The Order of Operations: Why Basic Calculators Get Math Wrong
Perform complex equations accurately using our Order-of-Operations-aware Basic Calculator, or read on to understand why cheap calculators fail at basic math.
If you type the equation 5 + 5 × 5 into a cheap, four-function desktop calculator, it will likely give you the answer 50.
If you type the exact same equation into a scientific calculator or a smartphone, it will give you the answer 30.
Why did two computers just give you completely different answers for the exact same basic math problem? The answer lies in the Order of Operations.
What is the Order of Operations?
Mathematics is a universal language. Just like English has grammar rules that dictate how a sentence must be read to make sense, mathematics has rules that dictate the order in which an equation must be solved.
If you read 5 + 5 × 5 strictly from left to right (like reading a book), you add the fives to get 10, then multiply by 5 to get 50.
But in mathematics, multiplication is always stronger than addition, regardless of where it appears in the sentence. You must solve the multiplication first (5 × 5 = 25), and then add the remaining 5 to get 30.
PEMDAS / BODMAS Explained
To help students remember the correct order, teachers use acronyms. In the United States, the acronym is PEMDAS (Please Excuse My Dear Aunt Sally). In the UK and other regions, it is BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction).
Both acronyms mean the exact same thing:
- Parentheses (or Brackets)
- Exponents (or Orders/Indices)
- Multiplication & Division (Left to Right)
- Addition & Subtraction (Left to Right)
The Most Common Mistake
Look closely at steps 3 and 4. Multiplication is NOT inherently stronger than Division. They have the exact same level of priority. If a sentence has both, you simply read them from left to right. The same applies to Addition and Subtraction.
Example: 10 - 4 + 2
Many people think you must do Addition before Subtraction because 'A' comes before 'S' in PEMDAS. If you do that, you get 4 + 2 = 6, and then 10 - 6 = 4. This is wrong.
Because addition and subtraction are equal, you just go left to right: 10 - 4 = 6, and then 6 + 2 = 8.
Why Cheap Calculators Fail
A basic, cheap physical calculator has no internal memory or understanding of grammar. It operates as a "Left-to-Right Execution Engine."
When you type 5 + 5, it immediately calculates 10. When you press × 5, it simply multiplies the current screen total by 5, resulting in 50. It physically cannot wait to see what the rest of the equation is before calculating the first part.
Scientific calculators (and our CalcUnit online calculators) are built with expression parsers. They read the entire mathematical sentence first, apply the rules of PEMDAS to determine priority, and then execute the math perfectly.
FAQ
Why do we use Parentheses?
Parentheses act as an override code. They tell the mathematician (or the computer) to ignore the standard rules of PEMDAS and prioritize whatever is inside the brackets first. If you actually wanted the answer to be 50 in our original example, you would write it as (5 + 5) × 5.
Never trust a cheap physical calculator with a multi-step equation. Get guaranteed mathematical accuracy using the CalcUnit Basic Calculator.