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Wednesday, July 15, 2009
lines and angels
Parallel Lines
Lines are parallel if they are always the same distance apart (called "equidistant"), and will never meet. Just remember:
Always the same distance apart and never touching.
The red line is parallel to the blue line in both these cases:
Parallel Example 1 Parallel Example 2
Example 1
Example 2
Parallel lines also point in the same direction.
Pairs of Angles
When parallel lines get crossed by another line (which is called a Transversal), you can see that many angles are the same, as in this example:
These angles can be made into pairs of angles which have special names.
Equation
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This article is about equations in mathematics. For the chemistry term, see chemical equation.
The first equation to ever be written, by Robert Recorde, who invented the equality sign, in its original form and in modern mathematic syntax.
An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent). Equations are written with an equal sign, as in
2 + 3 = 5\,.
9 - 2 = 7\,.
The equations above are examples of an equality: a proposition which states that two constants are equal. Equalities may be true or false.
While in arithmetic only numbers and their arithmetical operations (such as +, −, ×, ÷) occur, in algebra one also uses symbols (such as x and y, or a and b) to denote numbers. These are called variables. This is useful because:
* It allows the generalization of arithmetical equations (and inequalities) to be stated as laws (such as a + b = b + a for all a and b), and thus is the first step to the systematic study of the properties of the real number system.
* It allows reference to numbers which are not known. In the context of a problem, a variable may represent a certain value which is not yet known, but which may be found through the formulation and manipulation of equations.
* It allows the exploration of mathematical relationships between quantities (such as "if you sell x tickets, then your profit will be 3x − 10 dollars").
These three are the main strands of elementary algebra, which should be distinguished from abstract algebra, a more advanced topic generally taught to college students.
In elementary algebra, an "expression" may contain numbers, variables and arithmetical operations. These are usually written (by convention) with 'higher-power' terms on the left (see polynomial); a few examples are:
x + 3\,
y^{2} + 2x - 3\,
z^{7} + a(b + x^{3}) + 42/y - \pi.\,
In more advanced algebra, an expression may also include elementary functions.
A typical algebra problem.
An "equation" is the claim that two expressions are equal. Some equations are true for all values of the involved variables (such as a + b = b + a); such equations are called "identities". "Conditional" equations are true for only some values of the involved variables: x2 − 1 = 4. The values of the variables which make the equation true are called the "solutions" of the equation.
example :
x/4 = 3
x = 4 x 3
x = 12
Jump to: navigation, search
This article is about equations in mathematics. For the chemistry term, see chemical equation.
The first equation to ever be written, by Robert Recorde, who invented the equality sign, in its original form and in modern mathematic syntax.
An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent). Equations are written with an equal sign, as in
2 + 3 = 5\,.
9 - 2 = 7\,.
The equations above are examples of an equality: a proposition which states that two constants are equal. Equalities may be true or false.
While in arithmetic only numbers and their arithmetical operations (such as +, −, ×, ÷) occur, in algebra one also uses symbols (such as x and y, or a and b) to denote numbers. These are called variables. This is useful because:
* It allows the generalization of arithmetical equations (and inequalities) to be stated as laws (such as a + b = b + a for all a and b), and thus is the first step to the systematic study of the properties of the real number system.
* It allows reference to numbers which are not known. In the context of a problem, a variable may represent a certain value which is not yet known, but which may be found through the formulation and manipulation of equations.
* It allows the exploration of mathematical relationships between quantities (such as "if you sell x tickets, then your profit will be 3x − 10 dollars").
These three are the main strands of elementary algebra, which should be distinguished from abstract algebra, a more advanced topic generally taught to college students.
In elementary algebra, an "expression" may contain numbers, variables and arithmetical operations. These are usually written (by convention) with 'higher-power' terms on the left (see polynomial); a few examples are:
x + 3\,
y^{2} + 2x - 3\,
z^{7} + a(b + x^{3}) + 42/y - \pi.\,
In more advanced algebra, an expression may also include elementary functions.
A typical algebra problem.
An "equation" is the claim that two expressions are equal. Some equations are true for all values of the involved variables (such as a + b = b + a); such equations are called "identities". "Conditional" equations are true for only some values of the involved variables: x2 − 1 = 4. The values of the variables which make the equation true are called the "solutions" of the equation.
example :
x/4 = 3
x = 4 x 3
x = 12
Estimating
Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data. The parameters describe an underlying physical setting in such a way that the value of the parameters affects the distribution of the measured data. An estimator attempts to approximate the unknown parameters using the measurements.
For example, it is desired to estimate the proportion of a population of voters who will vote for a particular candidate. That proportion is the unobservable parameter; the estimate is based on a small random sample of voters.
Or, for example, in radar the goal is to estimate the location of objects (airplanes, boats, etc.) by analyzing the received echo and a possible question to be posed is "where are the airplanes?" To answer where the airplanes are, it is necessary to estimate the distance the airplanes are at from the radar station, which can provide an absolute location if the absolute location of the radar station is known.
In estimation theory, it is assumed that the desired information is embedded in a noisy signal. Noise adds uncertainty, without which the problem would be deterministic and estimation would not be needed.
Example Questions
(a) Estimate the cost of 21 packs of screws each costing £2.90.
The actual calculation is 21 × 2.9.
If we round the numbers to 1 significant figure we get 20 × 3, which we can do without a calculator.
Our estimated answer is £60 (which is quite close to the actual answer of £60.90).
(b) Estimate the length of 29 pieces of wood layed end-to-end if each is 1.48m long.
The actual calculation is 29 × 1.5.
If we round the numbers to 1 significant figure we get 30 × 1.
The estimated answer using 1 significant figure is 30m (which is quite different to the actual answer of 42.92m).
In this case, we need to be sensible about the rounding, for example we could calculate 30 × 1.5.
The estimated answer using more sensible rounding is 45m (which is much closer to the actual answer).
(c) James has worked out 31.5 × 49.6 and got the answer 156.24.
Use estimation to check whether his answer is likely to be correct.
If we round the numbers to 1 significant figure we get 30 × 50.
The estimated answer using 1 significant figure is 1500m (which is very different to the answer James got).
James has probably got the decimal point in the wrong place - the answer should be 1562.4
Practice Questions
(a) Estimate the weight of 38 boxes each weighing 26 kg.
(b) Jane has calculated 3.88 × 16.1 and got 62.468. Does her answer seem correct?
Answer :?
For example, it is desired to estimate the proportion of a population of voters who will vote for a particular candidate. That proportion is the unobservable parameter; the estimate is based on a small random sample of voters.
Or, for example, in radar the goal is to estimate the location of objects (airplanes, boats, etc.) by analyzing the received echo and a possible question to be posed is "where are the airplanes?" To answer where the airplanes are, it is necessary to estimate the distance the airplanes are at from the radar station, which can provide an absolute location if the absolute location of the radar station is known.
In estimation theory, it is assumed that the desired information is embedded in a noisy signal. Noise adds uncertainty, without which the problem would be deterministic and estimation would not be needed.
Example Questions
(a) Estimate the cost of 21 packs of screws each costing £2.90.
The actual calculation is 21 × 2.9.
If we round the numbers to 1 significant figure we get 20 × 3, which we can do without a calculator.
Our estimated answer is £60 (which is quite close to the actual answer of £60.90).
(b) Estimate the length of 29 pieces of wood layed end-to-end if each is 1.48m long.
The actual calculation is 29 × 1.5.
If we round the numbers to 1 significant figure we get 30 × 1.
The estimated answer using 1 significant figure is 30m (which is quite different to the actual answer of 42.92m).
In this case, we need to be sensible about the rounding, for example we could calculate 30 × 1.5.
The estimated answer using more sensible rounding is 45m (which is much closer to the actual answer).
(c) James has worked out 31.5 × 49.6 and got the answer 156.24.
Use estimation to check whether his answer is likely to be correct.
If we round the numbers to 1 significant figure we get 30 × 50.
The estimated answer using 1 significant figure is 1500m (which is very different to the answer James got).
James has probably got the decimal point in the wrong place - the answer should be 1562.4
Practice Questions
(a) Estimate the weight of 38 boxes each weighing 26 kg.
(b) Jane has calculated 3.88 × 16.1 and got 62.468. Does her answer seem correct?
Answer :?
Summary Decimal Places
Decimal notation
Decimal notation is the writing of numbers in a base-10 numeral system. Examples are Roman numerals, Brahmi numerals, and Chinese numerals, as well as the Hindu-Arabic numerals used by speakers of English. Roman numerals have symbols for the decimal powers (1, 10, 100, 1000) and secondary symbols for half these values (5, 50, 500). Brahmi numerals had symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100 and another for 1000. Chinese has symbols for 1–9, and fourteen additional symbols for higher powers of 10, which in modern usage reach 1044.
However, when people who use Hindu-Arabic numerals speak of decimal notation, they often mean not just decimal numeration, as above, but also decimal fractions, all conveyed as part of a positional system. Positional decimal systems include a zero and use symbols (called digits) for the ten values (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) to represent any number, no matter how large or how small. These digits are often used with a decimal separator which indicates the start of a fractional part, and with one of the sign symbols + (positive) or − (negative) in front of the numerals to indicate sign.
Positional notation uses positions for each power of ten: units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier (power of ten) multiplied with that digit—each position has a value ten times that of the position to its right. There are only two truly positional decimal systems in ancient civilization: the Chinese counting rod system and the Hindu-Arabic numeral system, which descended from Brahmi numerals.
Ten is the number which is the count of fingers and thumbs on both hands (or toes on the feet). In many languages the word digit or its translation is also the anatomical term referring to fingers and toes. In English, decimal (decimus < Lat.) means tenth, decimate means reduce by a tenth, and denary (denarius < Lat.) means the unit of ten. The symbols for the digits in common use around the globe today are called Arabic numerals by Europeans and Indian numerals by Arabs, the two groups' terms both referring to the culture from which they learned the system. However, the symbols used in different areas are not identical; for instance, Western Arabic numerals (from which the European numerals are derived) differ from the forms used by other Arab cultures.
[edit] Alternative notations
Some cultures do, or used to, use other numeral systems, including pre-Columbian Mesoamerican cultures such as the Maya, who use a vigesimal system (using all twenty fingers and toes), some Nigerians who use several duodecimal (base 12) systems, the Babylonians, who used sexagesimal (base 60), and the Yuki, who reportedly used quaternal (base 4).
Computer hardware and software systems commonly use a binary representation, internally (although a few of the earliest computers, such as ENIAC, did use decimal representation internally). For external use by computer specialists, this binary representation is sometimes presented in the related octal or hexadecimal systems. For most purposes, however, binary values are converted to the equivalent decimal values for presentation to and manipulation by humans.
Both computer hardware and software also use internal representations which are effectively decimal for storing decimal values and doing arithmetic. Often this arithmetic is done on data which are encoded using some variant of binary-coded decimal, especially in database implementations, but there are other decimal representations in use (such as in the new IEEE 754 Standard for Floating-Point Arithmetic). Decimal arithmetic is used in computers so that decimal fractional results can be computed exactly, which is not possible using a binary fractional representation. This is often important for financial and other calculations[1].
[edit] Decimal fractions
A decimal fraction is a fraction where the denominator is a power of ten.
Decimal fractions are commonly expressed without a denominator, the decimal separator being inserted into the numerator (with leading zeros added if needed), at the position from the right corresponding to the power of ten of the denominator. e.g., 8/10, 83/100, 83/1000, and 8/10000 are expressed as: 0.8, 0.83, 0.083, and 0.0008. In English-speaking and many Asian countries, a period (.) or raised period (•) is used as the decimal separator; in many other countries, a comma is used.
The integer part or integral part of a decimal number is the part to the left of the decimal separator (see also floor function). The part from the decimal separator to the right is the fractional part; if considered as a separate number, a zero is often written in front. Especially for negative numbers, we have to distinguish between the fractional part of the notation and the fractional part of the number itself, because the latter gets its own minus sign. It is usual for a decimal number whose absolute value is less than one to have a leading zero.
Trailing zeros after the decimal point are not necessary, although in science, engineering and statistics they can be retained to indicate a required precision or to show a level of confidence in the accuracy of the number: Whereas 0.080 and 0.08 are numerically equal, in engineering 0.080 suggests a measurement with an error of up to 1 part in two thousand (±0.0005), while 0.08 suggests a measurement with an error of up to 1 in two hundred (see Significant figures).
Other rational numbers
Any rational number which cannot be expressed as a finite decimal fraction has a unique infinite decimal expansion ending with recurring decimals.
The decimal fractions are those with denominator divisible by only 2 and or 5.
1/2 = 0.5
1/20 = 0.05
1/5 = 0.2
1/50 = 0.02
1/4 = 0.25
1/40 = 0.025
1/25 = 0.04
1/8 = 0.125
1/125= 0.008
1/10 = 0.1
1/3 = 0.333333… (with 3 repeating)
1/9 = 0.111111… (with 1 repeating)
100-1=99=9•11
1/11 = 0.090909… (with 09 or 90 repeating)
1000-1=9•111=27•37
1/27 = 0.037037037…
1/37 = 0.027027027…
1/111 = 0 .009009009…
also:
1/81= 0.012345679012… (with 012345679 repeating)
Other prime factors in the denominator will give longer recurring sequences, see for instance 7, 13.
That a rational number must have a finite or recurring decimal expansion can be seen to be a consequence of the long division algorithm, in that there are only q-1 possible nonzero remainders on division by q, so that the recurring pattern will have a period less than q. For instance to find 3/7 by long division:
Decimal notation is the writing of numbers in a base-10 numeral system. Examples are Roman numerals, Brahmi numerals, and Chinese numerals, as well as the Hindu-Arabic numerals used by speakers of English. Roman numerals have symbols for the decimal powers (1, 10, 100, 1000) and secondary symbols for half these values (5, 50, 500). Brahmi numerals had symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100 and another for 1000. Chinese has symbols for 1–9, and fourteen additional symbols for higher powers of 10, which in modern usage reach 1044.
However, when people who use Hindu-Arabic numerals speak of decimal notation, they often mean not just decimal numeration, as above, but also decimal fractions, all conveyed as part of a positional system. Positional decimal systems include a zero and use symbols (called digits) for the ten values (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) to represent any number, no matter how large or how small. These digits are often used with a decimal separator which indicates the start of a fractional part, and with one of the sign symbols + (positive) or − (negative) in front of the numerals to indicate sign.
Positional notation uses positions for each power of ten: units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier (power of ten) multiplied with that digit—each position has a value ten times that of the position to its right. There are only two truly positional decimal systems in ancient civilization: the Chinese counting rod system and the Hindu-Arabic numeral system, which descended from Brahmi numerals.
Ten is the number which is the count of fingers and thumbs on both hands (or toes on the feet). In many languages the word digit or its translation is also the anatomical term referring to fingers and toes. In English, decimal (decimus < Lat.) means tenth, decimate means reduce by a tenth, and denary (denarius < Lat.) means the unit of ten. The symbols for the digits in common use around the globe today are called Arabic numerals by Europeans and Indian numerals by Arabs, the two groups' terms both referring to the culture from which they learned the system. However, the symbols used in different areas are not identical; for instance, Western Arabic numerals (from which the European numerals are derived) differ from the forms used by other Arab cultures.
[edit] Alternative notations
Some cultures do, or used to, use other numeral systems, including pre-Columbian Mesoamerican cultures such as the Maya, who use a vigesimal system (using all twenty fingers and toes), some Nigerians who use several duodecimal (base 12) systems, the Babylonians, who used sexagesimal (base 60), and the Yuki, who reportedly used quaternal (base 4).
Computer hardware and software systems commonly use a binary representation, internally (although a few of the earliest computers, such as ENIAC, did use decimal representation internally). For external use by computer specialists, this binary representation is sometimes presented in the related octal or hexadecimal systems. For most purposes, however, binary values are converted to the equivalent decimal values for presentation to and manipulation by humans.
Both computer hardware and software also use internal representations which are effectively decimal for storing decimal values and doing arithmetic. Often this arithmetic is done on data which are encoded using some variant of binary-coded decimal, especially in database implementations, but there are other decimal representations in use (such as in the new IEEE 754 Standard for Floating-Point Arithmetic). Decimal arithmetic is used in computers so that decimal fractional results can be computed exactly, which is not possible using a binary fractional representation. This is often important for financial and other calculations[1].
[edit] Decimal fractions
A decimal fraction is a fraction where the denominator is a power of ten.
Decimal fractions are commonly expressed without a denominator, the decimal separator being inserted into the numerator (with leading zeros added if needed), at the position from the right corresponding to the power of ten of the denominator. e.g., 8/10, 83/100, 83/1000, and 8/10000 are expressed as: 0.8, 0.83, 0.083, and 0.0008. In English-speaking and many Asian countries, a period (.) or raised period (•) is used as the decimal separator; in many other countries, a comma is used.
The integer part or integral part of a decimal number is the part to the left of the decimal separator (see also floor function). The part from the decimal separator to the right is the fractional part; if considered as a separate number, a zero is often written in front. Especially for negative numbers, we have to distinguish between the fractional part of the notation and the fractional part of the number itself, because the latter gets its own minus sign. It is usual for a decimal number whose absolute value is less than one to have a leading zero.
Trailing zeros after the decimal point are not necessary, although in science, engineering and statistics they can be retained to indicate a required precision or to show a level of confidence in the accuracy of the number: Whereas 0.080 and 0.08 are numerically equal, in engineering 0.080 suggests a measurement with an error of up to 1 part in two thousand (±0.0005), while 0.08 suggests a measurement with an error of up to 1 in two hundred (see Significant figures).
Other rational numbers
Any rational number which cannot be expressed as a finite decimal fraction has a unique infinite decimal expansion ending with recurring decimals.
The decimal fractions are those with denominator divisible by only 2 and or 5.
1/2 = 0.5
1/20 = 0.05
1/5 = 0.2
1/50 = 0.02
1/4 = 0.25
1/40 = 0.025
1/25 = 0.04
1/8 = 0.125
1/125= 0.008
1/10 = 0.1
1/3 = 0.333333… (with 3 repeating)
1/9 = 0.111111… (with 1 repeating)
100-1=99=9•11
1/11 = 0.090909… (with 09 or 90 repeating)
1000-1=9•111=27•37
1/27 = 0.037037037…
1/37 = 0.027027027…
1/111 = 0 .009009009…
also:
1/81= 0.012345679012… (with 012345679 repeating)
Other prime factors in the denominator will give longer recurring sequences, see for instance 7, 13.
That a rational number must have a finite or recurring decimal expansion can be seen to be a consequence of the long division algorithm, in that there are only q-1 possible nonzero remainders on division by q, so that the recurring pattern will have a period less than q. For instance to find 3/7 by long division:
Summary significant figures
The significant figures (also called significant digits and abbreviated sig figs, sign.figs or sig digs) of a number are those digits that carry meaning contributing to its precision (see entry for Accuracy and precision). This includes all digits except:
* leading and trailing zeros (unless a decimal point is present) where they serve merely as placeholders to indicate the scale of the number.
* spurious digits introduced, for example, by calculations carried out to greater accuracy than that of the original data, or measurements reported to a greater precision than the equipment supports.
The concept of significant figures is often used in connection with rounding. Rounding to n significant figures is a more general-purpose technique than rounding to n decimal places, since it handles numbers of different scales in a uniform way. A practical calculation that uses any irrational number necessitates rounding the number, and hence the answer, to a finite number of significant figures. Computer representations of floating point numbers typically use a form of rounding to significant figures, but with binary numbers.
The rules for identifying significant digits when writing or interpreting numbers are as follows:
* All non-zero digits are considered significant. Example: the number 1 has one significant figure. In 20 and 300, the first figure is significant while the others may or may not be (see below). 123.45 has five significant figures: 1, 2, 3, 4 and 5.
* Zeros appearing anywhere between two non-zero digits are significant. Example: 101.12 has five significant figures: 1, 0, 1, 1 and 2.
* Leading zeros are not significant. For example, 0.00012 has two significant figures: 1 and 2.
* Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has six significant figures: 1, 2, 2, 3, 0 and 0. The number 0.000122300 still has only six significant figures (the zeros before the 1 are not significant). In addition, 120.00 has five significant figures. This convention clarifies the precision of such numbers; for example, if a result accurate to four decimal places is given as 12.23 then it might be understood that only two decimal places of accuracy are available. Stating the result as 12.2300 makes clear that it is accurate to four decimal places.
* Alternatively, the above can be summarized by three rules.
o 1. All non-zero digits are significant.
o 2. In a number without a decimal point, only zeros BETWEEN non-zero digits are significant (unless a bar indicates the last significant digit--see below).
o 3. In a number with a decimal point, all zeros to the right of the first non-zero digit are significant.
* The significance of trailing zeros in a number not containing a decimal point can be ambiguous. For example, it may not always be clear if a number like 1300 is accurate to the nearest unit (and just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundred due to rounding or uncertainty. Various conventions exist to address this issue:
* A bar may be placed over the last significant digit; any trailing zeros following this are insignificant. For example, 13 \bar{0} 0 has three significant figures (and hence indicates that the number is accurate to the nearest ten).
* The last significant figure of a number may be underlined; for example, "20000" has two significant figures.
* A decimal point may be placed after the number; for example "100." indicates specifically that three significant figures are meant.[1]
However, these conventions are not universally used, and it is often necessary to determine from context whether such trailing zeros are intended to be significant. If all else fails, the level of rounding can be specified explicitly. The abbreviation s.f. is sometimes used, for example "20 000 to 2 s.f." or "20 000 (2 sf)". Alternatively, the uncertainty can be stated separately and explicitly, as in 20 000 ± 1%, so that significant-figures rules do not apply.
A number with all zero digits (e.g. 0.000) has no significant digits, because the uncertainty is larger than the actual measurement.
example :
Postage Scale 3 ±1 g 1 significant figure
Two-pan balance 2.53 ±0.01 g 3 significant figures
Analytical balance 2.531 ±0.001 g 4 significant figures
Rules for counting significant figures are summarized below.
Zeros within a number are always significant. Both 4308 and 40.05 contain four significant figures.
Zeros that do nothing but set the decimal point are not significant. Thus, 470,000 has two significant figures.
Trailing zeros that aren't needed to hold the decimal point are significant. For example, 4.00 has three significant figures.
If you are not sure whether a digit is significant, assume that it isn't. For example, if the directions for an experiment read: "Add the sample to 400 mL of water," assume the volume of water is known to one significant figure.
* leading and trailing zeros (unless a decimal point is present) where they serve merely as placeholders to indicate the scale of the number.
* spurious digits introduced, for example, by calculations carried out to greater accuracy than that of the original data, or measurements reported to a greater precision than the equipment supports.
The concept of significant figures is often used in connection with rounding. Rounding to n significant figures is a more general-purpose technique than rounding to n decimal places, since it handles numbers of different scales in a uniform way. A practical calculation that uses any irrational number necessitates rounding the number, and hence the answer, to a finite number of significant figures. Computer representations of floating point numbers typically use a form of rounding to significant figures, but with binary numbers.
The rules for identifying significant digits when writing or interpreting numbers are as follows:
* All non-zero digits are considered significant. Example: the number 1 has one significant figure. In 20 and 300, the first figure is significant while the others may or may not be (see below). 123.45 has five significant figures: 1, 2, 3, 4 and 5.
* Zeros appearing anywhere between two non-zero digits are significant. Example: 101.12 has five significant figures: 1, 0, 1, 1 and 2.
* Leading zeros are not significant. For example, 0.00012 has two significant figures: 1 and 2.
* Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has six significant figures: 1, 2, 2, 3, 0 and 0. The number 0.000122300 still has only six significant figures (the zeros before the 1 are not significant). In addition, 120.00 has five significant figures. This convention clarifies the precision of such numbers; for example, if a result accurate to four decimal places is given as 12.23 then it might be understood that only two decimal places of accuracy are available. Stating the result as 12.2300 makes clear that it is accurate to four decimal places.
* Alternatively, the above can be summarized by three rules.
o 1. All non-zero digits are significant.
o 2. In a number without a decimal point, only zeros BETWEEN non-zero digits are significant (unless a bar indicates the last significant digit--see below).
o 3. In a number with a decimal point, all zeros to the right of the first non-zero digit are significant.
* The significance of trailing zeros in a number not containing a decimal point can be ambiguous. For example, it may not always be clear if a number like 1300 is accurate to the nearest unit (and just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundred due to rounding or uncertainty. Various conventions exist to address this issue:
* A bar may be placed over the last significant digit; any trailing zeros following this are insignificant. For example, 13 \bar{0} 0 has three significant figures (and hence indicates that the number is accurate to the nearest ten).
* The last significant figure of a number may be underlined; for example, "20000" has two significant figures.
* A decimal point may be placed after the number; for example "100." indicates specifically that three significant figures are meant.[1]
However, these conventions are not universally used, and it is often necessary to determine from context whether such trailing zeros are intended to be significant. If all else fails, the level of rounding can be specified explicitly. The abbreviation s.f. is sometimes used, for example "20 000 to 2 s.f." or "20 000 (2 sf)". Alternatively, the uncertainty can be stated separately and explicitly, as in 20 000 ± 1%, so that significant-figures rules do not apply.
A number with all zero digits (e.g. 0.000) has no significant digits, because the uncertainty is larger than the actual measurement.
example :
Postage Scale 3 ±1 g 1 significant figure
Two-pan balance 2.53 ±0.01 g 3 significant figures
Analytical balance 2.531 ±0.001 g 4 significant figures
Rules for counting significant figures are summarized below.
Zeros within a number are always significant. Both 4308 and 40.05 contain four significant figures.
Zeros that do nothing but set the decimal point are not significant. Thus, 470,000 has two significant figures.
Trailing zeros that aren't needed to hold the decimal point are significant. For example, 4.00 has three significant figures.
If you are not sure whether a digit is significant, assume that it isn't. For example, if the directions for an experiment read: "Add the sample to 400 mL of water," assume the volume of water is known to one significant figure.
Summary Rounding
Rounding involves reducing the number of significant digits in a number. The result of rounding is a "shorter" number having fewer non-zero digits yet similar in magnitude. The result is less precise but easier to use.
For example: 73 rounded to the nearest ten is 70, because 73 is closer to 70 than to 80.
Researchers may analyze rounding as a form of quantization.
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