Orders of Operation in Math and Magic: A Shared Logic of Numbers and Potions

In the great halls of Candlekeep, scholars often debate the structure of thought. Some argue that mathematics is the purest language, while others claim alchemy and spellcraft hold the deeper truth. Yet both disciplines share a single, crucial principle: order matters. Just as the Waterdeep Trading Company must post its ledgers in the right sequence to keep its coffers balanced, so too must wizards, merchants, and potion-brewers follow a strict order of operations, lest their results turn to chaos.

The Order of Numbers: From Scrolls to Schools

Modern arithmetic across Faerûn mirrors the conventions of our age: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction, all working left to right. An apprentice scribe in Baldur’s Gate, faced with the sum 6 + 7 × 2, is taught to find 20, not 26.

Yet this has not always been so. In early ages, mathematicians wrote their problems in words, not symbols, and most reckoning was done from left to right. Early mechanical calculators, gnomish clockworks from Lantan in the 1600s DR, also marched one step at a time, yielding 26 in the example above.

By the 1800s DR, however, standardized teaching from Silverymoon to Waterdeep had settled on the modern sequence. Schools, guilds, and later even enchanted calculators adopted the familiar PEMDAS rule. Order brought certainty to trade, engineering, and spellcraft alike.

Alternate Conventions: When Rules Shift

Different traditions still echo through history:

• Strict left-to-right: Found in early scrolls and basic counting tools.
• Forced grouping: With parentheses or magical brackets, results vary by design.
• Reverse Polish Notation: Favored by gnomish artificers, where symbols follow numbers (6 7 2 × + = 20).

These alternate rules remind us that clarity in notation prevents costly mistakes, whether in a sum of coins or in a contract for planar trade.

The Order of Potions: A Magical Precedence

Alchemy and potion-making mirror mathematics in their need for sequence. A healing draught brewed without structure will curdle into poison or burst into flame. Master apothecaries agree on six stages of precedence:

• Containment – vessels, seals, and wards to hold volatile energy.
• Catalyst – application of heat, moonlight, or ley energy.
• Primary Ingredients – herbs, minerals, monster parts.
• Stabilizers – salts, oils, or arcane binders.
• Essences & Distillates – phoenix tears, celestial extracts, rare alchemical spirits.
• Final Enchantments – chants, sigils, or breathwork to bind the brew.

Skipping a step is like ignoring parentheses in mathematics: it invites disaster. The glow of a proper healing potion depends entirely on faithful order.

The “Magical PEMDAS” Mnemonic

To guide apprentices, the guilds of Waterdeep teach the mnemonic:

C-C-P-S-E-F Order
“Careful Cauldrons Prevent Sudden Explosions, Fool.”

• Careful = Containment
• Cauldrons = Catalyst
• Prevent = Primary
• Sudden = Stabilizers
• Explosions = Essences
• Fool = Final Enchantments

Much as a scribe remembers PEMDAS for sums, an alchemist recalls C-C-P-S-E-F for potions. Both disciplines demand rigor.

Worked Example: Formulation and Route of a Healing Draught

Just as an accountant breaks down a ledger entry into debits and credits, an alchemist or apothecary breaks down a potion into its formulation (ingredients and costs) and route (the ordered sequence of preparation).

Below is how the Waterdeep Trading Company would record the production of a standard Healing Draught in both detail and sequence.

Result: A properly prepared Healing Draught that restores vitality and knits minor wounds when consumed.

A Shared Lesson for Trade and Craft

The lesson is plain: numbers and magic alike require structure. The Waterdeep Trading Company, having abandoned quill-and-scroll accounts for the precise ledgers of Dynamics 365, knows this truth well. Just as its clerks follow accounting rules when recording coins, so too must potion-brewers follow the order of their craft.

An error in one yields a wrong answer on parchment; an error in the other may yield an explosion in the apothecary. In both cases, order is everything.

Final Thoughts

Mathematics and magic are not separate languages, but dialects of order. To disregard precedence is to invite chaos, whether in a failed equation, a broken ledger, or a firebomb masquerading as a healing potion. For students, merchants, and alchemists alike, the rule remains eternal: follow the sequence, or face the consequences.

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