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Period: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">PeriodPeriod: Definition and ExamplesTable of ContentsPeriod in Math Definition of Period in Mathematics

Period in mathematics has multiple meanings across different contexts. In function analysis, a period refers to the interval at which a function repeats itself. A function is considered periodic with period T>0T > 0T>0 if f(x+T)=f(x)f(x + T) = f(x)f(x+T)=f(x) for all values of xxx in the domain. The smallest positive value of TTT is called the fundamental period. Trigonometric functions like sine and cosine are classic examples of periodic functions.

Periods also appear in other mathematical contexts. In place value systems, a period represents a group of three digits separated by commas when writing numbers in standard form, making large numbers easier to read. In decimal representations, a period refers to the recurring part of a repeating decimal. For example, in 0.3‾0.\overline{3}0.3 (or 0.333...0.333...0.333... ), the digit 333 is the period with a length of 111. The term also relates to time measurement, representing the completion of one cycle.

Examples of Period in Mathematics Example 1: Finding the Period of a Fraction as a Decimal Problem:

Find the period of 56\frac{5}{6}65​.

Step-by-step solution: Step 1, Convert the fraction into decimal form. Step 2, Write out the division: 56=0.833333333...=0.83‾\frac{5}{6} = 0.833333333... = 0.8\overline{3}65​=0.833333333...=0.83 Step 3, Identify the recurring part, which is 333. Step 4, Determine the period: The period of 56\frac{5}{6}65​ is 333. Step 5, Find the length of the period: Length of period =1= 1=1 (since only one digit repeats). Example 2: Identifying Periods in Place Value Chart Problem:

Find the period of 7 in 85,476,280.

Step-by-step solution: Step 1, Write the number with commas to separate periods: 85,476,280 Step 2, Identify the periods from right to left: ones period (280), thousands period (476), millions period (85) Step 3, Locate the digit 7 in the number: It appears in the thousands period Step 4, Determine the place value of 7 in 85,476,280: It equals 70,000 Example 3: Calculating the Period of a Trigonometric Function Problem:

Find the period of f(x)=cos(3x)f(x) = cos(3x)f(x)=cos(3x) if the period of cos(x)cos(x)cos(x) is 2π2\pi2π.

Step-by-step solution: Step 1, Recall that the period of the parent function cos(x)cos(x)cos(x) is 2π2\pi2π. Step 2, Identify the coefficient of xxx in the given function: The coefficient is 333. Step 3, Apply the period formula for functions: Period = Period of parent function∣Coefficient of x∣\frac{\text{Period of parent function}}{|\text{Coefficient of x}|}∣Coefficient of x∣Period of parent function​ Step 4, Substitute the values into the formula: Period = 2π3\frac{2\pi}{3}32π​ Step 5, Therefore, the period of the function f(x)=cos(3x)f(x) = cos(3x)f(x)=cos(3x) is 2π3\frac{2\pi}{3}32π​ Comments(2)PPlantParentHankNovember 6, 2025This glossary def of 'period' is great! I've used it to explain to my students. It really cleared up their confusion. Thanks!

MCMs. CarterSeptember 17, 2025I’ve used this clear definition of 'period' to help my students understand repeating decimals and trig functions better. The examples really brought it to life for them—thanks for making math less intimidating!

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