Decimal

• 1. Introduction

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  Decimal is the numeral system that underlies virtually every number you encounter in daily life, from a grocery receipt to a scientific measurement to the year on a calendar. Ten digits, a separator mark, and a minus sign for negatives: that is the entire toolkit. Yet behind this apparent simplicity lies a long, contentious history stretching back to ancient Egypt and the Indus Valley, passing through medieval China and the Islamic world, and arriving only in the 16th century at the notation Europeans recognise today.

  How did ten become the universal base? Why does writing 0.080 instead of 0.08 actually mean something different to a scientist? And what does a 13th-century Chinese book about rod calculus have to do with the decimal fractions on your bank statement? Those are the threads this documentary follows.

• 2. Why Ten?

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  Egyptian hieroglyphs have been using a decimal system since around 3000 BCE, and the reason is almost certainly anatomical. Two human hands carry ten fingers, and counting on fingers is the most natural first step toward a formal number system.

  The Indus Valley Civilisation, dated roughly 3300-1300 BCE, shows just how early this logic took hold. Standardised weights from that culture follow ratios built on ten: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500. Their measuring rule, the Mohenjo-daro ruler, was divided into ten equal parts.

  The Minoan Linear A script, in use around 1800-1450 BCE, and the Mycenaean Linear B script, around 1400-1200 BCE, both operated on decimal principles. The Únětice culture of central Europe, active from roughly 2300 to 1600 BCE, used standardised weights and a decimal scheme in trade. So did the number system of classical Greece, Roman numerals, and the Hittite hieroglyphs that appear from at least the 15th century BCE.

  Archimedes, the polymath who lived around 287-212 BCE, pushed the concept further. In his Sand Reckoner he invented a decimal positional system based on 10 to the power of 8, specifically to express numbers large enough to count the grains of sand that could fill the universe. The finger-count logic had, by then, grown ambitious enough to contemplate the cosmos.

• 3. Positional Versus Non-Positional

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  Egyptian numerals needed separate symbols for 10, for each of the multiples of 10 from 20 to 90, for 100, for each multiple of 100 from 200 to 900, for 1,000, and so on up to 10,000. That is an enormous inventory of symbols just to write a moderately large number, and multiplying or dividing such numbers was an exercise reserved for the most skilled mathematicians.

  The world's earliest positional decimal system, where the position of a digit carries its value, was the Chinese rod calculus. In a positional system, the same symbol means different things depending on where it sits: a 3 in the hundreds column is worth 300, not 3. That single insight collapses the need for dozens of distinct symbols.

  The Hindu-Arabic numeral system, the one now used globally, extended this positional logic to cover both integers and fractions. It built on the Brahmi numerals of India, was carried into the Islamic world, and eventually reached Europe. The ten Arabic digits, 0 through 9, are the direct descendants of that tradition.

  Non-positional decimal systems, including Roman numerals and Chinese numerals, never disappeared entirely. But the computational advantages of the positional Hindu-Arabic approach were decisive. Very large numbers, once the province of elite mathematicians, became routine.

• 4. Decimal Fractions Emerge

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  Chinese units of length were already being expressed in decimal fractions by the 3rd century CE, though still non-positionally. The Sunzi Suanjing, a text from the 3rd to 5th century CE, describes performing calculations with these fractional lengths using positional counting rods.

  The 5th-century mathematician Zu Chongzhi calculated pi to seven decimal digits, a precision that stood as a world record for centuries. Qin Jiushao's Mathematical Treatise in Nine Sections, written in 1247, goes further: it explicitly writes a decimal fraction as a number in its own right, not merely as a measurement. The number 0.96644 appears in that text using counting rods.

  Historians of Chinese science have speculated that knowledge of decimal fractions may have traveled from China to the Middle East. Al-Khwarizmi brought fractions into Islamic scholarship in the early 9th century, writing numerators above denominators but without a horizontal dividing bar. That form persisted for centuries.

  Positional decimal fractions appear for the first time in a European-adjacent context in a book by the Arab mathematician Abu'l-Hasan al-Uqlidisi, written in the 10th century. The Jewish mathematician Immanuel Bonfils worked with decimal fractions around 1350 but devised no notation for them. Then came the Persian mathematician Jamshid al-Kashi in the 15th century. In his work The Key to Arithmetic, known in Arabic as Miftah al-Hisab, al-Kashi provided the first systematic and comprehensive treatment of decimal fractions as a complete system, predating comparable European work by nearly 175 years.

• 5. European Notation Arrives

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  Simon Stevin, a Flemish mathematician working in the 16th century, wrote a short book that changed how Europeans recorded numbers. His booklet De Thiende, which translates roughly as "the art of tenths," was first published in Dutch in 1585. A French translation, La Disme, followed soon after. Stevin's notation was more cumbersome than what we use today, but the underlying principle, that fractions could be expressed as extensions of the integer system using a separator, was the forerunner of modern decimal notation.

  The clean separation we take for granted, a period or a comma between the whole part and the fractional part, was standardised by John Napier. In his book on constructing logarithm tables, published posthumously in 1620, Napier introduced the period as the decimal separator.

  Today the separator varies by country: a dot in most English-speaking nations, a comma in many others. The convention is typographic, not mathematical. The underlying value of 3.14159 and 3,14159 is identical; only the symbol differs.

• 6. Precision and Measurement

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  0.080 and 0.08 represent the same real number. Yet in a scientific report they carry different meanings. Writing 0.080 signals that the measurement is reliable to within 0.001; writing 0.08 signals reliability only to within 0.01.

  The same logic applies to expressing mass. A measurement given as 1.32 milligrams implies the true value lies somewhere between 1.315 and 1.325 milligrams. Expressed as 1.320 milligrams, the implied range tightens to between 1.3195 and 1.3205 milligrams. The trailing zero is not decorative; it is a claim about accuracy.

  This convention extends into pure mathematics. Computing the square root of 22 to two decimal places gives 4.69. Computing it to three decimal places gives 4.690. The extra zero signals that the computation was carried out to that additional digit, and that the digit is in fact zero rather than unknown.

  Real numbers in general cannot be represented exactly as finite decimals. A decimal can, however, approximate any real number to any desired precision by adding more digits. The number pi, for instance, is approximated by 3.14159 with an error smaller than 10 to the power of negative 5. By extending the decimal expansion, the error can be made arbitrarily small.

• 7. Repeating and Terminating Decimals

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  A decimal that reaches a point where all remaining digits are zero is called a terminating decimal. The fraction 1/4 terminates as 0.25. Whether a fraction terminates depends on its denominator: a fraction terminates in decimal form if and only if the denominator, in its fully reduced form, is a product of a power of 2 and a power of 5.

  The fraction 1/3 does not terminate, because 3 is neither a power of 2 nor a power of 5. Instead it produces the repeating decimal 0.333..., where the digit 3 continues without end. Every rational number, meaning every ratio of two integers, produces either a terminating decimal or a repeating one. And the converse holds: any decimal that eventually settles into a repeating pattern is rational.

  The source text gives a worked example with a number that starts 0.4156156156... To convert it to a fraction, the method uses the repeating block's length. Multiplying by 10,000 and by 10 and subtracting isolates the repeating part, eventually yielding a fraction. The final reduced form is found by dividing numerator and denominator by their common factor of 6.

  A more striking case is 0.012345679012..., where the group 012345679 repeats indefinitely. The missing 8 in that sequence is not a typo; it is a genuine feature of how that particular rational number unfolds in decimal form.

• 8. Counting in Other Bases

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  Base ten is not the only option humans have tried, and several living languages retain the evidence. Pre-Columbian Mesoamerican cultures, including the Maya, used base 20, most likely because they counted fingers and toes together.

  The Yuki language of California and the Oto-Pamean languages of Mexico use base 8. Their speakers count the spaces between the fingers, not the fingers themselves, which yields eight intervals rather than ten digits.

  The Huli language of Papua New Guinea has base-15 numbers. The word ngui means 15, ngui ki means 30, and ngui ngui means 225. The Ndom language, also from Papua New Guinea, operates in base 6: mer is 6, nif is 36, and nif thef is 72.

  Early Germanic languages carry traces of a base-12 or base-120 counting system. Words meaning "count in tens" or "tenty-wise" appear in those languages, which scholars read as a sign that decimal counting was at one point considered unusual enough to need labeling. The "long hundred" of 120 and the "long thousand" of 1,200 persisted in Scotland through the Middle Ages, where records show calculations in which one hundred carried a value of 120. Goodare's research documents this with examples from Scottish accounts, including usages of "vii score" to sidestep the ambiguous hundred altogether. The Chumashan languages of California used base 4, structuring their number names around multiples of 4 and 16. The Huli base-15 system, by contrast, points to a world where the universal adoption of ten was never inevitable, only dominant.