Editing Openai/68a11068-3fe8-8007-bd18-674462ff2b41 (section)

=== Assistant: Perfect — let’s trace the history of why implicit multiplication sometimes “feels stronger” and why conventions split. ===
Perfect — let’s trace the history of why implicit multiplication sometimes “feels stronger” and why conventions split.

==== - In early Greek, Indian, and Islamic mathematics, multiplication was described in words, not symbols. - Example: “two multiplied by a thing” = 2x2x2x. ====
* Juxtaposition (writing things next to each other) became the natural shorthand.

👉 Already, 2x2x2x meant a single object — it was never read as “2÷x2 \div x2÷x” or anything ambiguous.

==== - François Viète and René Descartes popularized symbols. ====
* Descartes (1637) introduced juxtaposition as the preferred notation for multiplication (no ×). - Example: ababab, 2x2x2x, πr2\pi r^2πr2.
* Division was often written with a stacked bar: abcd\frac{ab}{cd}cdab not with a slash.

👉 No ambiguity, because a horizontal bar clearly grouped numerator and denominator.

==== - Printing made it easier to show fraction bars, so most algebra books avoided the slash /. ====
* Implicit multiplication (ababab) was still dominant.
* Students were taught: “things next to each other form a product.”
* Division written with / was considered “sloppy” shorthand, mainly used on chalkboards.

👉 Implicit multiplication was understood as tighter than a loose slash.

==== - Pocket calculators introduced the / key for division. ====
* They needed a linear, one-line grammar: - Multiplication × and division / treated as equal precedence. - Evaluated strictly left to right.
* Example: 50/2×5 = (50/2)×5 = 125.

👉 This flattened the “implicit is stronger” convention into something simpler for machines.

==== - Programming languages (Fortran, C, Python, etc.) adopted calculator-style precedence: - * and / have equal precedence, left to right. ====
* But since implicit multiplication (2x) isn’t valid syntax in most languages, the historical nuance was lost.

👉 Now students meet computers before deep algebra, so they learn the programming convention, not the historical one.

==== - School & programming: use PEMDAS, treat × and / equally. ====
* Algebra & professional math: avoid /, use fraction bars or parentheses, keep implicit multiplication “visually tight.”

That’s why we get debates like:
* 50/2(1+4)50/2(1+4)50/2(1+4): ambiguous, depending on whether you follow the calculator rule (125) or the algebra habit (5).
* ac/bcac/bcac/bc: ambiguous, because people see a stacked fraction even if the left-to-right rule says otherwise.

✅ Summary:
* Implicit multiplication feels stronger because historically it was always written to show a single unit (e.g., 2x2x2x, πr2\pi r^2πr2).
* The / slash is a modern, linear compromise invented for calculators and computers, not classical math.
* Professional math avoids the problem altogether: use stacked fractions or parentheses.

Would you like me to draw up a timeline diagram showing this evolution visually (ancient → Descartes → textbooks → calculators → programming)?