3.3 Mathematical Expressions

What you need to know

• Operators: +, -, *, /, and MOD. Standard precedence applies: parentheses first, then * / MOD left to right, then + - left to right.
• a MOD b is the remainder when a is divided by b. 17 MOD 5 = 2, 20 MOD 4 = 0, 3 MOD 10 = 3 (if a < b, the result is a).
• MOD uses: n MOD 2 = 0 tests for even; n MOD 10 gets the last digit; i MOD LENGTH(list) wraps an index around; seconds MOD 60 converts to minutes and seconds.
• / in AP pseudocode is regular division that can produce decimals (7 / 2 = 3.5). There is no integer-division operator on the reference sheet.
• A sequence of statements executes in order, one after another. The order changes the result: x ← x * 2 then x ← x + 1 is not the same as the reverse.
• Arithmetic on values of the wrong type (e.g., adding a number to a string) is an error in most languages.

Worked example

a ← 17
b ← a MOD 5
c ← a / 5
d ← (a + 3) MOD 4
DISPLAY(b)
DISPLAY(c)
DISPLAY(d)

b = 17 MOD 5 = 2. c = 17 / 5 = 3.4. d = 20 MOD 4 = 0. Output: 2 3.4 0.

Going deeper

The nuance, edge cases, and connections that turn a 3 into a 5.

• MOD's precedence is the same as multiplication and division — higher than addition and subtraction. 10 + 7 MOD 3 is 10 + 1 = 11, not 17 MOD 3 = 2. When operators are the same precedence, evaluate left to right.
• MOD for wraparound: if you're cycling through positions 1 to n and want to go from n back to 1, the formula is (i MOD n) + 1. Try it: i = n gives (0) + 1 = 1. i = 1 gives 2. This shows up in circular list and clock problems.
• MOD for digit extraction: n MOD 10 is the last digit; n MOD 100 is the last two digits. To get other digits you'd need division, and the exam may define an integer-division procedure for that purpose within a question.
• MOD with a smaller left operand: 3 MOD 7 is 3 — 7 goes in zero times, remainder 3. Students sometimes answer 0 or 4. The remainder of a small number divided by a bigger one is the small number itself.
• AP pseudocode's / is real division: 7 / 2 is 3.5. Unlike Java or Python 2, there's no integer division operator on the reference sheet. If a question needs integer division, it will define a procedure or use MOD to get the same effect.
• Expressions can nest: ((a + b) * c) MOD d. Work inside-out, innermost parentheses first.

Mistakes that cost points

• Applying MOD last. a + b MOD c means a + (b MOD c). Students compute (a + b) MOD c. Same precedence as *, so it goes before +.
• Returning the quotient instead of the remainder. 17 MOD 5 is 2 (remainder), not 3 (quotient). MOD is always the leftover.
• Assuming / truncates. It doesn't in AP pseudocode. 9 / 2 is 4.5.
• Ignoring statement order. x ← x + 1 then x ← x * 2 is not the same as the reverse. Trace in order.

Practice questions

Written in the style of the real exam. Try each one before revealing the answer.

x ← 4
x ← x * 3
x ← x + 2
DISPLAY(x MOD 5)