, with p any PRIME NUMBER. Then, 1/p equals a DECIMAL FRACTION with a REPEATER of LENGTH p - 1, or a FACTOR OF p - 1.

We can learn from this how to deal with THE FAILURE OF THE LOGARITHMIC INVERSE OF EXPONENTIATION to be TOTAL.

We learn how to SHORTCUT a LENGTHY GP by a simple "algebraic" fraction". Remember? In the above cases, we found two numbers with the same mantissas so that differencing will ELIMINATE THE MANTISSAS, leaving only THE CHARACTERISTIC.

  1. Formally write the GENERAL SUMMING GP as: S = a + ar + ar2 + ar3 + ... + ar(n - 1).

  2. To obtain another SUM with all the "mid terms", we multiply S above by r:
    rS = ar + ar2 + ar3 + ar4 + ... + ar(n - 1) + arn.

  3. The SUMS S and rS DIFFER (on the right-hand side) ONLY IN THEIR FIRST AND LAST TERMS. So, DIFFERENCING YIELDS: S - rS = a - arn.

  4. We can FACTOR OUT THE COMMON TERMS, on left and right: S(1 - r) = a(1 - rn).

  5. We could solve for S by dividing both side by (1 - r), but we need to make sure we aren't DIVIDING BY ZERO. Let r != 0. Then we have: S = a (1 - rn)/(1 - r) = a(1 - r) - arn/(1 - r).

  6. Look at the term, rn/(1 - r). If, say, r = 1/2, r2 = 1/4, r3 = 1/8, etc. As n INCREASES, rn DECREASES -- DECREASING THE - arn/(1 - r) TERM TOWARDS ZERO.

  7. As "n goes to infinity", S "goes to the LIMIT a(1 - r)", a DEFINITE NUMBER.

We ,A NAME="learn">learn from this that THE INFINITE GP SUMS, REPRESENTED BY "DECIMALS", HAVE LIMITS AS RATIONAL NUMBERS!

Are there NONREPEATER INFINITE SUMS or DECIMALS THAT HAVE LIMITS? We can CONSTRUCT one.

Take 1/9 = 0.1111111... = 0.|1|. Let's use that to CONSTRUCT a NONREPEATING DEFINITE NUMBER: after the first "1" place one "0"; after the second "1" place two zeros; after the third "1", three zeros; after the nth "1" in 1/9's expansion place n zeros; etc. Thus, m = 0.10100100010000....

But m is a MEANINGFUL NUMBER. Just as we can PREDICT ANY DIGIT OF THE DECIMAL EXPANSION OF A RATIONAL, so we can predict any digit of m. The "1" in m occurs at every n(n + 1)/2 position; otherwise, the digit is "0". But we note that m has a LIMIT, since, for example, m < 2.

We can show this for that famous IRRATIONAL NUMBER, (2)1/2:

  • 12 = 1 < 2 < 22 = 4;
  • (1.4)2 = 1.96 < 2 < (1.5)2 = 2.25;
  • (1.41)2 = 1.9881 < 2 < (1.42)2 = 2.0264;
  • (1.414)2 = 1.999396 < 2 < (1.415)2 = 2.002225;
  • (1.4142)2 = 1.99996164 < 2 < (1.4143)2 = 2.00024449;
  • Etc.

SUMMARY

  1. Every nonterminating decimal expression (repeating or not) represents a sequence with a "limit".

  2. Such a "limit" DEFINES A REAL NUMBER.
  3. Hence, every nonterminating decimal expression (repeating or not) is a REAL NUMBER. (Thus, m = 0.10100100010000.... is a REAL NUMBER.)

  4. Therefore, every RATIONAL NUMBER can only be represented as
    • a finite terminating decimal expression
    • or a nonterminating periodic decimal expression.

  5. A number such as m = 0.10100100010000...., being neither, is not a RATIONAL NUMBER.

  6. Hence, REAL NUMBERS exist which are not RATIONAL, that is, which are IRRATIONAL.

How do we FIND or CONSTRUCT a NUMBER SYSTEM containing "m", (2)1/2, p, ETC.?

Return to LIMIT to learn more.