"Rounding means replacing a number with an approximate value that has a shorter, simpler, or more explicit representation."
The process of approximating a decimal to a given place value.
Decimal notation: Understanding the basics of the decimal system, including place value, and how decimals differ from whole numbers.
Rounding: The process of rounding a decimal to a certain number of decimal places or to the nearest whole number.
Rules for rounding: The guidelines for rounding decimals, including rounding up or down based on the value in the next significant digit.
Significant digits: Identifying the digits in a number that contribute to its precision or accuracy, and understanding how to round based on these digits.
Decimal places: The number of digits after the decimal point in a decimal number, and how to round based on this value.
Rounding errors: The potential for errors or inaccuracies to arise when rounding decimal numbers, and how to minimize these errors.
Estimation: Using rounding to make quick, approximate calculations, and understanding how this technique can be useful in everyday life.
Real-world applications: Understanding how rounding decimals is used in real-world scenarios, such as banking, science, and measurement.
Standard Rounding: This method of rounding involves looking at the digit to the right of the rounding point and rounding the digit based on its value. If the digit to the right is 5 or greater, the rounding digit increases by one.
Rounding Up: This method of rounding involves rounding up to the nearest whole number, regardless of the value of the digit to the right of the rounding point.
Rounding Down: This method of rounding involves rounding down to the nearest whole number, regardless of the value of the digit to the right of the rounding point.
Symmetrical Rounding: This method of rounding rounds to the nearest even number if the digit to the right of the rounding point is 5. For example, 0.5 would be rounded to 0, and 1.5 would be rounded to 2.
Truncation: This method of rounding simply truncates all digits after the rounding point. For example, 3.14159 to one decimal place would be 3.1.
Ceiling Rounding: This method of rounding rounds up to the next whole number, regardless of the value of the digit to the right of the rounding point.
Floor Rounding: This method of rounding rounds down to the next whole number, regardless of the value of the digit to the right of the rounding point.
Significant Figures Rounding: This method of rounding involves determining the number of significant figures in the number being rounded and adjusting the rounding point to that value. For example, if rounding to two significant figures, 1.234 would be rounded to 1.2.
Alternate Rounding: This method of rounding involves rounding to alternating numbers, such as 0 and 5. For example, 3.15 would be rounded to 3.2 using this method.
"Rounding is often done to obtain a value that is easier to report and communicate than the original."
"Rounding can also be important to avoid misleadingly precise reporting of a computed number, measurement, or estimate."
"On the other hand, rounding of exact numbers will introduce some round-off error in the reported result."
"In a sequence of calculations, these rounding errors generally accumulate, and in certain ill-conditioned cases they may make the result meaningless."
"The number of extra digits that need to be calculated to resolve whether to round up or down cannot be known in advance."
"A wavy equals sign (≈, approximately equal to) is sometimes used to indicate rounding of exact numbers."
- Rounding should be done by a function. - Calculations done with rounding should be close to those done without rounding. - The output from rounding should be close to its input. - Rounding should preserve symmetries that already exist between the domain and range. - Rounding should have utility in computer science or human arithmetic and consider speed.
"As a general rule, rounding is idempotent; i.e., once a number has been rounded, rounding it again will not change its value."
"Rounding functions are also monotonic; i.e., rounding a larger number gives a larger or equal result than rounding a smaller number."
"With finite precision (or a discrete domain), this translates to removing bias."
"Rounding has many similarities to the quantization that occurs when physical quantities must be encoded by numbers or digital signals."
"The number of extra digits that need to be calculated to resolve whether to round up or down cannot be known in advance. This problem is known as 'the table-maker's dilemma'."
"To obtain a value that is easier to report and communicate than the original."
"A quantity that was computed as 123,456 but is known to be accurate only to within a few hundred units is usually better stated as 'about 123,500'."
"Rounding is almost unavoidable when reporting many computations – especially when dividing two numbers in integer or fixed-point arithmetic; when computing mathematical functions such as square roots, logarithms, and sines; or when using a floating-point representation with a fixed number of significant digits."
"In certain ill-conditioned cases, rounding errors may make the result meaningless."
"Rounding means replacing a number with an approximate value that has a shorter, simpler, or more explicit representation."
"Rounding is often done to obtain a value that is easier to report and communicate than the original."
"Rounding can be important to avoid misleadingly precise reporting of a computed number, measurement, or estimate."