I am as a sure student wish to master lecturing items maximally. To master that ability hence me learn in this university, Yogyakarta State University. One of the items which I study here is language of English.
A compulsion for me to master Ianguage of English, to have a competence in English, specially in mathematics education.
According to me Ianguage of English is vital importance, because Ianguage of English represent interracial communications Ianguage ( international language). Besides, Ianguage of English earn to access science or knowledge. Most science like for example mathematics, physics, biological, chemistry, and technology overspread in Ianguage of English.
Considering most science reading generally have Ianguage of English, hence to obtain science easily and quickly from more nations go forward to be needed by us, specially I am, as the rising generation of Indonesia taft in English. Ably adequate English, we will easy to access / obtaining information / new science of nations go forward.
Globalization era have peeped out very interracial tight emulation. Nation owning ability compete will obtain advantage conversely nation which do not have ability compete will harvest loss. Ability compete very determined by strength of power factor emulate. Among many power factor emulate, one other especial is human resource. Excellence of technology will only earn to be reached through the ownership of strong human resource in hand sciences constitutoing technology, one of them that is English. Excellence of power factor emulate human resource represent key because human resource represent the single active resource, while rest resource is passive. Human resource owning excellence will be able to compete healthyly in global chess which have, is, and will take place tightly.
So that we have to have ability in Ianguage of English well. I have to have a good competence in English for mathematics education. Because some years later I will be a techer, and I will give all of my competence in English and mathematics for my student.
I will develop my ability which I had have now.
It is true many people say in this globalization era of vital importance to master at least Ianguage of English. But although that thing is correctness, we also have to be proud fixed with our own identity, that is Indonesian.
Kamis, 11 Juni 2009
What I have done and what I will do about English for Mathematics
I have study English in general since long time ago. This is an interesting lesson but also difficult.
Then now, English that I study is specially, that is English for Mathematics.
Since entering into mathematics education subject in Yogyakarta State University, I was getting the items of English start from the first semester. At that opportunity, the items that I got was just a little.
There was an interesting experience that I feel. I am belong to lucky people because I got an uncommon chance. From the TOEFL test was held at the first semester, my score was enough so that I’m passed. And because in Yogyakarta State University will hold a new program (WCU), so 20 students including me were got like a training. That was very useful for me. I could explore my English and also I could get more friends from another class. Because of WCU training was held by Faculty of Languages and Arts, Yogyakarta State University, every student should be more speaking and speaking.
In the second semester, I got the English items again, that is English II which is Mr Marsigit as the lecture.
Although maybe still a little, but I am sure that my knowledge about English for Mathematics was developing. In this semester Mr Marsigit also use the technology media, that is blogger. From many works that given by Mr Marsigit, automatically I often to browsing internet and searching items that correlating with English for Mathematics.
I will develop my knowledge about English for Mathematics. Because English is international language, so it’s very important. I will learn and learn from many sources.
And also I surely always pray to God, in order to God gives me an amenity during I study, especially in study English for Mathematics.
If one time I become a teacher, I will transfer my knowledge as many as possible to all of my student. I will applicate the methods I thinking can make easier in transfer of English, maybe also including the method was applicated by Mr Marsigit.
I like English. And I hope that I can really understanding English some times later.
Then now, English that I study is specially, that is English for Mathematics.
Since entering into mathematics education subject in Yogyakarta State University, I was getting the items of English start from the first semester. At that opportunity, the items that I got was just a little.
There was an interesting experience that I feel. I am belong to lucky people because I got an uncommon chance. From the TOEFL test was held at the first semester, my score was enough so that I’m passed. And because in Yogyakarta State University will hold a new program (WCU), so 20 students including me were got like a training. That was very useful for me. I could explore my English and also I could get more friends from another class. Because of WCU training was held by Faculty of Languages and Arts, Yogyakarta State University, every student should be more speaking and speaking.
In the second semester, I got the English items again, that is English II which is Mr Marsigit as the lecture.
Although maybe still a little, but I am sure that my knowledge about English for Mathematics was developing. In this semester Mr Marsigit also use the technology media, that is blogger. From many works that given by Mr Marsigit, automatically I often to browsing internet and searching items that correlating with English for Mathematics.
I will develop my knowledge about English for Mathematics. Because English is international language, so it’s very important. I will learn and learn from many sources.
And also I surely always pray to God, in order to God gives me an amenity during I study, especially in study English for Mathematics.
If one time I become a teacher, I will transfer my knowledge as many as possible to all of my student. I will applicate the methods I thinking can make easier in transfer of English, maybe also including the method was applicated by Mr Marsigit.
I like English. And I hope that I can really understanding English some times later.
How to proof square root of 2 is irrational number
Greek ancient find square root of 2 wasn’t rational although square root of 2 was the length of hypotenuse with upright side triangles was 1. The number cannot write as result from integer number. So the square root of 2 is irrational.
The proof:
Rational number is the number which can be stated in ratio a over b, a and b the integer number which do not have factor partner and b unequal 0
If square root of 2 is rational number, then :
Square root of 2 equals a over b times r
¬2 equals a square over b square
¬a square equals 2 times b square (2b square is the integer number, integer times 2 is even number )
¬a square equals even
¬a equals even………(1)
¬a equals 2n (n is integer number)
¬a square equals 4n square
¬2 times b square equals 4n square
¬b square equals 2n square
¬b square is even………(2)
¬b equals even
From equation 1 and 2 we know that a and b are even.
a and b are even, so they have factor partner that is 2. The entire step is right, that is the opposite with the definition of the rational number. It means the proof is wrong, so square root of 2 is irrational.
The proof:
Rational number is the number which can be stated in ratio a over b, a and b the integer number which do not have factor partner and b unequal 0
If square root of 2 is rational number, then :
Square root of 2 equals a over b times r
¬2 equals a square over b square
¬a square equals 2 times b square (2b square is the integer number, integer times 2 is even number )
¬a square equals even
¬a equals even………(1)
¬a equals 2n (n is integer number)
¬a square equals 4n square
¬2 times b square equals 4n square
¬b square equals 2n square
¬b square is even………(2)
¬b equals even
From equation 1 and 2 we know that a and b are even.
a and b are even, so they have factor partner that is 2. The entire step is right, that is the opposite with the definition of the rational number. It means the proof is wrong, so square root of 2 is irrational.
How to determine abc formula
The general formula : ax2+ bx+ c
a, b, c is real number
a, b equals coefficients
c equals constanta
Then to get the formula:
The general formula: ax2+ bx+ c
1. the equation square divided by a, the formula become : x square plus bx over a plus c over a equals 0
2. then the formula plus square of b over 2a in each part, the formula become : x square plus bx over a plus square of b over 2a plus c over a equals square of b over 2a
3. the formula become : square of x plus b over 2a equals b square over 4a square minus c over a
equals b square minus 4ac all over 4a square
4. then the formula become : x plus b over 2a equals ± square root of b square minus 4ac all over 4a square
5. the formula become : x equals minus b over 2a plus square root of b square minus 4ac all over 4a square, or
x equals plus b over 2a plus square root of b square minus 4ac all over 4a square.
a, b, c is real number
a, b equals coefficients
c equals constanta
Then to get the formula:
The general formula: ax2+ bx+ c
1. the equation square divided by a, the formula become : x square plus bx over a plus c over a equals 0
2. then the formula plus square of b over 2a in each part, the formula become : x square plus bx over a plus square of b over 2a plus c over a equals square of b over 2a
3. the formula become : square of x plus b over 2a equals b square over 4a square minus c over a
equals b square minus 4ac all over 4a square
4. then the formula become : x plus b over 2a equals ± square root of b square minus 4ac all over 4a square
5. the formula become : x equals minus b over 2a plus square root of b square minus 4ac all over 4a square, or
x equals plus b over 2a plus square root of b square minus 4ac all over 4a square.
How to find ח (phi)
Egypt have found ח value with 3,16. They got it from area of circle equals square of 8/9 times diameter.
We know diameter of circle is the length of the line through the center and touching two points on its edge.
Diameter equals 2R; R is radius of the circle.
Radius is the distance from the center to any point of edge.
The formula becomes:
¬ Area of circle equals square of 8/9 times 2 radiuses,
¬ Area of circle equals 64/81 times 4 radiuses square,
¬ Area of circle equals 3,16 times radiuses square.
We get ח equals 3,16.
Many mathematicians examined carefully about ח with method analytic or use computer till get the value of ח is 3,14.
Now we use the value of ח equals 3,14.
We know diameter of circle is the length of the line through the center and touching two points on its edge.
Diameter equals 2R; R is radius of the circle.
Radius is the distance from the center to any point of edge.
The formula becomes:
¬ Area of circle equals square of 8/9 times 2 radiuses,
¬ Area of circle equals 64/81 times 4 radiuses square,
¬ Area of circle equals 3,16 times radiuses square.
We get ח equals 3,16.
Many mathematicians examined carefully about ח with method analytic or use computer till get the value of ח is 3,14.
Now we use the value of ח equals 3,14.
VIDEOS OF MATHEMATICS
Video 1
“Do You Believe?”
In the first video there is an explanation about trusting someone. In this video, there is a children who orating and give some questions for audience, like : Do you believe in me? Do you believe in me? over and over again. He said that he can be believed and don`t underestimating him. He said that he can do anything and also can be anything. Then all in the room give applause for his oration.
Not only adult people can make a change in this world but also moppet can do that. So we must believe everyone, belong children. The important matter is we also need to believe in ourself.
Video 2
“What You Know about Math”
There are a lot of knowledges in mathematics. In this video there are two people who singing a song about mathematics. In the song, they explain about mathematics and some items in mathematics, like significant figures, graphic, trigonometry, exponent, calculator, phi, ln x, etc.
Video 3
“English Solving Problem“
There are some questions :
1. If the function h(x)=g(2x)+2, what the value of h(1)?
Answer:
h(x)=g(2x)+2
h(1)=g(2x1)+2 g(2x1) = 1
h(1)=3
2. The graph of f(x) = x + 1 if 2f(p) = 20. What the value of f(3p)?
Answer:
2f(p) = 20
f(p) = 10
f(x) = x + 1
We substitution p to f(x) = x + 1, become f(p) = p + 1
f(p) = p + 1 = 10
p = 10 – 1 = 9
f(3p)????
3p = 3(9) = 27
f(x) = x + 1
f(3p) = 3p + 1 = 27 + 1 = 28
So the value of f(3p) equals 28
3. In the xy-coordinate plane, the graph of x equals y square minus four intersects line l at (0, p) and (5, t). What is the greatest possible value of the slope of graph?
x equals y square minus four
Line l: m equals ytwo minus yone all over xtwo minus xone
Equals t minus p all over five minus 0
Video 4
“Properties of Logarithms”
Log x to the base b equals y similar with b to the power of y equals x
Log x to the base ten equals log x
Log x to the base e equals ln x (natural logarithm)
Example :
1. Log one hundred to the base ten equals x
Ten to the power of x equals one hundred
Ten to the power of two equals one hundred
x equals two
2. Log x to the base two equals three
Two to the power of three equals x
eight equals x
So log eight to the base two equals three
3. Log one forty nine to the base seven equals x
Seven to the power of x equals one forty nine
Seven to the power of x equals one seven to the power of two
Seven to the power of x equals seven to the power of negative two
x equals negative two
Log M times N to the base b equals log M to the base b plus log N to the base b
Log M over N to the base b equals log M to the base b minus log N to the base b
Log x to the power of n to the base b equals n log x to the base b
Expand:
Log x square times y plus one in bracket all over z to the power of three to the base three
Equals log x square times y plus one in bracket to the base three minus log z to the power of three to the base three
Equals log x square to the base three plus log y plus one in bracket to the base three minus log z to the power of three to the base three
Equals two times log x to the base three plus log y plus one in bracket to the base three minus three times log z to the base three
Video 5
“Graph of A Rational Function”
Graphs of a rational function
Let the function f be defined by f(x) equals x plus two all over x minus one, if x equals one
We can substitute one to that function
So, f(x) equals one plus two all over one minus one
And we get that denominator equals zero
We know that if the denominator equals zero, so the result can’t be definitioned. Because the denominator of a rational function can’t be zero.
Example:
y equals x square minus x minus six all over x minus three, if x equals three
Answer:
1. Substitute three to that function
y equals three square minus three minus six all over three minus three
Then, we get that denominator equals zero
We know that if the denominator is zero, so the result can’t be definitioned.
2. Factored the function
y equals x square minus x minus six all over x minus three
y equals x minus three in bracket times x plus two in bracket all over x minus three
y equals x plus two
y equals three plus two
y equals five
With use the second way, we can get the answer of the function.
Video 6
“Trigonometry“
Trigonometry’s story is right triangle. Relationship between the sides to the angles. So, let’s make a right triangle. The angles we call theta. The side of line is 3, 4, and 5 in the hypotenuse.
Sin theta equals?
To solve those problem, we must use the trick in order to we can solve : soh cah toa
It means :
soh: sin is opposite over hypotenuse
cah: cos is adjust over hypotenuse
toa: tan is opposite over adjust
So, sin theta equals 4 as opposite over 5 as hypotenuse.
Cos theta equals 3 as adjust over 5 as hypotenuse.
Then let see, one left. Tan theta equals 4 as opposite over 3 as adjust
Then, we named the other angle is x.
So, tan x is also opposite over adjust, tan x equals 3 as opposite over 4 as adjust. It means that tan x is inverse of tan theta.
“Do You Believe?”
In the first video there is an explanation about trusting someone. In this video, there is a children who orating and give some questions for audience, like : Do you believe in me? Do you believe in me? over and over again. He said that he can be believed and don`t underestimating him. He said that he can do anything and also can be anything. Then all in the room give applause for his oration.
Not only adult people can make a change in this world but also moppet can do that. So we must believe everyone, belong children. The important matter is we also need to believe in ourself.
Video 2
“What You Know about Math”
There are a lot of knowledges in mathematics. In this video there are two people who singing a song about mathematics. In the song, they explain about mathematics and some items in mathematics, like significant figures, graphic, trigonometry, exponent, calculator, phi, ln x, etc.
Video 3
“English Solving Problem“
There are some questions :
1. If the function h(x)=g(2x)+2, what the value of h(1)?
Answer:
h(x)=g(2x)+2
h(1)=g(2x1)+2 g(2x1) = 1
h(1)=3
2. The graph of f(x) = x + 1 if 2f(p) = 20. What the value of f(3p)?
Answer:
2f(p) = 20
f(p) = 10
f(x) = x + 1
We substitution p to f(x) = x + 1, become f(p) = p + 1
f(p) = p + 1 = 10
p = 10 – 1 = 9
f(3p)????
3p = 3(9) = 27
f(x) = x + 1
f(3p) = 3p + 1 = 27 + 1 = 28
So the value of f(3p) equals 28
3. In the xy-coordinate plane, the graph of x equals y square minus four intersects line l at (0, p) and (5, t). What is the greatest possible value of the slope of graph?
x equals y square minus four
Line l: m equals ytwo minus yone all over xtwo minus xone
Equals t minus p all over five minus 0
Video 4
“Properties of Logarithms”
Log x to the base b equals y similar with b to the power of y equals x
Log x to the base ten equals log x
Log x to the base e equals ln x (natural logarithm)
Example :
1. Log one hundred to the base ten equals x
Ten to the power of x equals one hundred
Ten to the power of two equals one hundred
x equals two
2. Log x to the base two equals three
Two to the power of three equals x
eight equals x
So log eight to the base two equals three
3. Log one forty nine to the base seven equals x
Seven to the power of x equals one forty nine
Seven to the power of x equals one seven to the power of two
Seven to the power of x equals seven to the power of negative two
x equals negative two
Log M times N to the base b equals log M to the base b plus log N to the base b
Log M over N to the base b equals log M to the base b minus log N to the base b
Log x to the power of n to the base b equals n log x to the base b
Expand:
Log x square times y plus one in bracket all over z to the power of three to the base three
Equals log x square times y plus one in bracket to the base three minus log z to the power of three to the base three
Equals log x square to the base three plus log y plus one in bracket to the base three minus log z to the power of three to the base three
Equals two times log x to the base three plus log y plus one in bracket to the base three minus three times log z to the base three
Video 5
“Graph of A Rational Function”
Graphs of a rational function
Let the function f be defined by f(x) equals x plus two all over x minus one, if x equals one
We can substitute one to that function
So, f(x) equals one plus two all over one minus one
And we get that denominator equals zero
We know that if the denominator equals zero, so the result can’t be definitioned. Because the denominator of a rational function can’t be zero.
Example:
y equals x square minus x minus six all over x minus three, if x equals three
Answer:
1. Substitute three to that function
y equals three square minus three minus six all over three minus three
Then, we get that denominator equals zero
We know that if the denominator is zero, so the result can’t be definitioned.
2. Factored the function
y equals x square minus x minus six all over x minus three
y equals x minus three in bracket times x plus two in bracket all over x minus three
y equals x plus two
y equals three plus two
y equals five
With use the second way, we can get the answer of the function.
Video 6
“Trigonometry“
Trigonometry’s story is right triangle. Relationship between the sides to the angles. So, let’s make a right triangle. The angles we call theta. The side of line is 3, 4, and 5 in the hypotenuse.
Sin theta equals?
To solve those problem, we must use the trick in order to we can solve : soh cah toa
It means :
soh: sin is opposite over hypotenuse
cah: cos is adjust over hypotenuse
toa: tan is opposite over adjust
So, sin theta equals 4 as opposite over 5 as hypotenuse.
Cos theta equals 3 as adjust over 5 as hypotenuse.
Then let see, one left. Tan theta equals 4 as opposite over 3 as adjust
Then, we named the other angle is x.
So, tan x is also opposite over adjust, tan x equals 3 as opposite over 4 as adjust. It means that tan x is inverse of tan theta.
MY DIFFICULT WORDS TO EXPRESS MATHEMATICAL IDEAS
*PRINTSTORMING
Axiom
Binomial
Continuous function
Composite number
Diameter of a circle or sphere
Even function
Explicit function
Hypotenuse
Identity function
Uncountably infinite
Vector
Theorem
Transformations
Two dimensional
Rational equation
Range
Quadrants
Perfect square
Factorial
Probability
One-to-One function
Acute angle
Acute triangle
Bisector
Combination
Matrix
Modular numbers
Exponent
Hyperbola
Horizontal
*MEANINGS OF MY DIFFICULT WORDS
1. Axiom is a statement accepted as true without proof. An axiom should be so simple and direct that it is unquestionably true.
2. Binomial is a polynomial with two terms which are not like terms.
3. Continuous function is a function with a connected graph.
4. Composite number is a positive integer that has factors other than just 1 and the number itself. For example, 4, 6, 8, 9, 10, 12, etc. are all composite numbers. The number 1 is not composite.
5. Diameter of a circle or sphere is a line segment between two points on the circle or sphere which passes through the center.
6. Even function is a function with a graph that is symmetric with respect to the y-axis. A function is even if and only if f(–x) = f(x).
7. Explicit function is a function in which the dependent variable can be written explicitly in terms of the independent variable.
8. Hypotenuse is the side of a right triangle opposite the right angle. Note: The hypotenuse is the longest side of a right triangle.
9. Identity function is the function f(x) = x. More generally, an identity function is one which does not change the domain values at all.
10. Uncountably infinite
Describes a set which contains more elements than the set of integers. Formally, an uncountably infinite set is an infinite set that cannot have its elements put into one-to-one correspondence with the set of integers. For example, the set of real numbers is uncountably infinite.
11. Vector is a quantity, drawn as an arrow, with both direction and magnitude. For example, force and velocity are vectors.
12. Theorem is an assertion that can be proved true using the rules of logic. A theorem is proven from axioms, postulates, or other theorems already known to be true.
13. Transformations is operations that alter the form of a figure. The standard transformations are translations, reflections, dilations (stretches), compressions (contractions or shrinks), and rotations.
14. Two dimensional is the property of a plane that indicates that motion can take place in two perpendicular directions.
15. Rational equation is an equation which has a rational expression on one or both sides of the equal sign. Sometimes rational equations have extraneous solution.
16. Range is the set of y-values of a function or relation. More generally, the range is the set of values assumed by a function or relation over all permitted values of the independent variable(s).
17. Quadrants are the four sections into which the x-y plane is divided by the x- and y-axes.
18. Perfect square is any number that is the square of a rational number.
19. Factorial is the result of all numbers smaller natural or equal to natural number that expressed specifically. Example, 4! = 1.2.3.4 = 24.
20. Probability is the likelihood of the occurrence of an event. The probability of event A is written P(A). Probabilities are always numbers between 0 and 1, inclusive.
21. One-to-One function is a function for which every element of the range of the function corresponds to exactly one element of the domain. One-to-one is often written 1-1.
22. Acute Angle is an angle that has measure less than 90°.
23. Acute Triangle is a triangle for which all interior angles are acute.
24. Bisector is a line segment, line, or plane that divides a geometric figure into two congruent halves.
25. Combination is a selection of objects from a collection. Order is irrelevant.
26. Matrix is a rectangular (or square) array of numbers. Matrices can be written using brackets or parentheses.
27. Modular numbers is the value of an integer modulo n is equal to the remainder left when the number is divided by n. Modulo n is usually written mod n.
28. Exponent is x in the expression ax. For example, 3 is the exponent in 23.
29. Hyperbola is a conic section that can be thought of as an inside-out ellipse. Formally, a hyperbola can be defined as follows: For two given points, the foci, a hyperbola is the locus of points such that the difference between the distances to each focus is constant.
30. Horizontal is perfectly flat and level. For example, the horizon is horizontal. So is the floor.
Axiom
Binomial
Continuous function
Composite number
Diameter of a circle or sphere
Even function
Explicit function
Hypotenuse
Identity function
Uncountably infinite
Vector
Theorem
Transformations
Two dimensional
Rational equation
Range
Quadrants
Perfect square
Factorial
Probability
One-to-One function
Acute angle
Acute triangle
Bisector
Combination
Matrix
Modular numbers
Exponent
Hyperbola
Horizontal
*MEANINGS OF MY DIFFICULT WORDS
1. Axiom is a statement accepted as true without proof. An axiom should be so simple and direct that it is unquestionably true.
2. Binomial is a polynomial with two terms which are not like terms.
3. Continuous function is a function with a connected graph.
4. Composite number is a positive integer that has factors other than just 1 and the number itself. For example, 4, 6, 8, 9, 10, 12, etc. are all composite numbers. The number 1 is not composite.
5. Diameter of a circle or sphere is a line segment between two points on the circle or sphere which passes through the center.
6. Even function is a function with a graph that is symmetric with respect to the y-axis. A function is even if and only if f(–x) = f(x).
7. Explicit function is a function in which the dependent variable can be written explicitly in terms of the independent variable.
8. Hypotenuse is the side of a right triangle opposite the right angle. Note: The hypotenuse is the longest side of a right triangle.
9. Identity function is the function f(x) = x. More generally, an identity function is one which does not change the domain values at all.
10. Uncountably infinite
Describes a set which contains more elements than the set of integers. Formally, an uncountably infinite set is an infinite set that cannot have its elements put into one-to-one correspondence with the set of integers. For example, the set of real numbers is uncountably infinite.
11. Vector is a quantity, drawn as an arrow, with both direction and magnitude. For example, force and velocity are vectors.
12. Theorem is an assertion that can be proved true using the rules of logic. A theorem is proven from axioms, postulates, or other theorems already known to be true.
13. Transformations is operations that alter the form of a figure. The standard transformations are translations, reflections, dilations (stretches), compressions (contractions or shrinks), and rotations.
14. Two dimensional is the property of a plane that indicates that motion can take place in two perpendicular directions.
15. Rational equation is an equation which has a rational expression on one or both sides of the equal sign. Sometimes rational equations have extraneous solution.
16. Range is the set of y-values of a function or relation. More generally, the range is the set of values assumed by a function or relation over all permitted values of the independent variable(s).
17. Quadrants are the four sections into which the x-y plane is divided by the x- and y-axes.
18. Perfect square is any number that is the square of a rational number.
19. Factorial is the result of all numbers smaller natural or equal to natural number that expressed specifically. Example, 4! = 1.2.3.4 = 24.
20. Probability is the likelihood of the occurrence of an event. The probability of event A is written P(A). Probabilities are always numbers between 0 and 1, inclusive.
21. One-to-One function is a function for which every element of the range of the function corresponds to exactly one element of the domain. One-to-one is often written 1-1.
22. Acute Angle is an angle that has measure less than 90°.
23. Acute Triangle is a triangle for which all interior angles are acute.
24. Bisector is a line segment, line, or plane that divides a geometric figure into two congruent halves.
25. Combination is a selection of objects from a collection. Order is irrelevant.
26. Matrix is a rectangular (or square) array of numbers. Matrices can be written using brackets or parentheses.
27. Modular numbers is the value of an integer modulo n is equal to the remainder left when the number is divided by n. Modulo n is usually written mod n.
28. Exponent is x in the expression ax. For example, 3 is the exponent in 23.
29. Hyperbola is a conic section that can be thought of as an inside-out ellipse. Formally, a hyperbola can be defined as follows: For two given points, the foci, a hyperbola is the locus of points such that the difference between the distances to each focus is constant.
30. Horizontal is perfectly flat and level. For example, the horizon is horizontal. So is the floor.
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