The rule is also applicable in several practical applications, such as determining which way to turn a screw, etc. There is also a left-hand rule, which exhibits opposite chirality. 
Friday, 28 October 2011
Vectors
Vector
- Equal Vector, Negative Vector, Parallel Vector, Free Vector (Fixed Vector & Position Vector)
Free vector:
Vectors in which there is no restriction to choose the origin of a vector.
Position (vector)
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector which represents the position of a point P in space in relation to an arbitrary reference origin O. Usually denoted s or r, it corresponds to the displacement from O to P.
http://emweb.unl.edu/math/mathweb/vectors/vectors.html
Vectors and vector addition:
Addition of two vectors is accomplished by laying the vectors head to tail in sequence to create a triangle such as is shown in the figure.
A vector can be resolved along any two directions in a plane containing it. The figure shows how the parallelogram rule is used to construct vectors a and b that add up to c.
Wednesday, 26 October 2011
Geometric Constructions
"Construction" in Geometry means to draw shapes, angles or lines accurately.
This is the "pure" form of geometric construction - no numbers involved!
Learn these two first, they are used a lot:
Line Bisectorhttp://www.mathsisfun.com/geometry/construct-linebisect.html
Angle Bisectorhttp://www.mathsisfun.com/geometry/construct-anglebisect.html
Tuesday, 25 October 2011
Sunday, 23 October 2011
isometric transformation
http://www.mi.sanu.ac.rs/vismath/jadrbookhtml/part03.html
Isometry
n.
1. Equality of measure.
2. Equality of elevation above sea level.
3. Mathematics A function between metric spaces which preserves distances, such as a rotation or translation in a plane.
n
1. (Mathematics) Maths rigid motion of a plane or space such that the distance between any two points before and after this motion is unaltered
2. (Earth Sciences / Physical Geography) equality of height above sea level
1. Equality of measure.
2. Equality of elevation above sea level.
3. A function between two metric spaces (such as two coordinate systems) which preserves distances. A rotation or translation in a plane is an isometry, since the distances between two points on the plane remain the same after the rotation or translation.
Metric
Mathematics A geometric function that describes the distances between pairs of points in a space.
(Mathematics) Maths denoting or relating to a set containing pairs of points for each of which a non-negative real number ρ(x, y) (the distance) can be defined, satisfying specific conditions
a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement
The first steps in the development of the theory of symmetry in the 18th century stem from basic isometric transformations - mirror reflections
A typical example of isometric transformation (transformation of congruence
Isometry
n.
1. Equality of measure.
2. Equality of elevation above sea level.
3. Mathematics A function between metric spaces which preserves distances, such as a rotation or translation in a plane.
n
1. (Mathematics) Maths rigid motion of a plane or space such that the distance between any two points before and after this motion is unaltered
2. (Earth Sciences / Physical Geography) equality of height above sea level
1. Equality of measure.
2. Equality of elevation above sea level.
3. A function between two metric spaces (such as two coordinate systems) which preserves distances. A rotation or translation in a plane is an isometry, since the distances between two points on the plane remain the same after the rotation or translation.
Metric
Mathematics A geometric function that describes the distances between pairs of points in a space.
(Mathematics) Maths denoting or relating to a set containing pairs of points for each of which a non-negative real number ρ(x, y) (the distance) can be defined, satisfying specific conditions
a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement
The first steps in the development of the theory of symmetry in the 18th century stem from basic isometric transformations - mirror reflections
A typical example of isometric transformation (transformation of congruence
Their combinations create all kinds of plane isometric transformations (transformations of congruence, or simply, isometries). Geometric transformation is an isometry (transformation of congruence) if it preserves the distance (metrics), i.e., if each pair of points X and Y are transformed into points X1 and Y1 in such a way that the distance between points X and Y is congruent to the distance between X1 and Y1 (Lopandic, 1979, Martin, 1982). This means that for every pair of the points X,Y and their images X1=t(X), Y1=t(Y) holds XY @ X1Y1. A typical example of isometric transformation (transformation of congruence) is the physical motion of a solid, where the distance between any two of its points remains unchanged (congruent) and consequently, the whole solid itself remains unchanged. However, the motion of fluids (for example, the vapor in clouds) does not have this characteristic.
Saturday, 22 October 2011
Significant Figures in Measurements and Calculations
Significant Figures
http://www.chem.tamu.edu/class/fyp/mathrev/mr-sigfg.html
There are two kinds of numbers in the world:
exact:
example: There are exactly 12 eggs in a dozen.
example: Most people have exactly 10 fingers and 10 toes.
inexact numbers:
example: any measurement.
If I quickly measure the width of a piece of notebook paper, I might get 220 mm (2 significant figures). If I am more precise, I might get 216 mm (3 significant figures). An even more precise measurement would be 215.6 mm (4 significant figures).
PRECISION VERSUS ACCURACY
Accuracy refers to how closely a measured value agrees with the correct value.
Precision refers to how closely individual measurements agree with each other.
The significant figures (also called significant digits) of a number are those digits that carry meaning contributing to its precision.
This includes all digits except:
.leading and trailing zeros 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 digits is often used in connection with rounding. Rounding to n significant digits is a more general-purpose technique than rounding to n decimal places, since it handles numbers of different scales in a uniform way. For example, the population of a city might only be known to the nearest thousand and be stated as 52,000, while the population of a country might only be known to the nearest million and be stated as 52,000,000. The former might be in error by hundreds, and the latter might be in error by hundreds of thousands, but both have two significant digits (5 and 2). This reflects the fact that the significance of the error (its likely size relative to the size of the quantity being measured) is the same in both cases.
Computer representations of floating point numbers typically use a form of rounding to significant digits, but with binary numbers.
The term "significant digits" can also refer to a crude form of error representation based around significant-digit rounding; for this use, see significance arithmetic.
Identifying significant digits
The rules for identifying significant digits when writing or interpreting numbers are as follows:
All non-zero digits are considered significant. For example, 91 has two significant digits (9 and 1), while 123.45 has five significant digits (1, 2, 3, 4 and 5).
Zeros appearing anywhere between two non-zero digits are significant. Example: 101.12 has five significant digits: 1, 0, 1, 1 and 2.
Leading zeros are not significant. For example, 0.00052 has two significant digits: 5 and 2.
Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has six significant digits: 1, 2, 2, 3, 0 and 0. The number 0.000122300 still has only six significant digits (the zeros before the 1 are not significant). In addition, 130.00 has five significant digits. This convention clarifies the accuracy 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.
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, has three significant digits (and hence indicates that the number is accurate to the nearest ten).
The last significant digit of a number may be underlined; for example, "2000" has one significant digit.
A decimal point may be placed after the number; for example "100." indicates specifically that three significant digits 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.
Scientific notation
Generally, the same rules apply to numbers expressed in scientific notation. However, in the normalized form of that notation, placeholder leading and trailing digits do not occur, so all digits are significant. For example, 0.00012 (two significant digits) becomes 1.2×10−4, and 0.00122300 (six significant digits) becomes 1.22300×10−3. In particular, the potential ambiguity about the significance of trailing zeros is eliminated. For example, 1300 to four significant digits is written as 1.300×103, while 1300 to two significant digits is written as 1.3×103.
Rounding
To round to n significant digits:
If the first non-significant digit is a 5 followed by other non-zero digits, round up the last significant digit (away from zero). For example, 1.2459 as the result of a calculation or measurement that only allows for 3 significant digits should be written 1.25.
If the first non-significant digit is a 5 not followed by any other digits or followed only by zeros, rounding requires a tie-breaking rule. For example, to round 1.25 to 2 significant digits, Round half up rounds up to 1.3, while Round half to even rounds to the nearest even number 1.2.
Replace any non-significant digits by zeros.
Arithmetic
Main article: Significance arithmetic
For multiplication and division, the result should have as many significant digits as the measured number with the smallest number of significant digits.
For addition and subtraction, the result should have as many decimal places as the measured number with the smallest number of decimal places.
When performing a calculation, do not follow these guidelines for intermediate results; keep as many digits as is practical to avoid rounding errors.[2]
Significant Figures and Rounding Rules
http://www.angelfire.com/oh/cmulliss/
http://www.chem.tamu.edu/class/fyp/mathrev/mr-sigfg.html
There are two kinds of numbers in the world:
exact:
example: There are exactly 12 eggs in a dozen.
example: Most people have exactly 10 fingers and 10 toes.
inexact numbers:
example: any measurement.
If I quickly measure the width of a piece of notebook paper, I might get 220 mm (2 significant figures). If I am more precise, I might get 216 mm (3 significant figures). An even more precise measurement would be 215.6 mm (4 significant figures).
PRECISION VERSUS ACCURACY
Accuracy refers to how closely a measured value agrees with the correct value.
Precision refers to how closely individual measurements agree with each other.
accurate
(the average is accurate)
not precise
precise
not accurate
accurate
and
precise
The number of significant figures is the number of digits believed to be correct by the person doing the measuring. It includes one estimated digit.
So, does the concept of significant figures deal with precision or accuracy?
Conclusion: The number of significant figures is directly linked to a measurement.
So, does the concept of significant figures deal with precision or accuracy? Hopefully, you can see that it really deals with precision only. Consider measuring the length of a metal rod several times with a ruler. You will get essentially the same measurement over and over again with a small reading error equal to about 1/10 of the smallest division on the ruler. You have determined the length with high precision. However, you don't know if the ruler was accurate to begin with. Perhaps it was a plastic ruler left in the hot Texas sun and was stretched. You don't know the accuracy of your measuring device unless you calibrate it, i.e. compare it against a ruler you knew was accurate. Note: in the laboratory, a good analytical chemist always calibrates her volumetric glassware before using it by weighing a known volume of liquid dispensed from the glassware. By dividing the mass of the liquid by its density, she can determine the actual volume and hence the accuracy of the glassware.
Rules for Working with Significant Figures:
Leading zeros are never significant.
Imbedded zeros are always significant.
Trailing zeros are significant only if the decimal point is specified.
Hint: Change the number to scientific notation. It is easier to see.
Leading zeros are never significant.
Imbedded zeros are always significant.
Trailing zeros are significant only if the decimal point is specified.
Hint: Change the number to scientific notation. It is easier to see.
The significant figures (also called significant digits) of a number are those digits that carry meaning contributing to its precision.
This includes all digits except:
.leading and trailing zeros 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 digits is often used in connection with rounding. Rounding to n significant digits is a more general-purpose technique than rounding to n decimal places, since it handles numbers of different scales in a uniform way. For example, the population of a city might only be known to the nearest thousand and be stated as 52,000, while the population of a country might only be known to the nearest million and be stated as 52,000,000. The former might be in error by hundreds, and the latter might be in error by hundreds of thousands, but both have two significant digits (5 and 2). This reflects the fact that the significance of the error (its likely size relative to the size of the quantity being measured) is the same in both cases.
Computer representations of floating point numbers typically use a form of rounding to significant digits, but with binary numbers.
The term "significant digits" can also refer to a crude form of error representation based around significant-digit rounding; for this use, see significance arithmetic.
Identifying significant digits
The rules for identifying significant digits when writing or interpreting numbers are as follows:
All non-zero digits are considered significant. For example, 91 has two significant digits (9 and 1), while 123.45 has five significant digits (1, 2, 3, 4 and 5).
Zeros appearing anywhere between two non-zero digits are significant. Example: 101.12 has five significant digits: 1, 0, 1, 1 and 2.
Leading zeros are not significant. For example, 0.00052 has two significant digits: 5 and 2.
Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has six significant digits: 1, 2, 2, 3, 0 and 0. The number 0.000122300 still has only six significant digits (the zeros before the 1 are not significant). In addition, 130.00 has five significant digits. This convention clarifies the accuracy 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.
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, has three significant digits (and hence indicates that the number is accurate to the nearest ten).
The last significant digit of a number may be underlined; for example, "2000" has one significant digit.
A decimal point may be placed after the number; for example "100." indicates specifically that three significant digits 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.
Scientific notation
Generally, the same rules apply to numbers expressed in scientific notation. However, in the normalized form of that notation, placeholder leading and trailing digits do not occur, so all digits are significant. For example, 0.00012 (two significant digits) becomes 1.2×10−4, and 0.00122300 (six significant digits) becomes 1.22300×10−3. In particular, the potential ambiguity about the significance of trailing zeros is eliminated. For example, 1300 to four significant digits is written as 1.300×103, while 1300 to two significant digits is written as 1.3×103.
Rounding
To round to n significant digits:
If the first non-significant digit is a 5 followed by other non-zero digits, round up the last significant digit (away from zero). For example, 1.2459 as the result of a calculation or measurement that only allows for 3 significant digits should be written 1.25.
If the first non-significant digit is a 5 not followed by any other digits or followed only by zeros, rounding requires a tie-breaking rule. For example, to round 1.25 to 2 significant digits, Round half up rounds up to 1.3, while Round half to even rounds to the nearest even number 1.2.
Replace any non-significant digits by zeros.
Arithmetic
Main article: Significance arithmetic
For multiplication and division, the result should have as many significant digits as the measured number with the smallest number of significant digits.
For addition and subtraction, the result should have as many decimal places as the measured number with the smallest number of decimal places.
When performing a calculation, do not follow these guidelines for intermediate results; keep as many digits as is practical to avoid rounding errors.[2]
Significant Figures and Rounding Rules
http://www.angelfire.com/oh/cmulliss/
Statistics Tutorial: Rules of Probability
Often, we want to compute the probability of an event from the known probabilities of other events. This lesson covers some important rules that simplify those computations.
Definitions and Notation
Before discussing the rules of probability, we state the following definitions:
.Two events are mutually exclusive if they have no sample points in common.
.The probability that Event A occurs, given that Event B has occurred, is called a conditional probability.
.The conditional probability of A, given B, is denoted by the symbol P(A|B).
.The probability that event A will not occur is denoted by P(A').
Rule of Subtraction
The probability of a sample point ranges from 0 to 1.
The sum of probabilities of all the sample points in a sample space equals 1.
The rule of subtraction follows directly from these properties.
P(A) = 1 - P(A')
Rule of Multiplication
The rule of multiplication applies to the following situation. We have two events from the same sample space, and we want to know the probability that both events occur.
Rule of Multiplication If events A and B come from the same sample space, the probability that both A and B occur is equal to the probability the event A occurs times the probability that B occurs, given that A has occurred.
P(A ∩ B) = P(A) * P(B|A)
An urn contains 6 red marbles and 4 black marbles. Two marbles are drawn without replacement from the urn. What is the probability that both of the marbles are black?
Solution: Let A = the event that the first marble is black; and let B = the event that the second marble is black.
We know the following:
In the beginning, there are 10 marbles in the urn, 4 of which are black. Therefore, P(A) = 4/10.
After the first selection, there are 9 marbles in the urn, 3 of which are black. Therefore, P(B|A) = 3/9.
Therefore, based on the rule of multiplication:
P(A ∩ B) = P(A) P(B|A)
P(A ∩ B) = (4/10)*(3/9) = 12/90 = 2/15
Example 2
Suppose we repeat the experiment of Example 1; but this time we select marbles with replacement. That is, we select one marble, note its color, and then replace it in the urn before making the second selection. When we select with replacement, what is the probability that both of the marbles are black?
Solution: Let A = the event that the first marble is black; and let B = the event that the second marble is black. We know the following:
In the beginning, there are 10 marbles in the urn, 4 of which are black. Therefore, P(A) = 4/10.
After the first selection, we replace the selected marble; so there are still 10 marbles in the urn, 4 of which are black. Therefore, P(B|A) = 4/10.
Therefore, based on the rule of multiplication:
P(A ∩ B) = P(A) P(B|A)
P(A ∩ B) = (4/10)*(4/10) = 16/100 = 4/25
Rule of Addition
The rule of addition applies to the following situation. We have two events from the same sample space, and we want to know the probability that either event occurs.
Rule of Addition If events A and B come from the same sample space, the probability that event A and/or event B occur is equal to the probability that event A occurs plus the probability that event B occurs minus the probability that both events A and B occur.
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Note: Invoking the fact that P( A ∩ B ) = P( A )P( B | A ), the Addition Rule can also be expressed as
P(A ∪ B) = P(A) + P(B) - P(A) * P( B | A )
Example 1
A student goes to the library. The probability that she checks out (a) a work of fiction is 0.40, (b) a work of non-fiction is 0.30, , and (c) both fiction and non-fiction is 0.20. What is the probability that the student checks out a work of fiction, non-fiction, or both?
Solution: Let F = the event that the student checks out fiction; and let N = the event that the student checks out non-fiction. Then, based on the rule of addition:
P(F ∪ N) = P(F) + P(N) - P(F ∩ N)
P(F ∪ N) = 0.40 + 0.30 - 0.20 = 0.50
Example 2
A card is drawn randomly from a deck of ordinary playing cards. You win $10 if the card is a spade or an ace. What is the probability that you will win the game?
Solution: Let S = the event that the card is a spade; and let A = the event that the card is an ace. We know the following:
There are 52 cards in the deck.
There are 13 spades, so P(S) = 13/52.
There are 4 aces, so P(A) = 4/52.
There is 1 ace that is also a spade, so P(S ∩ A) = 1/52.
Therefore, based on the rule of addition:
P(S ∪ A) = P(S) + P(A) - P(S ∩ A)
P(S ∪ A) = 13/52 + 4/52 - 1/52 = 16/52 = 4/13
Definitions and Notation
Before discussing the rules of probability, we state the following definitions:
.Two events are mutually exclusive if they have no sample points in common.
.The probability that Event A occurs, given that Event B has occurred, is called a conditional probability.
.The conditional probability of A, given B, is denoted by the symbol P(A|B).
.The probability that event A will not occur is denoted by P(A').
Rule of Subtraction
The probability of a sample point ranges from 0 to 1.
The sum of probabilities of all the sample points in a sample space equals 1.
The rule of subtraction follows directly from these properties.
Rule of Subtraction The probability that event A will occur is equal to 1 minus the probability that event A will not occur.
P(A) = 1 - P(A')
Rule of Multiplication
The rule of multiplication applies to the following situation. We have two events from the same sample space, and we want to know the probability that both events occur.
Rule of Multiplication If events A and B come from the same sample space, the probability that both A and B occur is equal to the probability the event A occurs times the probability that B occurs, given that A has occurred.
P(A ∩ B) = P(A) * P(B|A)
Example 1
An urn contains 6 red marbles and 4 black marbles. Two marbles are drawn without replacement from the urn. What is the probability that both of the marbles are black?
Solution: Let A = the event that the first marble is black; and let B = the event that the second marble is black.
We know the following:
In the beginning, there are 10 marbles in the urn, 4 of which are black. Therefore, P(A) = 4/10.
After the first selection, there are 9 marbles in the urn, 3 of which are black. Therefore, P(B|A) = 3/9.
Therefore, based on the rule of multiplication:
P(A ∩ B) = P(A) P(B|A)
P(A ∩ B) = (4/10)*(3/9) = 12/90 = 2/15
Example 2
Suppose we repeat the experiment of Example 1; but this time we select marbles with replacement. That is, we select one marble, note its color, and then replace it in the urn before making the second selection. When we select with replacement, what is the probability that both of the marbles are black?
Solution: Let A = the event that the first marble is black; and let B = the event that the second marble is black. We know the following:
In the beginning, there are 10 marbles in the urn, 4 of which are black. Therefore, P(A) = 4/10.
After the first selection, we replace the selected marble; so there are still 10 marbles in the urn, 4 of which are black. Therefore, P(B|A) = 4/10.
Therefore, based on the rule of multiplication:
P(A ∩ B) = P(A) P(B|A)
P(A ∩ B) = (4/10)*(4/10) = 16/100 = 4/25
Rule of Addition
The rule of addition applies to the following situation. We have two events from the same sample space, and we want to know the probability that either event occurs.
Rule of Addition If events A and B come from the same sample space, the probability that event A and/or event B occur is equal to the probability that event A occurs plus the probability that event B occurs minus the probability that both events A and B occur.
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Note: Invoking the fact that P( A ∩ B ) = P( A )P( B | A ), the Addition Rule can also be expressed as
P(A ∪ B) = P(A) + P(B) - P(A) * P( B | A )
Example 1
A student goes to the library. The probability that she checks out (a) a work of fiction is 0.40, (b) a work of non-fiction is 0.30, , and (c) both fiction and non-fiction is 0.20. What is the probability that the student checks out a work of fiction, non-fiction, or both?
Solution: Let F = the event that the student checks out fiction; and let N = the event that the student checks out non-fiction. Then, based on the rule of addition:
P(F ∪ N) = P(F) + P(N) - P(F ∩ N)
P(F ∪ N) = 0.40 + 0.30 - 0.20 = 0.50
Example 2
A card is drawn randomly from a deck of ordinary playing cards. You win $10 if the card is a spade or an ace. What is the probability that you will win the game?
Solution: Let S = the event that the card is a spade; and let A = the event that the card is an ace. We know the following:
There are 52 cards in the deck.
There are 13 spades, so P(S) = 13/52.
There are 4 aces, so P(A) = 4/52.
There is 1 ace that is also a spade, so P(S ∩ A) = 1/52.
Therefore, based on the rule of addition:
P(S ∪ A) = P(S) + P(A) - P(S ∩ A)
P(S ∪ A) = 13/52 + 4/52 - 1/52 = 16/52 = 4/13
Friday, 21 October 2011
What's the difference between a frequency polygon and a frequency diagram
A plot of the cumulative frequency against the upper class boundary with the points joined by line segments. Any continuous cumulative frequency curve, including a cumulative frequency polygon, is called an ogive.
Frequency Density
Histograms
The following table shows the ages of 25 children on a school bus:
Age Frequency
5-10 6
11-15 15
16-17 4
> 17 0
If we are going to draw a histogram to represent the data, we first need to find the class boundaries. In this case they are 5, 11, 16 and 18. The class widths are therefore 6, 5 and 2.
The area of a histogram represents the frequency.
The areas of our bars should therefore be 6, 15 and 4.
Remember that in a bar chart the height of the bar represents the frequency. It is therefore correct to label the vertical axis 'frequency'.
However, as in a histogram, it is the area which represents the frequency.
It would therefore be incorrect to label the vertical axis 'frequency' and the label should be 'frequency density'.
To draw a histogram you will need to work out the frequency density. The frequency density can be calculated by using the following formula:
Frequency density = frequency ÷ class width
The class width is basically the width of the group.
The frequency density goes up the y-axis, and the area of each bar will represent the frequency.
The following table shows the ages of 25 children on a school bus:
Age Frequency
5-10 6
11-15 15
16-17 4
> 17 0
If we are going to draw a histogram to represent the data, we first need to find the class boundaries. In this case they are 5, 11, 16 and 18. The class widths are therefore 6, 5 and 2.
The area of a histogram represents the frequency.
The areas of our bars should therefore be 6, 15 and 4.
Remember that in a bar chart the height of the bar represents the frequency. It is therefore correct to label the vertical axis 'frequency'.
However, as in a histogram, it is the area which represents the frequency.
It would therefore be incorrect to label the vertical axis 'frequency' and the label should be 'frequency density'.
Frequency density = frequency ÷ class width
A histogram is usually drawn when you have continuous data and the groups in the frequency table are of unequal size
To draw a histogram you will need to work out the frequency density. The frequency density can be calculated by using the following formula:
Frequency density = frequency ÷ class width
The class width is basically the width of the group.
The frequency density goes up the y-axis, and the area of each bar will represent the frequency.
Wednesday, 19 October 2011
Mother of Sciences
Philosophy is considered the mother of sciences.
Mathematics, "the mother of science." - Plato.
The sharp growth of technology and the extensive development of science in the last quarter of the twentieth century are without any doubt due to the true applications of mathematics. Applied Mathematics has filled the gap between mathematics and other disciplines which existed in the past. Today everyone who intends to enter the infinite world of science must be familiar with the language of mathematics.
"Mathematics is the language of Economics and Economics is the Mother of Commerce."
Thales of Miletus
was also the first to define general principles and set forth hypotheses, and as a result has been dubbed the "Father of Science", though it is argued that Democritus is actually more deserving of this title
Mathematics, "the mother of science." - Plato.
The sharp growth of technology and the extensive development of science in the last quarter of the twentieth century are without any doubt due to the true applications of mathematics. Applied Mathematics has filled the gap between mathematics and other disciplines which existed in the past. Today everyone who intends to enter the infinite world of science must be familiar with the language of mathematics.
"Mathematics is the language of Economics and Economics is the Mother of Commerce."
Thales of Miletus
was also the first to define general principles and set forth hypotheses, and as a result has been dubbed the "Father of Science", though it is argued that Democritus is actually more deserving of this title
Factorize Quadratic Equation
Theory:
A quadratic expression is one where the highest power on the variable is 2. Here are some examples of quadratic expressions:
When these quadratic expressions are factorised, they are written as the product of two factors.
- x²,
- z² + 3z,
- 2x² + 3x - 5,
- a² - 1,
- p² - 4p - 6
This is how to factorise these quadratic expressions:
Example 1: x²
Write as a product: x×xExample 2: z² + 3z
Take 'z' out as a common factor: z(z + 3)Example 3: 2x² + 3x - 5
This expression can be factorised if the middle term: '3x', is replaced by '-2x+5x', like this:Example 4: a² - 1
2x² - 2x + 5x - 5
See the end of this article for an explanation of how to split the middle term and write down the factorised quadratic immediately without further steps.
After doing this, the first two terms and the last two terms can be factorised:
2x(x - 1) + 5(x - 1)
Now there are only two terms, and there is a common factor: (x-1), so the expression can be factorised again, like this:
(x - 1)(2x + 5)
This expression can be expressed as: a² - 1² which is a difference of perfect squares, a special pattern that you need to be able to recognise. It can be written immediately as:Example 5: p² - 4p - 6
(a + 1)(a - 1)
In general, any expression of the form: a² - b²
can then be written immediately as: (a + b)(a - b).
In this case, the middle term cannot be split to allow factorising as in example 3.
You can use Algematics to change this expression to a difference of perfect squares, as in example 4 above:
then factorise like this:
which simplifies to:
Sunday, 16 October 2011
Transformations
Transformation involves moving an object from its original position to a new position.The object in the new position is called the image. Each point in the object is mapped to another point in the image. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Rotation |  | Turn! |
|---|---|---|
| Reflection |  | Flip! |
| Translation |  | Slide! |
 | After any of those transformations (turn, flip or slide), the shape still has the same size, area, angles and line lengths. |
If one shape can become another using Turns, Flips and/or Slides, then the two shapes are called Congruent.
A reflection is a flip over a line
Resizing
The other important Transformation is Resizing (Dilation) (also called dilation, contraction, compression, enlargement or even expansion). The shape becomes bigger or smaller:| Resizing |  |
|---|
If you have to Resize to make one shape become another then the shapes are not Congruent, but they are Similar.
Congruent or Similar
So, if one shape can become another using transformation, the two shapes might be Congruent or just Similar| If you ... | Then the shapes are ... | |
|---|---|---|
| ... only Rotate, Reflect and/or Translate | | Congruent |
| ... need to Resize | | Similar |
Rotation
"Rotation" means turning around a center:The distance from the center to any point on the shape stays the same.Every point makes a circle around the center. |
 | Here a triangle is rotated around the point marked with a "+" |
Reflection
| the reflected image is always the same size, it just faces the other way: | ||
 |  | |
A reflection is a flip over a line
How Do I Do It Myself?
| Just approach it step-by-step. For each corner of the shape: | |||
| 1. Measure from the point to the mirror line (must hit the mirror line at a right angle) | 2. Measure the same distance again on the other side and place a dot. | 3. Then connect the new dots up! | |
 |  |  | |
Labels
| It is common to label each corner with letters, and to use a little dash (called a Prime) to mark each corner of the reflected image. Here the original is ABC and the reflected image is A'B'C' |  |
Some Tricks
X-AxisIf the mirror line is the x-axis, just change each (x,y) into (x,-y) |
Y-AxisIf the mirror line is the y-axis, just change each (x,y) into (-x,y) |  |
Fold the Paper
And if all else fails, just fold your sheet of paper along the mirror line and then hold it up to the light !Reflection Symmetry
Reflection Symmetry (sometimes called Line Symmetry or Mirror Symmetry) is easy to recognise, because one half is the reflection of the other half.Rotational Symmetry
With Rotational Symmetry, the shape or image can be rotated and it still looks the same.
| How many matches there are as you go once around is called the Order. If you think of propeller blades (like below) it makes it easier. |
Examples of Different Rotational Symmetry Order
| Order | Example Shape | Artwork |
|---|---|---|
(using Symmetry Artist)
| ||
 |  |  |
 |  |  |
 |  |  |
| ... and there is also Order 5, 6, 7, and ... | ||
 |  |  |
| ... and then there is Order 9, 10, and so on ... | ||
 | Is there Rotational Symmetry of Order 1 ?Not really! If a shape only matches itself once as you go around (ie it matches itself after one full rotation) there is really no symmetry at all, because the word "Symmetry" comes from syn- together and metron measure, and there can't be "together" if there is just one thing. |
W
Real World Examples
 |  |  |
| A Dartboard has Rotational Symmetry of Order 10 | The US Bronze Star Medal has Order 5 | The London Eye has Order ... oops, I lost count! |
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