The under and over approximation, is a numerical method used to establish the value of a number according to different scales of accuracy. For example, the number 235,623, is close to 235.6 by default and 235.7 by excess. If we consider the tenths as an error bound.
Approximating consists of replacing an exact figure with another, where said replacement should facilitate operations of a mathematical problem, preserving the structure and essence of the problem..
A ≈B
It reads; A approximate of B. Where "A" represents the exact value and "B" the approximate value.
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The values with which an approximate number is defined are known as significant figures. In the approximation of the example four significant figures were taken. The precision of a number is given by the number of significant figures that define it.
The infinite zeros that can be located both to the right and to the left of the number are not considered significant figures. The location of the comma does not play any role in defining the significant figures of a number.
750385
… 00.0075038500…
75.038500000 ...
750385000 ...
… 000007503850000…
The method is quite simple; choose the error bound, which is nothing other than the numerical range where you want to make the cut. The value of this range is directly proportional to the margin of error of the approximate number.
In the example above 235,623 owns thousandths (623). Then the approximation to the tenths has been made. The value for excess (235,7) corresponds to the most significant value in tenths immediately after the original number.
On the other hand the value for default (235,6) corresponds to the closest and most significant value in tenths that is before the original number.
The numerical approximation is quite common in practice with numbers. Other widely used methods are rounding and truncation; which respond to different criteria to assign the values.
When defining the numerical range that the number will cover after being approximated, we also define the error bound that accompanies the figure. This will be denoted with an existing or significant rational number in the assigned range.
In the initial example the values defined by excess (235.7) and by default (235,6) have an approximate error of 0.1. In statistical and probability studies, 2 types of errors are handled with respect to the numerical value; absolute error and relative error.
The criteria for establishing the approximation ranges can be highly variable and are closely related to the specifications of the element to be approximated. In countries with high inflation, excess approximations ignore some numerical ranges, because these are less than the inflationary scale.
In this way, in an inflation greater than 100% a seller will not adjust a product from $ 50 to $ 55 but will approximate it to $ 100, thus ignoring the units and tens when directly approaching the hundred.
Conventional calculators bring with them the FIX mode, where the user can configure the number of decimal places they want to receive in their results. This generates errors that must be considered when making exact calculations..
Irrational numbers approximation
Some values widely used in numerical operations belong to the set of irrational numbers, whose main characteristic is to have an indeterminate number of decimal places.
Values like:
They are common in experimentation and their values must be defined in a certain range, taking into consideration the possible errors generated..
In the case of division (1 ÷ 3), it is observed through experimentation, the need to establish a cut in the number of operations performed to define the number.
1 ÷ 3 = 0.333333…
1 ÷ 3 3/10 = 0.3
1 ÷ 3 33/100 = 0.33
1 ÷ 3 333/1000 = 0.333
1 ÷ 3 3333/10000 = 0.3333
1 ÷ 3 333333… / 10000… = 0.333333…
An operation is presented that can be perpetuated indefinitely, so it is necessary to approximate at some point.
In the case of:
1 ÷ 3 333333… / 10000… = 0.333333…
For any point established as a margin of error, a number less than the exact value of (1 ÷ 3) will be obtained. In this way, all the approximations made previously are default approximations of (1 ÷ 3).
Thousandths: The thousandths correspond to the first 3 digits after the comma, where after 999 comes the unit. We proceed to approximate 547,264.
Hundredths: Denoted by the first 2 digits after the comma, the hundredths must meet, 99 to reach unity. In this way by default it approximates 547.26.
Tens: In this case the error bound is much higher, because the range of the approximation is defined within the whole numbers. When approximating by default in the ten, we obtain 540.
Tenths: Refers to the first digit after the comma, where the unit is composed after 0.9. Approximating by excess to the tenths we obtain 1204.3.
Hundreds: Again an error bound is observed whose range is within the whole numbers of the figure. By excessively approximating the hundreds, we obtain 1300. This figure is considerably different from 1204,27317. Because of this, the approximations are not usually applied to integer values..
Units: By excessively approaching the unit, we obtain 1205.
Approximate the results by excess and defect.
The area of the flag is rectangular and is defined by:
A = side x side
side = A / side
side = 7855cmtwo / 135.3cm
side = 58.05617147 cm
Due to the appreciation of the rule we can obtain data up to millimeters, which corresponds to the range of decimals with respect to the centimeter.
Thus 58cm is a default approximation.
While 58.1 is an excess approximation.
34.07124 34.07108 34.07199
34.0719 34.07157 34.07135
34.0712 34.071001 34.07176
0.01291 0.012099 0.01202
0.01233 0.01223 0.01255
0.01201 0.0121457 0.01297
23.801 23.85555 23.81
23.89 23.8324 23.82
23.833 23.84 23.80004
58.3605 58.36001 58.36065
58,3655 58,362 58,363
58.3623 58.361 58.3634
Thousandths per default π = 3.141
Thousandths per excess π = 3.142
Hundredths per default π = 3.14
Hundredths per excess π = 3.15
Tenths per default π = 3.1
Tenths per excess π = 3.2
Thousandths per default e = 2.718
Thousandths per excess e = 2,719
Hundredths per default e = 2.71
Hundredths per excess e = 2.72
Tenths per default e = 2.7
Tenths per excess e = 2.8
Thousandths per default √2 = 1.414
Thousandths per excess √2 = 1,415
Hundredths per default √2= 1.41
Hundredths per excess √2 = 1.42
Tenths per default √2 = 1.4
Tenths per excess √2 = 1.5
Thousandths per default 1 ÷ 3 = 0.332
Thousandths per excess 1 ÷ 3 = 0.334
Hundredths per default 1 ÷ 3 = 0.33
Hundredths per excess 1 ÷ 3 = 0.34
Tenths per default 1 ÷ 3 = 0.3
Tenths per excess 1 ÷ 3 = 0.4
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