Prime Symbol — Definition, Formula & Examples
The prime symbol (′) is a small tick mark written as a superscript after a letter or number. It indicates a derivative in calculus (like ), a measurement in feet or arcminutes (like ), or a related-but-different version of a variable (like ).
The prime symbol, denoted ′, is a modifier placed to the upper right of a variable, function name, or numeral. In calculus, denotes the first derivative of with respect to . In measurement, a single prime indicates feet or arcminutes, while a double prime (″) indicates inches or arcseconds. In geometry and algebra, primes distinguish transformed or corresponding points, such as being the image of under a transformation.
How It Works
You write the prime symbol as a short diagonal stroke immediately after and slightly above the character it modifies. One prime mark (′) is called "prime," two marks (″) are called "double prime," and three (‴) are called "triple prime." In calculus, each additional prime mark indicates a higher-order derivative: is the first derivative, is the second, and is the third. In measurement contexts, primes and double primes always come after a number: means 6 feet 2 inches. Context determines which meaning applies — if you see the symbol on a function in a calculus course, it means a derivative; if you see it after a number with no function, it likely means a unit of length or angle.
Worked Example
Problem: A point is located at . After reflecting it over the -axis, write the coordinates of the image using prime notation.
Step 1: Reflecting a point over the -axis negates its -coordinate while keeping the -coordinate the same.
Step 2: Apply the reflection to point and label the image (read "A prime").
Answer: The image of is .
Why It Matters
You will encounter the prime symbol constantly in calculus when writing derivatives, in geometry when labeling transformed figures, and in physics or engineering when converting between feet/inches and degrees/minutes/seconds. Recognizing which meaning is intended from context is essential for reading mathematical notation fluently.
Common Mistakes
Mistake: Confusing the prime symbol (′) with an apostrophe (') or a single quotation mark.
Correction: The prime symbol is a distinct Unicode character (U+2032) that slants to the right and sits at superscript height. In handwriting and many typeset documents, an apostrophe is used as a substitute, but in precise notation they differ. More importantly, don't confuse the meaning — an apostrophe shows possession in English, while a prime mark carries mathematical meaning.