Arithmetic operators in PHP - Time & Space Complexity

Time Complexity: Arithmetic operators

O(n)

Understanding Time Complexity

When using arithmetic operators in PHP, it's helpful to know how the time to run your code changes as you work with bigger numbers or more calculations.

We want to see how the number of operations grows when using arithmetic operators repeatedly.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

$sum = 0;
for ($i = 1; $i <= $n; $i++) {
    $sum += $i;  // addition operation
}
echo $sum;
    

This code adds numbers from 1 up to n using the addition operator inside a loop.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Addition inside the loop ($sum += $i)
• How many times: Exactly n times, once for each number from 1 to n

How Execution Grows With Input

As n grows, the number of additions grows in the same way.

Input Size (n)	Approx. Operations
10	10 additions
100	100 additions
1000	1000 additions

Pattern observation: The number of operations grows directly with n, so doubling n doubles the work.

Final Time Complexity

Time Complexity: O(n)

This means the time to finish the additions grows in a straight line as the input size increases.

Common Mistake

[X] Wrong: "Arithmetic operations like addition take constant time no matter how many times they run, so the whole loop is constant time."

[OK] Correct: While each addition is quick, doing it n times means the total time grows with n, so the loop is not constant time.

Interview Connect

Understanding how simple arithmetic inside loops affects time helps you explain how your code scales, a key skill in many programming discussions.

Self-Check

"What if we replaced the addition inside the loop with a multiplication? How would the time complexity change?"

Practice

1. Which arithmetic operator in PHP is used to find the remainder of a division?

easy

A. The subtraction operator (-)

B. The modulus operator (%)

C. The multiplication operator (*)

D. The division operator (/)

Solution

1. Step 1: Understand the modulus operator and compare with other operators

   The modulus operator (%) returns the remainder after division of two numbers. Division (/) returns the quotient, multiplication (*) multiplies, and subtraction (-) subtracts.

2. Final Answer:

   The modulus operator (%) -> Option B

3. Quick Check:

   Remainder = % [OK]

Hint: Remainder means modulus (%) operator [OK]

Common Mistakes:

• Confusing division (/) with modulus (%)
• Using multiplication (*) instead of modulus
• Thinking subtraction (-) gives remainder

2. Which of the following is the correct syntax to add two variables $a and $b in PHP?

easy

A. $sum = $a / $b;

B. $sum = $a - $b;

C. $sum = $a * $b;

D. $sum = $a + $b;

Solution

1. Step 1: Identify the addition operator and check the syntax correctness

   In PHP, the plus sign (+) adds two numbers or variables. The syntax $sum = $a + $b; correctly assigns the sum of $a and $b to $sum.

2. Final Answer:

   $sum = $a + $b; -> Option D

3. Quick Check:

   Addition uses + [OK]

Hint: Use + for addition between variables [OK]

Common Mistakes:

• Using - instead of + for addition
• Using * or / which are multiplication/division
• Missing semicolon at end

3. What will be the output of the following PHP code?

$x = 10;
$y = 3;
echo $x % $y;

medium

A. 1

B. 3

C. 0

D. Syntax Error

Solution

1. Step 1: Calculate the modulus of 10 and 3 and understand the output of echo $x % $y;

   10 divided by 3 is 3 with a remainder of 1. The modulus operator (%) returns the remainder, so output is 1.

2. Final Answer:

   1 -> Option A

3. Quick Check:

   10 % 3 = 1 [OK]

Hint: Modulus gives remainder, 10 % 3 = 1 [OK]

Common Mistakes:

• Confusing modulus with division result
• Expecting 3 instead of remainder 1
• Syntax errors from missing semicolon

4. Identify the error in this PHP code snippet:

$a = 5;
$b = 0;
echo $a / $b;

medium

A. Incorrect use of multiplication operator

B. Syntax error due to missing semicolon

C. Division by zero error

D. No error, outputs 0

Solution

1. Step 1: Analyze the division operation and check for syntax correctness

   Dividing any number by zero is not allowed and causes an error in PHP. Semicolons are present; multiplication operator is not used here.

2. Final Answer:

   Division by zero error -> Option C

3. Quick Check:

   Division by zero causes error [OK]

Hint: Never divide by zero to avoid errors [OK]

Common Mistakes:

• Ignoring division by zero error
• Thinking output is 0
• Confusing syntax error with runtime error

5. You want to calculate the total cost after applying a 15% discount on a price stored in $price. Which PHP expression correctly calculates the discounted price?

hard

A. $discounted = $price - ($price * 0.15);

B. $discounted = $price + ($price * 0.15);

C. $discounted = $price * 0.15;

D. $discounted = $price / 0.15;

Solution

1. Step 1: Calculate the discount amount and subtract discount from original price

   Multiply the price by 0.15 to get 15% of the price. Subtract the discount amount from the original price to get the discounted price.

2. Final Answer:

   $discounted = $price - ($price * 0.15); -> Option A

3. Quick Check:

   Discounted price = price - 15% price [OK]

Hint: Subtract 15% of price for discount [OK]

Common Mistakes:

• Adding discount instead of subtracting
• Using only 15% value without subtracting
• Dividing price by 0.15 incorrectly