Formulas: order, substitution and rearranging

print(10 - 4 / 2 * 3)

Which part comes first

A calculation with several operators needs an agreed order, or two people get two answers. Python uses the same order as school math:

1. Brackets first.
2. Then powers (**).
3. Then multiplying and dividing (*, /, //, %), left to right.
4. Then adding and subtracting, left to right.

print(2 + 3 * 4)        # 3 * 4 first: 14
print((2 + 3) * 4)      # brackets first: 20
print(2 * 3 ** 2)       # the power first: 18
print(-3 ** 2)          # the power before the minus sign: -9

The last line catches nearly everyone: -3 ** 2 is −(3²), not (−3)². When in doubt, add brackets. They cost nothing, and they show the next reader exactly what you meant.

"Left to right" matters more than it looks. 8 / 4 / 2 is (8 / 4) / 2 = 1.0, not 8 / (4 / 2) = 4.0. Operators of the same rank are worked out in the order you read them, so if you mean the other grouping, you need brackets.

Letters stand for numbers

A formula uses letters for the numbers that change. "fare = 3.50 + 2.25 × miles" works for any trip: put the miles in, and out comes the fare. Putting a number in for a letter is called substituting, and Python variables are exactly this:

miles = 8
fare = 3.50 + 2.25 * miles
print(fare)

The order rules still apply after substituting: here the multiplication happens before the addition, so the 2.25 is charged per mile and the 3.50 only once. Many formulas give their letters short names, like w for a weight or n for a count. They're still just numbers waiting to be filled in.

miles = 8
fare = 3.50 + 2.25 * miles
print(fare)

input_tokens = 12000
output_tokens = 800
input_price = 3      # dollars per million input tokens
output_price = 15    # dollars per million output tokens

cost = 0

print("This call cost $", cost)

From paper to code

Formulas on paper leave out things that code has to spell out:

• 2x means 2 × x, written 2 * x in Python, and xy means x × y.
• A small raised number is a power: x² is x ** 2.
• A dot, ·, or a cross, ×, means multiply; ÷ and / both mean divide.
• A fraction bar groups everything above it and everything below it, as if each had brackets: (a + b) over 2 is (a + b) / 2 in code, never a + b / 2.

Copying a formula from a page into code is where most of these mistakes creep in, and Python won't warn you, because the wrong version is still valid code, so read each part of the formula aloud and check the brackets in your code match it.

Rearranging: finding the missing number

Sometimes you know the answer and need one of the inputs. A ride cost $21.50; how many miles was it? The formula gives the fare from the miles, so you turn it around. The rule: do the same thing to both sides, undoing the steps in reverse order.

The fare was worked out as "times 2.25, then add 3.50". To undo it, subtract 3.50 first, then divide by 2.25:

fare = 21.50
miles = (fare - 3.50) / 2.25
print(miles)
print(3.50 + 2.25 * miles)    # substitute back: the fare again

Substituting the answer back into the original formula is the best check there is: if you get the number you started with, the rearranging was right. The same steps turn y = 2x + 3 into x = (y − 3) / 2.

Percentages are where rearranging goes wrong most often. A jacket costs $48 after 20% off; what was the full price? The sale price was full × 0.8, so the full price is 48 ÷ 0.8 = $60. Adding 20% back, 48 × 1.2, gives $57.60, which is wrong: 20% of the smaller sale price is less than 20% of the original. Undo a multiplication by dividing.

price = 25
cost = 10
setup = 3000

profit_at_150 = 0
break_even = 0

print(profit_at_150, break_even)

Checking a formula

Three quick habits catch most formula mistakes:

• Units. Dollars per mile times miles gives dollars; if your units come out as "dollars squared", something is multiplied that shouldn't be.
• An easy value. Put in a number you can work out in your head. A 0-mile taxi ride should cost the fixed 3.50, and a 1-mile ride should cost 3.50 + 2.25 = 5.75. If either comes out differently, you've found the bug.
• A known case. If you know one real answer, like last month's bill, check that the formula reproduces it exactly, to the cent.

These checks take seconds, and they're the same ones you'll use on a model's loss function or a cost estimate in later phases. A formula that passes all three isn't guaranteed to be right, but one that fails any of them is certainly wrong, and finding that out early is cheap.

sales = 150
profit = sales * (25 - 10) - 3000
print(profit)

Formulas in AI

The calculator above is a neuron: y = w × x + b, with a weight and a bias that a model learns from data. A model's loss is a formula that scores how wrong its predictions are. The cost of an API call is a formula of tokens and prices. Even the laws that predict how good a model will get from more data and compute are formulas, with powers in them.

In Phase 2 you'll learn to read math written in symbols, like Σ for "add these up". It's the same three moves: know the order, substitute, and rearrange when you need to. Papers and documentation write formulas in symbols, and your code writes them in Python; moving between the two, carefully, is one of the most useful skills in AI engineering.

x = -3
print(x ** 2, -3 ** 2)