high school mathematics
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high school mathematics
The aim of this meta entry is to index entries suitable for high school students.
Basics
1. set, union, intersection
2. natural numbers, rational numbers, real numbers
3. associativity of multiplication
4. product of negative numbers
5. equation, inequality
6. proportion equation, proportionality of numbers
7. per cent
8. mathematical induction
9. proof by contradiction
10. converse
11. contrapositive
Algebra
1. opposite number, difference
2. inverse number, division
3. multiple, product
4. entries on rational numbers
5. irrational numbers
6. factorization of integers
7. linear equation
8. square of sum
9. difference of squares
10. grouping method for factoring polynomials
11. factoring a sum or difference of two cubes
12. zero rule of product
13. 14. even-even-odd rule
15. completing the square
16. square roots of rationals
17. quadratic formula
18. quadratic inequality
19. strange root
20. inequality with absolute values
21. absolute value inequalities
22. long division of polynomials
Geometry
1. basic geometric figures:
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points
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lines
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planes
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line segments
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rays
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angles
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triangles
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parallelograms
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rectangles
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trapezoids
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polygons
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regular polygons
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base and height of triangle
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circles
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parts of a ball
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cylinder
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solid cone
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2. basic geometric properties:
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intersections
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endpoints
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parallelism
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perpendicularity
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similarity
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similar triangles
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tangent of circle
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3. acute angles, convex angles, radian
4. complementary angles, supplementary angles, explementary angles
5. angle between two lines, angle of view
6. projection of point
7. locus
8. normal line, angle bisector, angle bisector as locus, center normal as locus
9. measurements (lengths, areas, and volumes) of basic geometric figures
10. compass and straightedge constructions:
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perpendicular bisector
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dropping the perpendicular from a point to a line
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erecting the perpendicular to a line at a point
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construction of tangent
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circle with given center and given radius
11. trisection of angle
12. axiomatic proofs in geometry:
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angles of an isosceles triangle
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determining from angles that a triangle is isosceles
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isosceles triangle theorem
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converse of isosceles triangle theorem
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parallelogram theorems
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regular polygon and circles
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Pythagorean theorem and its various proofs:
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construct the center of a given circle
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Thales’ theorem and its proof
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opposing angles in a cyclic quadrilateral are supplementary
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mid-segment theorem
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Analytic geometry
1. analytic geometry
2. Cartesian coordinates
3. coordinates of midpoint
4. slope
5. tangent line
6. condition of orthogonality
7. 8. conics (ellipse, hyperbola, parabola)
9. polar coordinates
Vectors and Matrices
1. sum of vectors (i.e. parallelogram principle), difference of vectors
2. Euclidean vectors
3. mutual positions of vectors
4. scalar product
5. matrices, addition and multiplication of matrices
Trigonometry
1. right triangle
2. regular triangle
3. isosceles triangle
4. altitudes
5. bisectors
6. ASA, SSS, SAS, SSA (triangle solving)
7. exact trigonometry tables
8. sohcahtoa
9. determining signs of trigonometric functions
10. addition formulas for sine and cosine
11. addition formula for tangent
12. goniometric formulae
13. trigonometric equation
Functions
1. definitions and operations of functions
2. argument
3. polynomial functions (including linear functions)
4. rational functions
5. functions involving radicals
6. exponential functions
7. Briggsian logarithms
8. trigonometric functions
9. limit of real number sequence
10. geometric sequence
11. sequences and series
Differential calculus
1. concept of a limit
2. limit rules of functions, improper limit
3. continuous
4. 5. intermediate value theorem
6. derivative
7. derivatives of sine and cosine
8. derivative of inverse function
9. related rates
10. minimum and maximum of functions (extrema)
11. least and greatest value of function
12. mean value theorem
13. Rolle’s theorem
Integral calculus
1. Riemann sum
2. integral
3. left hand rule
4. right hand rule
5. midpoint rule
6. 7. fundamental theorem of calculus
8. integration techniques
Complex numbers
1. complex numbers
2. complex function
Applications (word problems)
1. graphing of equations and inequalities
2. counting and basic probability
3. length, area, volume
4. distance, rate, speed, velocity
5. money, simple interest, compound interest
Mathematics Subject Classification
00A20 Dictionaries and other general reference works
Comments
Units digit
Whats the units digit in the expansion of
(15+sqrt220)^19 + (15+sqrt220)^82
Re: Units digit
Form the recursion
a_{n+2} = 30 a_{n+1} - 5 a_n.
The solution of this recusrion with initial conditions
a_0 = 2
and
a_1 = 30
is
a_n = (15 + sqrt 220)^n + (15 - sqrt 220)^n.
Now, 15 - sqrt 220 = 0.1676..., so the latter
term will be much less than 0.1 when n = 19 or
n=84. Hence, (15+sqrt220)^19 will have the
same units digit as a_19 and (15+sqrt220)^82
will have the same units digit as a_84, so to
obtain your answer we simply add these units digits.
How are we to detemine these units digits?
Consider the recursion modulo 10 --- the units
digits of a_n are all zero when n > 0, so the
answer is 0.
Re: Units digit
Thanks rspuzio, but I am not so brilliant that I can understand all these things in one shot. Can you please all these in detail
Why did you form the recusrsion
a_{n+2} = 30 a_{n+1} - 5 a_n.
On what basis did you take the initial conditions
a_0 = 2 and a_1 = 30
and how did you get the solution
a_n = (15 + sqrt 220)^n + (15 - sqrt 220)^n.
Also can you please explain in detail how did you get the units
digits of a_n all zero when n > 0
Re: Units digit
Now I will go a bit slower and explain the motivation for what
I did. We want to look at (15 + sqrt 220)^19 and
(15 + sqrt 220)^84. Since 19 and 84 are big, it would take
a long time to simply compute the powers, so it would be nice
to have a short-cut.
Since 15 + sqrt 220 is irrational, it is not so easy to work with it;
it would be a lot nicer if one could work with rational numbers
instead.
Thinking of Binet's formula for Fibonacci numbers, my memory having
been jogged by the recently added entry on linear recurrences in the
right-hand column of the webpage, an idea for how to do this came
to me --- consider the quantity
a_n = (15 + sqrt 220)^n + (15 - sqrt 220)^n.
On the one hand, since this is the sum of a number with its conjugate,
it will be rational, in fact, it will be an integer. On the other
hand, since 15 - sqrt 220 = 0.1676..., the second term will be
quite negligibly small when n = 19 or n = 84, especially since we
are only interested in the first decimal place.
Following the analogy with the Fibonacci sequence suggests that the
way to compute this quantity is by means of a recursion, we only
need to figure out which recursion. To this end, write
(15 + sqrt 220)^n = p_n + q_n sqrt 220.
To find the recursion, multiply both sides by 15 + sqrt 220,
simplify, and equate coefficients:
p_{n+1} + q_{n+1} sqrt 220 =
(15 + sqrt 220)^{n+1} =
(15 + sqrt 220) (15 + sqrt 220)^n =
(15 + sqrt 220) (p_n + q_n sqrt 220) =
15 p_n + 15 q_n sqrt 220 + p_n sqrt 220 + 220 q_n =
(15 p_n + 220 q_n) + (p_n + 15 q_n) sqrt 220
Hence, we have
p_{n+1} = 15 p_n + 220 q_n
and
q_{n+1} = p_n + 15 q_n,
whence, by elimination, we may conclude
p_{n+2} = 15 p_{n+1} + 220 q_{n+1} =
15 p_{n+1} + 220 (p_n + 15 q_n) =
15 p_{n+1} + 220 p_n + 3300 q_n =
15 p_{n+1} + 220 p_n + 3300 (p_{n+1} - 15 p_n) / 220 =
30 p_{n+1} - p_n.
Now, if
(15 + sqrt 220)^n = p_n + q_n sqrt 220
then, taking the other sign of the square root,
(15 - sqrt 220)^n = p_n - q_n sqrt 220,
so a_n = 2 p_n, and hence a_n will satisfy the same
recursion, to wit
a_{n+2} = 30 a_{n+1} - 5 a_n
since it is linear. As for the initial values, we
simply substitute n=0 and n=0 into the definition to
find them:
a_0 = 2
a_1 = 30
we may now use our recursion to obtain a few more values:
a_2 = 890
a_3 = 26550
a_4 = 792050
a_5 = 23628750
Note that the units digit of all these values is zero. Furthermore,
if the units digit of both a_n and a_{n+1} is zero, it follows
from the recursion that the units digit of a_{n+2} will also be
zero. Hence, without need of further computation, we see that the
units digits of a_19 and a_84 will also both be zero.
Re: Units digit:Thanks a lot rspuzio
Thanks rspuzio
Great work. You are really a genius. I really appreciate.
divisibility by 11
I am just confused with the ways to find divisibility by 11.
1) If sum of alternate digits is same then number is divisible by 11 -> this is sufficient but not necessary
2) If sum of digits is 11, then number is divisible by 11. This does not seem true always. For eg: 605 is divisible by 11 but 434 is not.
Can anyone please prove how to find divisibility by 11?
Re: divisibility by 11
Sure. You add up the odd-position digits, and add up the even-position digits. If the two sums differ by a multiple of 11, then the number is divisible by 11.
This is easy to see by writing out the decimal representation. Without any carries, it's trivial; you have to think about it just a little bit to convince yourself that it works when you have to carry in the multiplication. But it's all straightforward.
November Ponder This!!!!!
November's challenge is dedicated to IBM Fellow Benoît Mandelbrot, who died on October 14, 2010, at the age of 85.
Let's define a function f on five variables (i, j, k, l, and m) as follows:
If (i-m)*(j-m) is zero then
f(i,j,k,l,m) ↠((abs(j-(1-k+l)*m)-m+1)* (abs(i-(k+l)*m)-m+1))
else
f(i,j,k,l,m) ↠f(i mod m,
j mod m,
int(((i^j)&((m*l)^i))/m)^l^k^int(i/m),
int(((i^j)&((m*k)^i))/m)^l,
m/2)
Where
x^y means bitwise xor (exclusive or)
x&y means bitwise and (conjunction)
abs is the absolute value
mod is the modulo function
int means truncating downwards to an integer
What should be the input to the function f, and how do you need to interpret its output to associate it with Professor Mandelbrot?