DEFAULT AND EXCESS APPROXIMATION: WHAT IT IS AND EXAMPLES - MATHS

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 a bound of error.

Approximating consists of replacing one exact figure with another, where said replacement should facilitate operations of a mathematical problem, preserving the structure and essence of the problem.

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A ≈B

It reads; A Approximate B. Where "A" represents the exact value and "B" the approximate value.

Significant numbers

The values with which an approximate number is defined are known as significant figures. In the example approximation 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…..

What does it consist on?

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 excess value (235.7) corresponds to the most significant value in tenths immediately after the original number.

On the other hand, the default value (235.6) corresponds to the nearest 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.

The margin of error

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.

Scales

The criteria for establishing approximation ranges can be highly variable and are closely related to the specifications of the element to be approximated. In countries with high inflation, the excess approximations ignore some numerical ranges, since these are lower 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 by directly approaching the hundred.

Using the calculator

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.

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Values like:

π = 3.141592654….
e = 2.718281828…
√2 = 1.414213562…

They are common in experimentation and their values must be defined in a certain range, taking into consideration the possible errors generated.

What are they for?

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).

Examples

Example 1

Which of the following numbers is a default approximation of 0.0127

0.13
0.012; It is a default approximation of 0.0127
0.01; It is a default approximation of 0.0127
0.0128

Example 2

Which of the following numbers is an excess approximation of 23,435

24; is an approximation by excess of 23,435
23.4
23.44; is an approximation by excess of 23,435
23.5; is an approximation by excess of 23,435

Example 3

Define the following numbers using a default approximation, with the specified error bound.

547.2648…. For thousandths, hundredths and tens.

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, it approaches 547.26 by default .

Tens: In this case the error bound is much higher, because the range of the approximation is defined within the whole numbers. When you approximate by default in the ten you get 540.

Example 4

Define the following numbers using an excess approximation, with the specified error bound.

1204,27317 For tenths, hundreds and ones.

Tenths: Refers to the first digit after the comma, where the unit is composed after 0.9. Approaching the tenths in excess gives 1204.3.

Hundreds: Again an error bound is observed whose range is within the whole numbers of the figure. Approximating the hundreds by excess gives 1300. This figure is considerably different from 1204.27317. Because of this, approximations are not usually applied to integer values.

Units: By excessively approaching the unit, 1205 is obtained .

Example 5

A seamstress cuts a length of fabric 135.3 cm long to make a 7855 cm ² flag. How much the other side will measure if you use a conventional ruler that marks up to millimeters.

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 = 7855cm ² / 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.

Example 6

Define 9 values that can be exact numbers in each of the approximations:

34,071 results from approximate thousandths by default

34.07124 34.07108 34.07199

34.0719 34.07157 34.07135

34.0712 34.071001 34.07176

0.012 results from approximate thousandths by default

0.01291 0.012099 0.01202

0.01233 0.01223 0.01255

0.01201 0.0121457 0.01297

23.9 results from approximating tenths by excess

23.801 23.85555 23.81

23.89 23.8324 23.82

23,833 23,84 23,80004

58.37 is the result of approximating hundredths by excess

58.3605 58.36001 58.36065

58,3655 58,362 58,363

58.3623 58.361 58.3634

Example 7

Approximate each irrational number according to the indicated error bound:

π = 3.141592654….

Thousandths by default π = 3.141

Thousandths by excess π = 3.142

Hundredths by default π = 3.14

Hundredths in excess π = 3.15

Tenths by default π = 3.1

Tenths by excess π = 3.2

e = 2.718281828…

Thousandths by default e = 2.718

Thousandths by excess e = 2.719

Hundredths by default e = 2.71

Hundredths in excess e = 2.72

Tenths by default e = 2.7

Tenths by excess e = 2.8

√2 = 1.414213562…

Thousandths by default √2 = 1.414

Thousandths by excess √2 = 1.415

Hundredths by default √2 = 1.41

Hundredths in excess √2 = 1.42

Tenths by default √2 = 1.4

Tenths by excess √2 = 1.5

1 ÷ 3 = 0.3333333…..

Thousandths by default 1 ÷ 3 = 0.332

Thousandths in excess 1 ÷ 3 = 0.334

Hundredths by default 1 ÷ 3 = 0.33

Hundredths in excess 1 ÷ 3 = 0.34

Tenths by default 1 ÷ 3 = 0.3

Tenths by excess 1 ÷ 3 = 0.4

References

Problems in Mathematical Analysis. Piotr Biler, Alfred Witkowski. University of Wroclaw. Poland.
Introduction to Logic and to the Methodology of the Deductive Sciences. Alfred Tarski, New York Oxford. Oxford University press.
The Arithmetic Teacher, Volume 29. National Council of Teachers of Mathematics, 1981. University of Michigan.
Learning and teaching number theory: Research in cognition and instruction / edited by Stephen R. Campbell and Rina Zazkis. Ablex publishing 88 Post Road West, Westport CT 06881.
Bernoulli, J. (1987). Ars Conjectandi- 4ème partie. Rouen: IREM.

DEFAULT AND EXCESS APPROXIMATION: WHAT IT IS AND EXAMPLES - MATHS - 2024