Understanding lower and upper bounds in mathematics is crucial for defining the range of possible values for rounded numbers. These bounds help in determining the error interval and maintaining the accuracy of numerical estimations. The text explains how to calculate these bounds and apply them in various mathematical operations, such as addition, subtraction, multiplication, and division, as well as in practical scenarios like measuring lengths and calculating areas or speeds.
Show More
Lower and upper bounds are critical for delineating the range within which a rounded number may actually lie in mathematics
Definition of Lower Bound
The lower bound is the greatest conceivable number that, when rounded down, would result in the rounded estimate
Calculation of Lower Bound
The lower bound is found by subtracting half of the degree of accuracy from the rounded number
Definition of Upper Bound
The upper bound is the smallest number that, when rounded up, would produce the same estimate
Calculation of Upper Bound
The upper bound is found by adding half of the degree of accuracy to the rounded number
Error intervals are the span of all possible values that round to the estimated figure
Error intervals are often represented using inequality notation, which specifies the set of real numbers that could, upon rounding, equate to the estimated value
Error intervals are calculated by halving the degree of accuracy and adding/subtracting it from the rounded number to find the upper and lower bounds
For addition, the upper bound of the result is the sum of the upper bounds, and the lower bound is the sum of the lower bounds of the addends
For subtraction, the upper bound is the difference between the upper bound of the minuend and the lower bound of the subtrahend, while the lower bound is the difference between the lower bounds of the minuend and subtrahend
For multiplication, the upper bound of the product is the product of the upper bounds, and the lower bound is the product of the lower bounds of the multiplicands
For division, the upper bound of the quotient is the upper bound of the dividend divided by the lower bound of the divisor, and the lower bound is the lower bound of the dividend divided by the upper bound of the divisor
When rounding to the nearest 10, the lower bound is the largest number that would round down to the given number, and the upper bound is the smallest number that would round up to the next multiple of 10
When rounding measurements, the error interval is calculated by adding and subtracting half of the rounding unit from the measured value
In more intricate situations, such as extending the length of a rope or calculating the area of a rectangle, the bounds for the new value are calculated by applying the rules for addition and multiplication of bounds
When calculating speed, the upper bound is the upper bound of the distance divided by the lower bound of the time, and the lower bound is the lower bound of the distance divided by the upper bound of the time
00
In ______, the ______ bound is the highest number that rounds down to the estimated value.
mathematics
lower
01
The ______ bound in mathematics is the minimal number that rounds up to the same estimated figure.
upper
02
Determining Degree of Accuracy
Halve the degree of accuracy to find rounding error range.
