Numbers

The ancient Egyptians were possibly the first civilisation to practice the scientific arts. Indeed, the word chemistry is derived from the word Alchemy which is the ancient name for Egypt.Where the Egyptians really excelled was in medicine and applied mathematics. But although there is a large body of papyrus literature describing their achievements in medicine, there is no records of how they reached their mathematical conclusions. Of course they must have had an advanced understanding of the subject because their exploits in engineering, astronomy and administration would not have been possible without it.

The Egyptians had a decimal system using seven different symbols.1 is shown by a single stroke.10 is shown by a drawing of a hobble for cattle.100 is represented by a coil of rope.1,000 is a drawing of a lotus plant.10,000 is represented by a finger.100,000 by a tadpole or frog1,000,000 is the figure of a god with arms raised above his head.

Solids

A polyhedron is said to be regular if its faces are congruent regular polygons and if its polyhedral angles are all congruent. There is the tetrahedron with four triangular faces, the hexahedron, or cube, with six square faces, the octahedron with eight triangular faces, the dodecahedron with twelve pentagonal faces, and the icosahedron with twenty triangular faces. Plato mystically associates fire, earth, air, universe, and water to each regular solid.

Quadratic functions

For each of the following functions, find the vertex, axis, domain, range, intercepts, and sketch the graph.

Mathematical romance

One romantic evening in front of a warm fireplace, your sweetheart turns to you and says: “I will marry you if you can solve my mathematical challenge. Consider a multidigit integer N that is a power of 2. Moreover, each digit of N is also a power of 2. What is N ?”
You think about it, but no solution pop into your head.
Your lover sighs, massages the back of your neck, and finally whispers: “If you do not answer correctly, I will seek a more intelligent mate.”
Can you satisfy your potential spouse ?

The square root symbol

The Austrian mathematician Christoff Rudolff was the first to use the square root in print; it was published in 1525 in DIE COSS

Champernowne's number

He is a very famous number called Champernowne's number: 0,1234567891011121314...

Do you see how it is created ? What makes it so interesting ?

A new operational symbol has been developed in addition to +, - , x, :

This symbol is ~ and is represented by

or a time b divided by a minus b.
Find the value of (2~3)~4

Work

For babysitting Monica charges a flat fee of 3$, plus 5$ per hour. The graph shows how much she earns.

1. Write an equation in slope-intercept form of the line


2. What do you think the slope and y-intercept ?


3. How much money will she make if she babysits 5 hours ?


4. What are the effects if Monica changes her rates so that the equation is y = 6x + 2 ?

A famous female mathematician

Sophie Germain ( 1776 – 1831 ) made major contributions to number theory, acoustics, and elasticity. At age thirteen, Sophie read an account of death of Archimedes at the hands of Roman soldier. She was so moved by this story that she decided to become a mathematician. Sadly, her parents felt that her interest in mathematics was inappropriate, so at night she secretly studied the works of Isaac Newton and the mathematician Leonhard Euler.

A single solution ?

According to The Inquisitive Problem Solver, the only positive integer solution to

A r B r C = C r D r E = E r F r G is

8 r 1 r 9 = 9 r 2 r 4 = 4 r 6 r 3,

if we assume that each variable must be a simple digit.

System of equations

1. Describe the graph of 2x + y = 11 and 2x + y = 2. Determine the number of solutions.
2. Is the system consistent and independent,consistent and dependent or inconsistent?
   9x + 12y = 8 and
3. Solve the system using the linear combination method.
   3x - y - z = 4
   x - 3y + z = - 8
   -3x - 3y + z = - 4
   

Experimenting with the heart

Experimenting with real human hearts isn’t possible, but experimenting with accurate mathematical models of the human heart has led to a new understanding of its complex processes. Mathematics and the computers can replace years of experimentation in laboratories. For example, understanding resulting from mathematics greatly speeds up the design and implementation of artificial valves.

Equations based on Hooke’s Law model the geometry of the heart by representing muscle fibers as close curves of different elasticities. The Navier-Stokers Equations, which describe all fluid flows, model blood flow in and around the heart. The fact that the heart’s shape is constantly changing, however, makes the equations especially hard to solve, and a precise solution to the equation can’t be found. Approximate solutions are generated by computers.

Loyd’s „teacher” puzzle

Sam Loyd (1841-1911) was America's greatest puzzle expert and invented thousands of ingenious and tremendously popular puzzles. After his death, Loyd's son published the Cyclopedia of Puzzles, a huge collection of Loyd's puzzles which had appeared in various newspapers and magazines over the previous fifty years.

The teacher pictured in the figure is explaining to his class the remarkable fact that 2 times 2 gives the same answer as 2 plus 2. Although is the only positive number with this property, there are many pairs of different numbers that can be substituted for a and b in an equations on the right of the blackboard, namely:
a X b = y i a + b = y
Can you find a value for a and b ? For this puzzle, Sam Loyd asks us to give different values for a and b. They may be fractions, of course, but they must have a product that is exactly equal to their sum.

Lines

Figure shows the lines l1, l2 and l3.

Line l1 passes through the points A (5, 2 ) and B ( 7, 8 ).

a) Find an equation of the line l1

Line l2 is perpendicular to line l1 and also passes through the point A.

b) Find an equation of the line l2

Line l3 has equation x - 2y + 9 = 0 and intersects line l1 at point the B and line l2 at the point C.

c) Find the coordinates of the point C

d) Show that triangle is isosceles.

Relations

Biologists have found that the chirping rate of crickets is related to temperature, and that the relationship is close to linear. If one species of cricket chirps 113 times per minute at 70F and 173 times per minute at 80F, find a linear equation that models the temperature T as a function of the number N of chirps per minute.

What does the slope of this line represent?

Greater-than and greater-than or equal to symbols

The greater-than and less-than symbols ( > and < ) were introduced by the British mathematician Thomas Harriot in his Artis Analyticae Praxis, published in 1631.

The symbol ( greater-than or equal to) was first introduced by the French scientist Pierre Bouguer in 1734.

Ages

I have five wonderful friends, one of whom is quite young. The sum of all their ages is 109. If I add pairs of ages, I get the following:

How old are my friends ?

Origami

Origami -paper-folding - may not seem like a subject for mathematical investigation or one with sophisticated applications, yet anyone who has tried to fold a road map or wrap a present knows that origami is not trivial matter.
Mathematicians, computer scientist, and engineers have recently discovered that this centuries-old subjects can be used to solve many modern problems. The method of origami are now used to fold object such as automobile air bags and huge space telescopes efficiently, and may be related to how proteins fold.

( from www.ams.org)

What is a mathematician ?

"A mathematician is a blind man in a dark room looking for a black cat which is't there"

Charles Darwin

Diophantus

Diophantus was a Greek mathematician sometimes known as 'the father of algebra' who is best known for his Arithmetica. This had an enormous influence on the development of number theory.

Diophantus introduced symbols for subtraction, for an unknown, and for the degree of the variable. Although there were several solutions to some of his problems, he only looked for one positive integer solution. Now we call an equation to be solved in integers a diophantine equation. For example, Diophantus considered the equations
ax + by = c where the variables x and y are positive integers.

"Here lies Diophantus," the wonder behold . . .

Through art algebraic, the stone tells how old:

"God gave him his boyhood one-sixth of his life,

One twelfth more as youth while whiskers grew rife;

And then yet one-seventh ere marriage begun;

In five years there came a bouncing new son.

Alas, the dear child of master and sage

After attaining half the measure of his fathers lifechill fate took him.

After consoling his fate by this science of numbers for four years, he ended his life."

Find Diophantus' age at death.

Labyrinths

When solving a labyrinth, you are trying to find its center. If you know the secret of the labyrinth, obviously finding the center is much easier. These labyrinths shown existed in real life.

Based on the labyrinth in:Campton Courts - Henry VIII times

Based on the labyrinth in:The garden of Rouse Ball

For the second labyrinth, also try to reach the gray field, and supposing that the exit is closed, try to make the longest journey possible by crossing the same path as few times as possible.

Linear Functions

• The linear function F = 1.8C + 32 can be used to convert temperatures between ( C )Celsius and Fahrenheit ( F) .

• If a utility company charges a fixed monthly rate plus a constant rate for each unit of power consumed, a linear function will show the monthly cost of power. If the fixed rate is $25, and the cost for each unit of power is $0.02, the linear function is C = 0.02P + 25.

• The linear function I = 400C + 1,500 yields the total monthly income of a car salesman who makes a monthly base salary of $1,500 and receives $400 dollars for each car sold.

The Sieve of Eratosthenes

Eratosthenes (275-194 B.C., Greece) devised a "sieve" to discover prime numbers. A sieve is like a strainer that you drain spaghetti through when it is done cooking. The water drains out, leaving your spaghetti behind. Eratosthenes's sieve drains out composite numbers and leaves prime numbers behind.

To use the sieve of Eratosthenes to find the prime numbers up to 120, make a chart of the first one hundred twenty positive integers:

• Cross out 1, because it is not prime.


• Circle 2, because it is the smallest positive even prime. Now cross out every multiple of 2; in other words, cross out every second number.


• Circle 3, the next prime. Then cross out all of the multiples of 3; in other words, every third number. Some, like 6, may have already been crossed out because they are multiples of 2.

• Circle the next open number, 5. Now cross out all of the multiples of 5, or every 5th number.


• Continue doing this until all the numbers through 100 have either been circled or crossed out. You have just circled all the prime numbers from 1 to 120!

Tower of Hanoi

The Tower of Hanoi or Towers of Hanoi is a mathematical game or puzzle. It consists of three pegs, and a number of disks of different sizes which can slide onto any peg. The puzzle starts with the disks neatly stacked in order of size on one peg, smallest at the top, thus making a conical shape.
The objective of the game is to move the entire stack to another peg, obeying the following rules:

Only one disk may be moved at a time.

Each move consists of taking the upper disk from one of the
pegs and sliding it onto another peg, on top of the other

disks that may already be present on that peg.

No disk may be placed on top of a smaller disk.

Remaining numbers

Place the remaining numbers from 4 to 10 in the seven divisions of the above figure so that the outer divisions total 30 and each geometric figure totals 30.

Aristotle said:

The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful. Metaphysica, 3-1078b.

Even function

The function f(x) is an even function defined on the interval [ -3, 3 ].

Give that f(x) = 1, x [ 0 , 1 ] and f(x) = 2x -1 dla x ( 1 , 3 ].

a) sketch the graph of f(x) for x [ -3, 3 ].

b) find the value of x for which f(x) = 2

The multiplication symbol

In 1631, the multiplication symbol × was introduced by the English mathematician William Oughtred ( 1574-1660 ) in his book Keys to Mathematics, published in London. Incidentally, this Anglican minister is also famous for having invented the slide rule, which was used by generations of scientists and mathematicians. The slide rule’s doom in the mid-1970s, due to the pervasive influx of inexpensive pocket calculators, was rapid and unexpected.

Graph

Consider tha function:

• Plot the graph of the function.
• Write down the domain and the range.
• How many roots have this function ?
• Calculate f(-1) - f(2) .

KEIZO USHIO

Keizo Ushio is a Japanese sculptor who works with geometrical and topological figures, the Möbius Band and the Torus being the most remarkable of his work. By using mathematical calculations, Ushio fragments these shapes causing inusual figures with which he wants people to reflect on eternity and the passage of time.

How Old Is My Daughter?

My daughter is twice as old as my son and half as old as I am. In twenty-two years my son will be half my age. How old is my daughter?