THE MOTION OF BODIES IN THE UNIVERSE TAKES THE FORM OF THE CONIC SECTIONS- THERE ARE FOUR THE FOURTH THE CIRCLE IS DIFFERENT- IT TALKS ABOUT HOW APOLLONIUS CALLED IT THE FOURTH TYPE- EUCLID WROTE FOUR LOST BOOKS ON THE CONIC SECTIONS- ALSO FOUR POINTS

In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse. The circle is a special case of the ellipse, and is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, when Apollonius of Perga undertook a systematic study of their properties.

There are three types of conics, the ellipse, parabola, and hyperbola. The circle is a special kind of ellipse, although historically it had been considered as a fourth type (as it was by Apollonius). The circle and the ellipse arise when the intersection of the cone and plane is a closed curve. The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone – for a right cone, see diagram, this means that the cutting plane is perpendicular to the symmetry axis of the cone. If the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola. In this case, the plane will intersect both halves of the cone, producing two separate unbounded curves.

Euclid (fl. 300 BCE) is said to have written four books on conics but these were lost as well.

A (non-degenerate) conic is completely determined by five points in general position (no three collinear) in a plane and the system of conics which pass through a fixed set of four points (again in a plane and no three collinear) is called a pencil of conics. The four common points are called the base points of the pencil. Through any point other than a base point, there passes a single conic of the pencil. This concept generalizes a pencil of circles.

In a projective plane defined over an algebraically closed field any two conics meet in four points (counted with multiplicity) and so, determine the pencil of conics based on these four points. Furthermore, the four base points determine three line pairs (degenerate conics through the base points, each line of the pair containing exactly two base points) and so each pencil of conics will contain at most three degenerate conics.

FOUR POINTS PROPERTY OF A PARABOLA

4-points-property of a parabola

4-points-property of a parabola

Any parabola can be described in a suitable coordinate system by an equation

y

=

a

x

2

y=ax^{2}.

Let

P

1

=

(

x

1

,

y

1

)

,

P

2

=

(

x

2

,

y

2

)

,

P

3

=

(

x

3

,

y

3

)

,

P

4

=

(

x

4

,

y

4

)

{\displaystyle P_{1}=(x_{1},y_{1}),\,P_{2}=(x_{2},y_{2}),\,P_{3}=(x_{3},y_{3}),\,P_{4}=(x_{4},y_{4})} be four points of the parabola

y

=

a

x

2

y=ax^{2} and

Q

2

Q_{2} the intersection of the secant line

P

1

P

4

{\displaystyle P_{1}P_{4}} with the line

x

=

x

2

{\displaystyle x=x_{2}} and let be

Q

1

Q_{1} the intersection of the secant line

P

2

P

3

{\displaystyle P_{2}P_{3}} with the line

x

=

x

1

x=x_{1} (s. picture), then the secant line

P

3

P

4

{\displaystyle P_{3}P_{4}} is parallel to line

Q

1

Q

2

{\displaystyle Q_{1}Q_{2}}.

(The lines

x

=

x

1

x=x_{1} and

x

=

x

2

{\displaystyle x=x_{2}} are parallel to the axis of the parabola.)

Proof: straight forward calculation for the unit parabola

y

=

x

2

y=x^{2}.

Application: The 4-points-property of a parabola can be used for the construction of point

P

4

P_{4}, while

P

1

,

P

2

,

P

3

{\displaystyle P_{1},P_{2},P_{3}} and

Q

2

Q_{2} are given.

Remark: the 4-points-property of a parabola is an affine version of the 5-point-degeneration of Pascal's theorem.

THERE ARE FOUR UNARY OPERATORS AND 256 TERNARY OPERATORS----- 256 IS FOUR TO THE FOURTH POWER

A concrete function may be also referred to as an operator. In two-valued logic there are 2 nullary operators (constants), 4 unary operators, 16 binary operators, 256 ternary operators.

THERE ARE FOUR UNARY OPERATORS AND 256 TERNARY OPERATORS----- 256 IS FOUR TO THE FOURTH POWER- 16 BINARY OPERATORS- 16 IS FOUR SQUARED

A concrete function may be also referred to as an operator. In two-valued logic there are 2 nullary operators (constants), 4 unary operators, 16 binary operators, 256 ternary operators.

16 BOOLEAN FUNCTIONS- 16 SQUARES QMR

In two-valued logic, there are sixteen possible truth functions, also called Boolean functions, of two inputs P and Q. Any of these functions corresponds to a truth table of a certain logical connective in classical logic, including several degenerate cases such as a function not depending on one or both of its arguments. Truth and falsehood is denoted as 1 and 0 in the following truth tables, respectively, for sake of brevity.

16 POSSIBLE TRUTH FUNCTIONS- 16 ROWS 16 SQUARES QMR

Binary operations

There are 16 possible truth functions of two binary variables:

Truth table for all binary logical operators

Here is an extended truth table giving definitions of all 16 of the possible truth functions of two binary variables (P and Q are thus boolean variables: information about notation may be found in Bocheński (1959), Enderton (2001), and Quine (1982); for details about the operators see the Key below):

P Q F0 NOR1 Xq2 ¬p3 ↛4 ¬q5 XOR6 NAND7 AND8 XNOR9 q10 if/then11 p12 then/if13 OR14 T15

T T F F F F F F F F T T T T T T T T

T F F F F F T T T T F F F F T T T T

F T F F T T F F T T F F T T F F T T

F F F T F T F T F T F T F T F T F T

Com ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

L id F F T T T,F T F

R id F F T T T,F T F

where T = true and F = false. The Com row indicates whether an operator, op, is commutative - P op Q = Q op P. The L id row shows the operator's left identities if it has any - values I such that I op Q = Q. The R id row shows the operator's right identities if it has any - values I such that P op I = P.[note 1]

The four combinations of input values for p, q, are read by row from the table above. The output function for each p, q combination, can be read, by row, from the table.

Key:

The following table is oriented by column, rather than by row. There are four columns rather than four rows, to display the four combinations of p, q, as input.

p: T T F F

q: T F T F

There are 16 rows in this key, one row for each binary function of the two binary variables, p, q. For example, in row 2 of this Key, the value of Converse nonimplication ('

↚\nleftarrow ') is solely T, for the column denoted by the unique combination p=F, q=T; while in row 2, the value of that '

↚\nleftarrow ' operation is F for the three remaining columns of p, q. The output row for

↚\nleftarrow is thus

2: F F T F

and the 16-row key is

 operator Operation name

0 (F F F F)(p, q) ⊥ false, Opq Contradiction

1 (F F F T)(p, q) NOR p ↓ q, Xpq Logical NOR

2 (F F T F)(p, q) ↚ p ↚ q, Mpq Converse nonimplication

3 (F F T T)(p, q) ¬p, ~p ¬p, Np, Fpq Negation

4 (F T F F)(p, q) ↛ p ↛ q, Lpq Material nonimplication

5 (F T F T)(p, q) ¬q, ~q ¬q, Nq, Gpq Negation

6 (F T T F)(p, q) XOR p ⊕ q, Jpq Exclusive disjunction

7 (F T T T)(p, q) NAND p ↑ q, Dpq Logical NAND

8 (T F F F)(p, q) AND p ∧ q, Kpq Logical conjunction

9 (T F F T)(p, q) XNOR p If and only if q, Epq Logical biconditional

10 (T F T F)(p, q) q q, Hpq Projection function

11 (T F T T)(p, q) p → q if p then q, Cpq Material implication

12 (T T F F)(p, q) p p, Ipq Projection function

13 (T T F T)(p, q) p ← q p if q, Bpq Converse implication

14 (T T T F)(p, q) OR p ∨ q, Apq Logical disjunction

15 (T T T T)(p, q) ⊤ true, Vpq Tautology

FOUR OPERATORS OF PRIMITIVE RECURSIVE FUNCTIONS

In computer science and recursion theory the McCarthy Formalism (1963) of computer scientist John McCarthy clarifies the notion of recursive functions by use of the IF-THEN-ELSE construction common to computer science, together with four of the operators of primitive recursive functions: zero, successor, equality of numbers and composition. The conditional operator replaces both primitive recursion and the mu-operator.

FOUR TO SIXTEEN- 16 SQUARES QMR

A single address decoder with n address input bits can serve up to 2n devices. Several members of the 7400 series of integrated circuits can be used as address decoders. For example, when used as an address decoder, the 74154 provides four address inputs and sixteen (i.e., 24) device selector outputs. An address decoder is a particular use of a binary decoder circuit known as a "demultiplexer" or "demux" (the 74154 is commonly called a "4-to-16 demultiplexer"), which has many other uses besides address decoding.

List of ICs which provide multiplexing

The 7400 series has several ICs that contain multiplexer(s):

S.No. IC No. Function Output State

1 74157 Quad 2:1 mux. Output same as input given

2 74158 Quad 2:1 mux. Output is inverted input

0 74153 Dual 4:1 mux. Output same as input

5 74352 Dual 4:1 mux. Output is inverted input

9 74151A 8:1 mux. Both outputs available (i.e., complementary outputs)

6 74151 8:1 mux. Output is inverted input

7 74150 16:1 mux. Output is inverted input

List of ICs which provide demultiplexing

Fairchild 74F138

The 7400 series has several ICs that contain demultiplexer(s):

S.No. IC No. (7400) IC No. (4000) Function Output State

1 74139 Dual 1:4 demux. Output is inverted input

3 74156 Dual 1:4 demux. Output is open collector

4 74138 1:8 demux. Output is inverted input

5 74238 1:8 demux.

6 74154 1:16 demux. Output is inverted input

7 74159 CD4514/15 1:16 demux. Output is open collector and same as input

The 7400 chip, containing four NANDs. The second line of numbers (7645) is a date code; this chip was manufactured in the 45th week of 1976. The N suffix on the part number is a vendor-specific code indicating PDIP packaging.

Four-Phase Systems

(Redirected from Four-Phase Systems AL1)

Four-Phase Systems, Inc.

Industry Semiconductor

Fate Acquired by Motorola

Founded 1969

Founder Lee Boysel

Defunct 1981

Products Semiconductor main memory, LSI MOS logic, central processing unit, microprocessor

Owner Motorola

Four-Phase Systems was a computer company, founded by Lee Boysel and others, which built one of the earliest computers using semiconductor main memory and LSI MOS logic. The company was incorporated in February 1969 and had moderate commercial success. It was acquired by Motorola in 1981.

16 ARITHMETIC AND 16 LOGICAL OPERATIONS
https://en.wikipedia.org/wiki/74181
The 4-bit wide ALU can perform all the traditional add / subtract / decrement operations with or without carry, as well as AND / NAND, OR / NOR, XOR, and shift. Many variations of these basic functions are available, for a total of 16 arithmetic and 16 logical operations on two four-bit words. Multiply and divide functions are not provided but can be performed in multiple steps using the shift and add or subtract functions. Shift is not an explicit function but can be derived from several available functions including (A+B) plus A, A plus AB[clarification needed].

16 SQUARES QMR

In mathematics and computing, hexadecimal (also base 16, or hex) is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F (or alternatively a, b, c, d, e, f) to represent values ten to fifteen.

Use in Chinese culture

The traditional Chinese units of weight were base-16. For example, one jīn (斤) in the old system equals sixteen taels. The suanpan (Chinese abacus) could be used to perform hexadecimal calculations.

Primary numeral system

As with the duodecimal system, there have been occasional attempts to promote hexadecimal as the preferred numeral system. These attempts often propose specific pronunciation and symbols for the individual numerals. Some proposals unify standard measures so that they are multiples of 16.

An example of unified standard measures is hexadecimal time, which subdivides a day by 16 so that there are 16 "hexhours" in a day.

Base16 or hex (not to be confused with Intel HEX and the like) is one of the simplest binary-to-text encodings, which stores each byte as a pair of hexadecimal digits. Many variations of such format are possible, for example either uppercase (A-F) or lowercase (a-f) letters may be used for digits greater than 9; spaces, line breaks or other separators may be added between digit groups of different lengths; header and/or footer with metainformation may be added.

16s AND 32s (TWO 16s) QUADRANT NUMBERS

Qf: The "Q" prefix. For example, Q15 represents a number with 15 fractional bits. This notation is ambiguous since it does not specify the word length, however it is usually assumed that the word length is either 16 or 32 bits depending on the target processor in use.

Qm.f: The unambiguous form of the "Q" notation. Since the entire word is a 2's complement integer, a sign bit is implied. For example, Q1.30 describes a number with 1 integer bit and 30 fractional bits stored as a 32-bit 2's complement integer.

fxm.b: The "fx" prefix is similar to the above, but uses the word length as the second item in the dotted pair. For example, fx1.16 describes a number with 1 magnitude bit and 15 fractional bits in a 16 bit word.

s:m:f: Yet other notations include a sign bit, such as this one used in the PS2 GS User's Guide. It also differs from conventional usage by using a colon instead of a period as the separator. For example, in this notation, 0:8:0 represents an unsigned 8-bit integer.

libfixmath is a recent open-source library for the manipulation of fixed point numbers, it currently supports Q16.16 and Q0.32 formats and provides an interface similar to math.h.

Doom was the last first-person shooter title by id Software to use a 16.16 fixed point representation for all of its non-integer computations, including map system, geometry, rendering, player movement etc. This was done in order for the game to be playable on 386 and 486SX CPUs without an FPU. For compatibility reasons, this representation is still used in modern Doom source ports.

https://en.wikipedia.org/wiki/Data_Encryption_Standard
Key mixing: the result is combined with a subkey using an XOR operation. Sixteen 48-bit subkeys—one for each round—are derived from the main key using the key schedule (described below).

16 ROUNDS 16 SQUARES QMR

Differential cryptanalysis was rediscovered in the late 1980s by Eli Biham and Adi Shamir; it was known earlier to both IBM and the NSA and kept secret. To break the full 16 rounds, differential cryptanalysis requires 247 chosen plaintexts. DES was designed to be resistant to DC.

There have also been attacks proposed against reduced-round versions of the cipher, that is, versions of DES with fewer than 16 rounds. Such analysis gives an insight into how many rounds are needed for safety, and how much of a "security margin" the full version retains. Differential-linear cryptanalysis was proposed by Langford and Hellman in 1994, and combines differential and linear cryptanalysis into a single attack. An enhanced version of the attack can break 9-round DES with 215.8 chosen plaintexts and has a 229.2 time complexity (Biham and others, 2002).

16 SQUARES QMR- THERE ARE 16 STAGES/ROUNDS

The algorithm's overall structure is shown in Figure 1: there are 16 identical stages of processing, termed rounds. There is also an initial and final permutation, termed IP and FP, which are inverses (IP "undoes" the action of FP, and vice versa). IP and FP have no cryptographic significance, but were included in order to facilitate loading blocks in and out of mid-1970s 8-bit based hardware.

Before the main rounds, the block is divided into two 32-bit halves and processed alternately; this criss-crossing is known as the Feistel scheme. The Feistel structure ensures that decryption and encryption are very similar processes—the only difference is that the subkeys are applied in the reverse order when decrypting. The rest of the algorithm is identical. This greatly simplifies implementation, particularly in hardware, as there is no need for separate encryption and decryption algorithms.

A comparison between single data rate, double data rate, and quad data rate.

In QDR, the data lines operate at twice the frequency of the clock signal.

Quad data rate (QDR, or quad pumping) is a communication signaling technique wherein data are transmitted at four points in the clock cycle: on the rising and falling edges, and at two intermediate points between them. The intermediate points are defined by a second clock that is 90° out of phase from the first. The effect is to deliver four bits of data per signal line per clock cycle.

The first Pentium 4 cores, codenamed Willamette, were clocked from 1.3 GHz to 2 GHz. They were released on November 20, 2000, using the Socket 423 system. Notable with the introduction of the Pentium 4 was the 400 MT/s FSB. It actually operated at 100 MHz, but the FSB was quad-pumped, meaning that the maximal transfer rate was four times the base clock of the bus, so it was marketed to run at 400 MHz. The AMD Athlon's double-pumped FSB was running at 100 or 133 MHz (200 or 266 MT/s) at that time.

FOUR PROOFS

What is the number Tn of different trees that can be formed from a set of n distinct vertices? Cayley's formula gives the answer Tn = nn − 2. Aigner & Ziegler (1998) list four proofs of this fact; they write of the fourth, a double counting proof due to Jim Pitman, that it is “the most beautiful of them all.”

Just now

TETRA IS FOUR
Quadray coordinates, also known as tetray coordinates or Chakovian coordinates, were Invented by Darrel Jarmusch and further developed by David Chako, Tom Ace, Kirby Urner, et al., as another take on simplicial coordinates, a coordinate system using a simplex or tetrahedron as its basis polyhedron.

Contents [hide]
1 Geometric definition
2 Pedagogical significance
4 References
Geometric definition
The four basis vectors stem from the center of a regular tetrahedron and go to its four corners. Their coordinate addresses are (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0) and (0, 0, 0, 1) respectively. These may be scaled and linearly combined to span conventional XYZ space, with at least one of the four coordinates unneeded (set to zero) in any given quadrant.

THE FOURTH IS ALWAYS TRANSCENDENT/DIFFERENT

The fallacy of four terms (Latin: quaternio terminorum) is the formal fallacy that occurs when a syllogism has four (or more) terms rather than the requisite three. This form of argument is thus invalid.

The fallacy of four terms is a syllogistic fallacy. Types of syllogism to which it applies include statistical syllogism, hypothetical syllogism, and categorical syllogism, all of which must have exactly three terms. Because it applies to the argument's form, as opposed to the argument's content, it is classified as a formal fallacy.

Equivocation of the middle term is a frequently cited source of a fourth term being added to a syllogism; both of the equivocation examples above affect the middle term of the syllogism. Consequently this common error itself has been given its own name: the fallacy of the ambiguous middle. An argument that commits the ambiguous middle fallacy blurs the line between formal and informal (material) fallacies, however it is usually considered an informal fallacy because the argument's form appears valid.

DEGREES HIGHER THAN FOUR CANNOT BE SOLVED BY RADICALS- IT TOOK EXTREMELY LONG FOR QUARTIC EQUATION TO BE SOLVED--- THE FIRST THREE WERE SOLVED RELATIVELY EASY- FOUR WAS EXTREMELY HARD FOUR IS ALWAYS DIFFERENT- QUINTIC FIFTH IS IMPOSSIBLE FIFTH IS ULTRA TRANSCENDENT BUT AT LEAST FIFTH IS DIFFERENT THAN THE OTHERS ABOVE IT BASED ON PROOF

There may be several meanings of "solving an equation". One may want to express the solutions as explicit numbers; for example, the unique solution of 2x – 1 = 0 is 2/3. Unfortunately, this is, in general, impossible for equations of degree greater than one, and, since the ancient times, mathematicians have searched to express the solutions as algebraic expression; for example the golden ratio

(

1

+

5

)

/

2

(1+\sqrt 5)/2 is the unique positive solution ofx^{2}-x-1=0. In the ancient times, they succeeded only for degrees one and two. For quadratic equations, the quadratic formula provides such expressions of the solutions. Since the 16th century, similar formulas (using cube roots in addition to square roots), but much more complicated are known for equations of degree three and four (see cubic equation and quartic equation). But formulas for degree 5 and higher eluded researchers for several centuries. In 1824, Niels Henrik Abel proved the striking result that there are equations of degree 5 whose solutions cannot be expressed by a (finite) formula, involving only arithmetic operations and radicals (see Abel–Ruffini theorem). In 1830, Évariste Galois proved that most equations of degree higher than four cannot be solved by radicals, and showed that for each equation, one may decide whether it is solvable by radicals, and, if it is, solve it. This result marked the start of Galois theory and group theory, two important branches of modern algebra. Galois himself noted that the computations implied by his method were impracticable. Nevertheless, formulas for solvable equations of degrees 5 and 6 have been published (see quintic function and sextic equation).

DEGREE FOUR IS THE HIGHEST AND IT IS DIFFERENT THAN THE FIRST THREE IT TOOK EXTREMLY LONG TO DISCOVER

The degree four (quartic case) is the highest degree such that every polynomial equation can be solved by radicals.

UNSOLVABILITY OF EQUATIONS BEYOND FOUR (AND FOUR IS DIFFERENT TRANSCENDENT)

–Ruffini_theorem

The theorem was first nearly proved by Paolo Ruffini in 1799. He sent his proof to several mathematicians to get it acknowledged, amongst them Lagrange (who did not reply) and Augustin-Louis Cauchy, who sent him a letter saying: “Your memoir on the general solution of equations is a work which I have always believed should be kept in mind by mathematicians and which, in my opinion, proves conclusively the algebraic unsolvability of general equations of higher than fourth degree.”[

I DISCUSSED HOW FOIL METHOD FIRST OUTER INNER LAST IS FOUR PARTS- I ALSO DISCUSSED HOW THE GEOMETRIC PROOFS FOR CALCULUS AND ALGEBRA ARE LITERALLY FOUR PART QUADRANT MODELS FOURTH PART DIFFERENT

While grouping may not lead to a factorization in general, if the polynomial expression to be factored consists of four terms and is the result of multiplying two binomial expressions (by the FOIL method for instance), then the grouping technique can lead to a factorization, as in the above example.

CANNOT BE DONE BEYOND FOUR

Some but not all polynomial equations with rational coefficients have a solution that is an algebraic expression that can be found using a finite number of operations that involve only those same types of coefficients (that is, can be solved algebraically). This can be done for all such equations of degree one, two, three, or four; but for degree five or more it can only be done for some equations, not for all. A large amount of research has been devoted to compute efficiently accurate approximations of the real or complex solutions of a univariate algebraic equation (see Root-finding algorithm) and of the common solutions of several multivariate polynomial equations (see System of polynomial equations).

FOUR FUNDAMENTAL SUBSPACES

n columns) induces four fundamental subspaces. These fundamental subspaces are:

name of subspace definition containing space dimension basis

column space, range or image

im

(

A

)

\operatorname {im} (A) or

range

(

A

)

{\displaystyle \operatorname {range} (A)}

R

m

\mathbb {R} ^{m}

r

r (rank) The first

r

r columns of

U

U

nullspace or kernel

ker

(

A

)

{\displaystyle \ker(A)} or

null

(

A

)

{\displaystyle \operatorname {null} (A)}

R

n

\mathbb {R} ^{n}

n

r

n-r (nullity) The last

(

n

r

)

(n-r) columns of

V

V

row space or coimage

im

(

A

T

)

{\displaystyle \operatorname {im} (A^{\mathrm {T} })} or

range

(

A

T

)

{\displaystyle \operatorname {range} (A^{\mathrm {T} })}

R

n

\mathbb {R} ^{n}

r

r (rank) The first

r

r columns of

V

V

left nullspace or cokernel

ker

(

A

T

)

{\displaystyle \ker(A^{\mathrm {T} })} or

null

(

A

T

)

{\displaystyle \operatorname {null} (A^{\mathrm {T} })}

R

m

\mathbb {R} ^{m}

m

r

m-r (corank) The last

(

m

r

)

(m-r) columns of

EXTREMELY IMPORTANT IN STATISTICS ONE OF THE FIRST THINGS YOU LEARN IN FIRST YEAR OF STATISTICS CLASS

DIVIDE INTO FOUR EQUAL PARTS- QUAR IS FOUR

The interquartile range (IQR) is a measure of variability, based on dividing a data set into quartiles. Quartiles divide a rank-ordered data set into four equal parts. The values that separate parts are called the first, second, and third quartiles; and they are denoted by Q1, Q2, and Q3, respectively.

HYPOTHESIS TESTING AND ERROR TYPES ARE PRETTY MUCH THE ONLY THING TAUGHT IN FIRST YEAR OF STATISTICS- IT IS THE QUADRANT MODEL

Table of error types

Table of error types

Tabularised relations between truth/falseness of the null hypothesis and outcomes of the test:

Table of error types Null hypothesis (H0) is

True False

Decision About Null Hypothesis (H0) Reject Type I error

(False Positive) Correct inference

(True Positive)

Fail to reject Correct inference

(True Negative) Type II error

(False Negative)

THE "FOURFOLD SCHEME OF PROPOSITIONS"- THIS IS ALL LOGIC- THIS IS THE BASIS OF ALL LOGIC IT IS THE QUADRANT MODEL

A proposition may be universal or particular, and it may be affirmative or negative. Traditionally, the four kinds of propositions are:

A-type: Universal and affirmative ("Every philosopher is mortal")

I-type: Particular and affirmative ("Some philosophers are mortal")

E-type: Universal and negative ("Every philosopher is not immortal")

O-type: Particular and negative ("Some philosophers are not immortal")

This was called the fourfold scheme of propositions (see types of syllogism for an explanation of the letters A, I, E, and O in the traditional square). Aristotle's original square of opposition, however, does not lack existential import.

SYLLOGISTIC LOGIC FOUR TERMS

The essential feature of the syllogistic is that, of the four terms in the two premises, one must occur twice. Thus

All Greeks are men

All men are mortal.

LACAN FOUR FORMULAE OF SEXUATION- LACANS FOUR PRINCIPALS UNDERLIED HIS BOOK WAS EVEN THE TITLE

A more complex set of mathemes are Lacan's 'formulae of sexuation,' which he popularized in the March 13, 1973 session of his Seminar XX. Composed of two pairs of propositions written in a unique logico-mathematical shorthand inspired by Gottlob Frege, one pair was dubbed 'masculine' and the other 'feminine.' These formulae Lacan originally constructed from Aristotelian logic over the course of an entire year of study.

The Aristotelian logical system had a formidable influence on the late-philosophy of the French psychoanalyst Jacques Lacan. In the early 1970s, Lacan reworked Aristotle's term logic by way of Frege and Jacques Brunschwig to produce his four formulae of sexuation. While these formulae retain the formal arrangement of the square of opposition, they seek to undermine the universals of both qualities by the 'existence without essence' of Lacan's particular negative proposition.

FAMOUS PUZZLE- FOUR LINES FOURTH DIFFERENT

The puzzle proposed an intellectual challenge—to connect the dots by drawing four straight, continuous lines that pass through each of the nine dots, and never lifting the pencil from the paper. The conundrum is easily resolved, but only by drawing the lines outside the confines of the square area defined by the nine dots themselves. The phrase "thinking outside the box" is a restatement of the solution strategy. The puzzle only seems difficult because people commonly imagine a boundary around the edge of the dot array. The heart of the matter is the unspecified barrier that people typically perceive.

EVERY FINITE SIMPLE GROUP BELONGS TO ONE OF FOUR CLASSES

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four broad classes described below

FOUR GROUP AXIOMS

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, applies much more widely. It allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.

A group is a set, G, together with an operation • (called the group law of G) that combines any two elements a and b to form another element, denoted a • b or ab. To qualify as a group, the set and operation, (G, •), must satisfy four requirements known as the group axioms:

Closure

For all a, b in G, the result of the operation, a • b, is also in G.b[›]

Associativity

For all a, b and c in G, (a • b) • c = a • (b • c).

Identity element

There exists an element e in G such that, for every element a in G, the equation e • a = a • e = a holds. Such an element is unique (see below), and thus one speaks of the identity element.

Inverse element

For each a in G, there exists an element b in G, commonly denoted a−1 (or −a, if the operation is denoted "+"), such that a • b = b • a = e, where e is the identity element.

The result of an operation may depend on the order of the operands. In other words, the result of combining element a with element b need not yield the same result as combining element b with element a; the equation

THE EGYPTIANS ADMIRED THE 3:4:5 TRIANGLE- THEY SAW THE SIDE OF LENGTH FOUR AS ONE OF THE MAIN EGYPTIAN GODS----- THE FOUR SIDE IS 4 SQUARED WHICH IS 16---- THEY ACTUALLY CALLED THIS SIDE THEIR MAIN GOD THE 16 SQUARES- I POSTED AN ARTICLE ON THIS A WHILE AGO

The 3 : 4 : 5 triangles are the only right triangles with edges in arithmetic progression. Triangles based on Pythagorean triples are Heronian, meaning they have integer area as well as integer sides.

The possible use of the 3 : 4 : 5 triangle in Ancient Egypt, with the supposed use of a knotted rope to lay out such a triangle, and the question whether Pythagoras theorem was known at that time, has been much debated. It was first conjectured by the historian Moritz Cantor in 1882. It is known that right angles were laid out accurately in Ancient Egypt; that their surveyors did use ropes for measurement; that Plutarch recorded in Isis and Osiris (around 100 AD) that the Egyptians admired the 3 : 4 : 5 triangle; and that the Berlin Papyrus 6619 from the Middle Kingdom of Egypt (before 1700 BC) stated that "the area of a square of 100 is equal to that of two smaller squares. The side of one is ½ + ¼ the side of the other." The historian of mathematics Roger L. Cooke observes that "It is hard to imagine anyone being interested in such conditions without knowing the Pythagorean theorem." Against this, Cooke notes that no Egyptian text before 300 BC actually mentions the use of the theorem to find the length of a triangle's sides, and that there are simpler ways to construct a right angle. Cooke concludes that Cantor's conjecture remains uncertain: he guesses that the Ancient Egyptians probably did know the Pythagorean theorem, but that "there is no evidence that they used it to construct right angles".

THE TRANSCENDENT PYTHAGOREAN QUADRUPLE BEYOND THE PYTHAGOREAN TRIPLE

SO ON THE LAST POST I DISCUSSED HOW THE EGYPTIANS SAW THE FOUR SQUARED IN THE FIRST PYTHAGOREAN TRIPPLE AS THEIR MAIN GOD (THEY SAW THE 16 SQUARES AS THEIR MAIN GOD)- BUT THERE IS NOT JUST PYTHAGOREAN TRIPPLES THERE IS THE TRANSCENDENT PYTHAGOREAN QUADRUPLES- I WATCHED A STRING THEORY THING ON HOW THE PYTHAGOREAN THEOREM GOES TO THE FOURTH DIMENSION TRANSCENDENT AND EXTRA SQUARE AND ALL THAT AND IT WAS THE QUADRANT MODEL BUT I FORGET HOW TO EXPLAIN IT NOW

A set of four positive integers a, b, c and d such that a2 + b2+ c2 = d2 is called a Pythagorean quadruple. The simplest example is (1, 2, 2, 3), since 12 + 22 + 22 = 32. The next simplest (primitive) example is (2, 3, 6, 7), since 22 + 32 + 62 = 72.

All quadruples are given by the formula

.

(m^{2}+n^{2}-p^{2}-q^{2})^{2}+(2mq+2np)^{2}+(2nq-2mp)^{2}=(m^{2}+n^{2}+p^{2}+q^{2})^{2}.

THE TRANSCENDENT PYTHAGOREAN QUADRUPLE BEYOND THE PYTHAGOREAN TRIPPLE

A Pythagorean quadruple is a tuple of integers a, b, c and d, such that a2 + b2 + c2 = d2. They are solutions of a Diophantine equation and often only positive integer values are considered. However, to provide a more complete geometric interpretation, the integer values can be allowed to be negative and zero (thus allowing Pythagorean triples to be included) with the only condition being that d > 0. In this setting, a Pythagorean quadruple (a, b, c, d) defines a cuboid with integer side lengths | a |, | b |, and | c |, whose space diagonal has integer length d. Pythagorean quadruples, with this interpretation are thus also called Pythagorean boxes. In this article we will assume, unless otherwise stated, that the values of a Pythagorean quadruple are all positive integers.

THE GREAT QUATENARY- SUM IS 36

It is the number of the Great Quaternary of the pythagoricians, obtained by the sum of the first four odd numbers, 1 + 3 + 5 + 7 = 16, and the first four even numbers, 2 + 4 + 6 + 8 = 20. Also: 36 = 13 + 23 + 33.

FOUR METHODS- 128 IS 16 TIMES 8

In mathematics, negative numbers in any base are represented by prefixing them with a minus ("−") sign. However, in computer hardware, numbers are represented only as sequences of bits, without extra symbols. The four best-known methods of extending the binary numeral system to represent signed numbers are: sign-and-magnitude, ones' complement, two's complement, and excess-K. Some of the alternative methods use implicit instead of explicit signs, such as negative binary, using the base −2. Corresponding methods can be devised for other bases, whether positive, negative, fractional, or other elaborations on such themes.

Base −2

In conventional binary number systems, the base, or radix, is 2; thus the rightmost bit represents 20, the next bit represents 21, the next bit 22, and so on. However, a binary number system with base −2 is also possible. The rightmost bit represents (−2)0 = +1, the next bit represents (−2)1 = −2, the next bit (−2)2 = +4 and so on, with alternating sign. The numbers that can be represented with four bits are shown in the comparison table below.

The range of numbers that can be represented is asymmetric. If the word has an even number of bits, the magnitude of the largest negative number that can be represented is twice as large as the largest positive number that can be represented, and vice versa if the word has an odd number of bits.

Comparison table

The following table shows the positive and negative integers that can be represented using four bits.

Four-bit integer representations

Decimal Unsigned Sign and magnitude Ones' complement Two's complement Excess-8 (biased) Base −2

+16 N/A N/A N/A N/A N/A N/A

+15 1111 N/A N/A N/A N/A N/A

+14 1110 N/A N/A N/A N/A N/A

+13 1101 N/A N/A N/A N/A N/A

+12 1100 N/A N/A N/A N/A N/A

+11 1011 N/A N/A N/A N/A N/A

+10 1010 N/A N/A N/A N/A N/A

+9 1001 N/A N/A N/A N/A N/A

+8 1000 N/A N/A N/A N/A N/A

+7 0111 0111 0111 0111 1111 N/A

+6 0110 0110 0110 0110 1110 N/A

+5 0101 0101 0101 0101 1101 0101

+4 0100 0100 0100 0100 1100 0100

+3 0011 0011 0011 0011 1011 0111

+2 0010 0010 0010 0010 1010 0110

+1 0001 0001 0001 0001 1001 0001

+0 0000 0000 0000 0000 1000 0000

−0 1000 1111

−1 N/A 1001 1110 1111 0111 0011

−2 N/A 1010 1101 1110 0110 0010

−3 N/A 1011 1100 1101 0101 1101

−4 N/A 1100 1011 1100 0100 1100

−5 N/A 1101 1010 1011 0011 1111

−6 N/A 1110 1001 1010 0010 1110

−7 N/A 1111 1000 1001 0001 1001

−8 N/A N/A N/A 1000 0000 1000

−9 N/A N/A N/A N/A N/A 1011

−10 N/A N/A N/A N/A N/A 1010

−11 N/A N/A N/A N/A N/A N/A

16 ROWS 16 SQUARES QMR

Ludwig Wittgenstein introduced a version of the 16-row truth table as proposition 5.101 of Tractatus Logico-Philosophicus (1921)

16 SQUARES QUADRANT MODEL--- 256 IS FOUR TO THE FOURTH POWER- THE 16 PRINCIPAL ODU OF THE IFA

Ifá divination 

Sixteen Principal Odú

Ogbe   I I I I   Ogunda   I I I II

Oyẹku   II II II II   Ọsa   II I I I

Iwori   II I I II   Ika   II I II II

Odi   I II II I   Oturupọn   II II I II

Irosun   I I II II   Otura   I II I I

Iwọnrin   II II I I   Irẹtẹ   I I II I

Ọbara   I II II II   Ọsẹ   I II I II

Ọkanran   II II II I   Ofun   II I II I

Ifá is the ancient system of divination and literary corpus of the Yoruba people of Nigeria. In Yoruba religion, the rite provides a means of communication with spiritual divinity. The Orisa Ifá or Orunmila ("Grand Priest") permits access to an initiated priest, a Babalawo ("father of the secrets") who generates binary values using sacred palm nuts. In wood powder, these are recorded as single and double lines. There are 16 principal Odú that are said to compose the 256 Odú. From memory alone, a Babalawo must be able to recite four to ten verses for each of the 256 Odú Ifá: generally, orisa lore, traditional medicine, and ritual advice. In 2005, UNESCO listed Ifá in the Masterpieces of the Oral and Intangible Heritage of Humanity.

AND gates are available in IC packages. 7408 IC is a famous QUAD 2-Input AND GATES and contains four independent gates each of which performs the logic AND function.

FOUR GATES

16 SQUARES QMR

A 16-bit Fibonacci LFSR. The feedback tap numbers shown correspond to a primitive polynomial in the table, so the register cycles through the maximum number of 65535 states excluding the all-zeroes state. The state shown, 0xACE1 (hexadecimal) will be followed by 0x5670.

64 IS FOUR 16S

The most commonly used version of the Mersenne Twister algorithm is based on the Mersenne prime 219937−1. The standard implementation of that, MT19937, uses a 32-bit word length. There is another implementation that uses a 64-bit word length, MT19937-64; it generates a different sequence.

FOUR FUDAMENTAL ARITHMETIC OPERATIONS

Leibniz's Stepped Reckoner was the first calculator that could perform all four arithmetic operations.

The basic arithmetic operations are addition, subtraction, multiplication and division, although this subject also includes more advanced operations, such as manipulations of percentages, square roots, exponentiation, and logarithmic functions. Arithmetic is performed according to an order of operations. Any set of objects upon which all four arithmetic operations (except division by 0) can be performed, and where these four operations obey the usual laws, is called a field.

Modern methods for four fundamental operations (addition, subtraction, multiplication and division) were first devised by Brahmagupta of India. This was known during medieval Europe as "Modus Indoram" or Method of the Indians. Positional notation (also known as "place-value notation") refers to the representation or encoding of numbers using the same symbol for the different orders of magnitude (e.g., the "ones place", "tens place", "hundreds place") and, with a radix point, using those same symbols to represent fractions (e.g., the "tenths place", "hundredths place"). For example, 507.36 denotes 5 hundreds (102), plus 0 tens (101), plus 7 units (100), plus 3 tenths (10−1) plus 6 hundredths (10−2).

PEIRCE IS ONE OF THE MOST FAMOUS LOGICIANS OF ALL TIME- HE CONTINUOUSLY REPEATED FOUR

Against Cartesianism

Peirce drew on the methodological implications of the four incapacities — no genuine introspection, no intuition in the sense of non-inferential cognition, no thought but in signs, and no conception of the absolutely incognizable — to attack philosophical Cartesianism, of which he said that:

1. "It teaches that philosophy must begin in universal doubt" — when, instead, we start with preconceptions, "prejudices [...] which it does not occur to us can be questioned", though we may find reason to question them later. "Let us not pretend to doubt in philosophy what we do not doubt in our hearts."

2. "It teaches that the ultimate test of certainty is...in the individual consciousness" — when, instead, in science a theory stays on probation till agreement is reached, then it has no actual doubters left. No lone individual can reasonably hope to fulfill philosophy's multi-generational dream. When "candid and disciplined minds" continue to disagree on a theoretical issue, even the theory's author should feel doubts about it.

3. It trusts to "a single thread of inference depending often upon inconspicuous premisses" — when, instead, philosophy should, "like the successful sciences", proceed only from tangible, scrutinizable premisses and trust not to any one argument but instead to "the multitude and variety of its arguments" as forming, not a chain at least as weak as its weakest link, but "a cable whose fibers", soever "slender, are sufficiently numerous and intimately connected".

4. It renders many facts "absolutely inexplicable, unless to say that 'God makes them so' is to be regarded as an explanation" — when, instead, philosophy should avoid being "unidealistic", misbelieving that something real can defy or evade all possible ideas, and supposing, inevitably, "some absolutely inexplicable, unanalyzable ultimate", which explanatory surmise explains nothing and so is inadmissible.

FOUR METHODS

In "The Fixation of Belief", Peirce characterized inquiry in general not as the pursuit of truth per se but as the struggle to settle disturbances or conflicts of belief, irritating, inhibitory doubts, belief being that on which one is willing to act. That let Peirce frame scientific inquiry not only as a special kind of inquiry in a broader spectrum, but also, like inquiry generally, as based on actual doubts, not mere verbal doubts (such as hyperbolic doubt), which he held to be fruitless, and it let him also frame it, by the same stroke, as requiring that proof rest on propositions free from actual doubt, rather than on ultimate and absolutely indubitable propositions. He outlined four methods, ordered from least to most successful in achieving a secure fixation of belief:

The method of tenacity (policy of sticking to initial belief) — which brings comforts and decisiveness, but leads to trying to ignore contrary information and others' views, as if truth were intrinsically private, not public. The method goes against the social impulse and easily falters since one may well fail to avoid noticing when another's opinion is as good as one's own initial opinion. Its successes can be brilliant but tend to be transitory.

The method of authority — which overcomes disagreements but sometimes brutally. Its successes can be majestic and long-lasting, but it cannot regulate people thoroughly enough to suppress doubts indefinitely, especially when people learn about other societies present and past.

The method of the a priori — which promotes conformity less brutally but fosters opinions as something like tastes, arising in conversation and comparisons of perspectives in terms of "what is agreeable to reason." Thereby it depends on fashion in paradigms and goes in circles over time. It is more intellectual and respectable but, like the first two methods, sustains accidental and capricious beliefs, destining some minds to doubt it.

The method of science — the only one whereby inquiry can, by its own account, go wrong (fallibilism), and purposely tests itself and criticizes, corrects, and improves itself.

The following example demonstrates the issue. Consider the power set of a 4-element set ordered by inclusion

⊆\subseteq . Below are four different Hasse diagrams for this partial order. Each subset has a node labelled with a binary encoding that shows whether a certain element is in the subset (1) or not (0):

Hypercubeorder binary.svg Hypercubecubes binary.svg Hypercubestar binary.svg Hypercubematrix binary.svg

The first diagram makes clear that the power set is a graded poset. The second diagram has the same graded structure, but by making some edges longer than others, it emphasizes that the 4-dimensional cube is a union of two 3-dimensional cubes. The third diagram shows some of the internal symmetry of the structure. In the fourth diagram the vertices are arranged like the elements of a 4×4 matrix.

TETRA IS FOUR- INDIA AND TIBET 64 POSSIBILITIES- ROYAL GAME OF UR

A four-sided tetrahedral die resting on its "1" face

Four-sided die

A four-sided tetrahedral die resting on its "1" face

Four-sided dice, abbreviated d4, are often used in tabletop role-playing games to obtain random integers in the range 1–4. Two forms exist of this die: a tetrahedron (pyramid shape) with four equilateral triangle-shaped faces, and an elongated long die with four faces. The former type does not roll well and is thus usually thrown into the air or shaken in a box.

Historical

Four-sided dice were among the gambling and divination tools used by early man who carved them from nuts, wood, stone, ivory and bone. Six-sided dice were invented later but four-sided dice continued to be popular in Asia. In Ancient Rome, elongated four-sided dice were called tali while the six-sided cubic dice were tesserae. In India and Tibet, three four-sided long dice were rolled sequentially as an oracle, to produce 1 of 64 possible outcomes. The ancient Jewish dreidel is a four-sided long die with one end changed into a handle, to allow it to be spun like a top.

The ancient Mesopotamian Royal Game of Ur uses eight four-sided pyramid-shaped dice made out of rock, half of them colored white, and half black. The Scandinavian game daldøs uses a four-sided long die.

Modern gaming

Popular role-playing games involving four-sided tetrahedral dice include Dungeons & Dragons and Ironclaw. The d20 System includes a four-sided tetrahedral die among other dice with 6, 8, 10, 12 and 20 faces. Tetrahedral dice are peculiar in that there is no topmost face when a die comes to rest. There are several common ways of indicating the value rolled. On some tetrahedral dice, three numbers are shown on each face. The number rolled is indicated by the number shown upright at all three visible faces—either near the midpoints of the sides around the base or near the angles around the apex. Another configuration places only one number on each face, and the rolled number is taken from the downward face.

USES FOUR TETRAHEDRAL DIE- QUADRANT GRID

The Royal Game of Ur was played with two sets, one black and one white, of seven markers and four tetrahedral dice

THE FOUR TYPES OF FOURIER TRANSFORMS (STUFF IS VERY BIG ITS TAUGHT A LOT IN COLLEGE)

I don't think any of the Fourier transform articles should be merged. I think it is important to make it absolutely clear that there are four separate and distinct types of FT's:

Continuous-time Fourier Transform (CTFT)

Discrete-time Fourier Transform (DTFT)

Discrete Fourier Transform (DFT)

Fourier series

Obviously, there are relationships between all four of these types of FT, but each one is different from the others, and each serves its own purpose. -- Metacomet 06:02, 24 December 2005 (UTC)

THIS IS HUGE IN COLLEGE- I WENT TO A LOT OF DIFFERENT CLASSES AND THEY TALKED ABOUT FOURIER TRANSFORMS AND I GOT DVDS AND LISTENED TO THEM AND THEY TALKED ABOUT IT- FOUR TYPES

 Fourier Transforms Summary

The following table recaps the four basic forms discussed above, highlighting the duality of the properties of discreteness and periodicity. I.e., if the signal representation in one domain has either (or both) of those properties, then its transform representation to the other domain has the other property (or both).

Name Time domain Frequency domain Function's

Domain property Function property Domain property Function property Energy Average Power

(Continuous) Fourier transform (FT) Continuous Aperiodic Continuous Aperiodic Finite Infinitesimal

Discrete-time Fourier transform (DTFT) Discrete Aperiodic Continuous Periodic (ƒs) Finite Infinitesimal

Fourier Series (FS) Continuous Periodic (τ) Discrete Aperiodic Infinite Finite

Discrete Fourier Series (DFS) Discrete Periodic (N) Discrete Periodic (N) Infinite Finite

FOUR SETS- 16 DARK RED DOTS

–Folkman_lemma.svg

–Folkman_lemma

The Shapley–Folkman lemma is illustrated by the Minkowski addition of four sets. The point (+) in the convex hull of the Minkowski sum of the four non-convex sets (right) is the sum of four points (+) from the (left-hand) sets—two points in two non-convex sets plus two points in the convex hulls of two sets. The convex hulls are shaded pink. The original sets each have exactly two points (shown as red dots).

Minkowski addition and convex hulls. The sixteen dark-red points (on the right) form the Minkowski sum of the four non-convex sets (on the left), each of which consists of a pair of red points. Their convex hulls (shaded pink) contain plus-signs (+): The right plus-sign is the sum of the left plus-signs.

COMPLETING THE SQUARE- A QUADRANT PROOF INVOVLING FOUR SQUARES IS USED TO PROVE CALCULUS AS WELL AS ALGEBRA

Let’s construct the following graphic. It’s made of a square of side c with four of our original triangles distributed around it.

ptstep1

I’ve only labelled the outer sides, and it’s pretty obvious that the square in the middle has an area of c2. Now we’re going to move a bunch of the triangles around. Let the dance unfold!

ptstep2

ptstep3

ptstep4

Here we’re left with the same total area, just rearranged to show, or prove, that the area c2 is the same as the sum of a2+b2. The proof is perfectly transparent to all, without a shred of math. This series of pictures takes the place of a couple pages of algebra and is very beautiful (at least to nerds like me.)

Large Numbers Squared- INVOLVES FOUR PARTS

Here, Prof. Strogatz quotes a truly great mind. That of Richard Feynman, a very notable physicist with quite an interesting personal history. When Dr. Feynman was at Los Alamos during the Manhattan Project, he came across a problem that needed to know the square of 48. Hans Bethe said “That’s 2,300. If you want to know exactly, it’s 2,304.” So how did Bethe know?

largesquares

Bethe used this formula with x = -2 to do the math is his head. Let’s look at a picture of why this is so. Professor Strogatz has the reader imagine a square of carpet about 50 feet on a side:

largesquares2

The large square in the upper left is 2,500 square units. The two long rectangles are each 50x square units, and the small square at the bottom is x2 square units. The night that Alison and I went over this chapter, I made the point that one can use this to find the square of any large number. The very next day at school, she used this trick to impress her classmates and teacher.

GEOMETRIC PROOF INVOLVES DIVIDING INTO QUADRANTS- THEN FINALLY INTO SIXTEENTHS- 16 SQUARES IN QUADRANT MODEL

The Area of a Circle

Now, I proved this (as did pretty much every introductory calculus student) by forming an integral that swept the area of a small triangle around a full circle, adding the area of an infinite number of infinitesimal equilateral triangles with long length r and short length rdθ over 360 degrees.

circleintegral

And yes, it’s a pretty basic integral, but for those that aren’t familiar with calculus, I think their eyes must have already glossed over. I hope they just skipped to the pictures, and didn’t just click over to Facebook!

This geometric proof is very approachable, and in fact techniques like this were used to get arbitrarily accurate numbers before the invention of calculus. And while it’s easy to follow this proof, it contains the seeds of some pretty useful concepts: Most notable is that of limits…. Here we go!

Here we show a circle cut into quadrants. Since the circumference of a circle is 2πr, the length of each quadrant arc is ½ πr.

foursection

The total arc-length for the bottom is πr, and the length of the straight edge is r. We can now look at it with 8 segments:

eightsegments

The straight edge is still r, and the length of the scalloped edge is still πr. For 16 slices, we have:

sixteensegment

Each progression to thinner and thinner slices reduces the “bumpiness” of the scalloped edge, yet the length is always πr. The straight edge stays r, just getting more and more vertical. In the limit that we slice the circle into an infinite number of pizza slices, we simply end up with the rectangle with length πr and height r…. And area πr²!

IF YOU LOOK THE PROOF INVOLVES FOUR SQUARES (rectangles are two of them) FOUR PARTS A QUADRANT- COMPLETING THE SQUARE

Extra Credit: The Binomial Theorem

As one progresses in math, one gets told that finding the roots of quadratic equations is really, really important (rarely are students told why, it’s really just presented as some sort of gospel.) After fighting, uh, learning how with easy examples, the binomial theorem mantra is presented and dutifully memorized. When I first encountered it in 6th grade, my teacher didn’t know where it came from and couldn’t show us how to prove it, so we just accepted it as a truth from above. This was a shame, really, as for many, this just reinforced the idea that math was hard, something beyond the reach of mere mortals. But it’s not that hard, and I think if Mr Brown had seen this proof, he could have shown those that were interested where the theorem came from and why it was true all while introduction the idea of geometric reasoning.

This involves both algebra and geometry, and I know that there are some who won’t want to deal with the equations so I’m presenting it here. Let’s begin!

All quadratic equations take the form of

The two roots of this are found by setting y=0 and solving for x. The answer is given by the binomial theorem and looks like

quadratic7But where does this come from? Once again, geometric proof comes to the rescue! This time, we do something called “completing the square”. But before we do that, I’m going to manipulate the equation a bit….

So why did I do this last step? The is called “completing the square.” It makes taking the square root of the side with x² really easy and is the key to unlocking the theorem. This picture (should be familiar, it’s a lot like the one from squaring big numbers) shows why:

completingthesquare

By adding b²/4a² to both sides of the equation, it becomes obvious that the left is just the square of (x+b/2a) and the right is just what it is.

But now we can take the square root of both sides to show that

And finally