جزءصحیح floor function

Floor and ceiling functions
Floor function
Ceiling function

In mathematics and computer science, the floor function is the function that takes as input a real number  and gives as output the greatest integer less than or equal to , denoted . Similarly, the ceiling function maps  to the least integer greater than or equal to , denoted 

The fractional part is the sawtooth function, denoted by  for real x and defined by the formula[11]

For all x,

Examples

x Floor  Ceiling 
2 2 2 0
2.4 2 3 0.4
2.9 2 3 0.9
−2.7 −3 −2 0.3
−2 −2 −2 0

Order of operations تقدم اعمال ریاضی

In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression.

For example, in mathematics and most computer languages, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation.[1][2] Thus, the expression 2 + 3 × 4 is interpreted to have the value 2 + (3 × 4) = 14, not (2 + 3) × 4 = 20. With the introduction of exponents in the 16th and 17th centuries, they were given precedence over both addition and multiplication and could be placed only as a superscript to the right of their base.[1] Thus 3 + 52 = 28 and 3 × 52 = 75.

These conventions exist to eliminate ambiguity while allowing notation to be as brief as possible. Where it is desired to override the precedence conventions, or even simply to emphasize them, parentheses ( ) (sometimes replaced by brackets [ ] or braces { } for readability) can indicate an alternate order or reinforce the default order to avoid confusion. For example, (2 + 3) × 4 = 20 forces addition to precede multiplication, and (3 + 5)2 = 64 forces addition to precede exponentiation.

Examples

A horizontal fractional line also acts as a symbol of grouping:

For ease in reading, other grouping symbols, such as curly braces { } or square brackets [ ], are often used along with parentheses ( ). For example:

Matrix ماتریس

The m rows are horizontal and the ncolumns are vertical. Each element of a matrix is often denoted by a variable with two subscripts. For example, a2,1represents the element at the second row and first column of a matrix A.

In mathematics, a matrix (plural: matrices) is a rectangular arrayof numberssymbols, or expressions, arranged in rows and columns.For example, the dimensions of the matrix below are 2 × 3 (read “two by three”), because there are two rows and three columns:

Addition, scalar multiplication and transposition

Operation Definition Example
Addition The sum A+B of two m-by-nmatrices A and B is calculated entrywise:

(A + B)i,j = Ai,j + Bi,j, where 1 ≤ i ≤ m and 1 ≤ j≤ n.
Scalar multiplication The product cA of a number c (also called a scalar in the parlance of abstract algebra) and a matrix A is computed by multiplying every entry of A by c:

(cA)i,j = c · Ai,j.

This operation is called scalar multiplication, but its result is not named “scalar product” to avoid confusion, since “scalar product” is sometimes used as a synonym for “inner product“.

Transposition The transpose of an m-by-nmatrix A is the n-by-m matrix AT (also denoted Atr or tA) formed by turning rows into columns and vice versa:

(AT)i,j = Aj,i.
}

Familiar properties of numbers extend to these operations of matrices: for example, addition is commutative, that is, the matrix sum does not depend on the order of the summands: A + B = B + A.[12] The transpose is compatible with addition and scalar multiplication, as expressed by (cA)T = c(AT) and (A + B)T = AT + BT. Finally, (AT)T = A.

Ulam number patterns

یک تجدید آرایش از اعداد به سبک استنسلاو اولام

Stanisław Marcin Ulam ; (13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics. He participated in the Manhattan Project, originated the Teller–Ulam design of thermonuclear weapons, discovered the concept of cellular automaton, invented the Monte Carlo method of computation, and suggested nuclear pulse propulsion. In pure and applied mathematics, he proved some theorems and proposed several conjectures.

A smiling man in a hat and heavy winter coat and scarf, carrying a portfolio tucked under his arm

زاویه ها Common angles

زاویه های معروف و مهم

The radian (SI symbol rad) is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees(expansion at OEISA072097). The unit was formerly an SI supplementary unit, but this category was abolished in 1995 and the radian is now considered an SI derived unit.

رادیان زاویه مرکزی مقابل به کمانی از دایره است که طول آن با شعاع دایره برابر است. یعنی زاویه مرکزیِ متناظر با محیط دایره، مساویِ  رادیان و اندازه زاویه نیم صفحه،  رادیان و اندازه زاویه قائمه،  رادیان است.

هر رادیان برابر  درجه است. بنابر این با ضرب در رادیان، درجه به دست می‌آید. به عبارت دیگر با ضرب زاویه بر حسب رادیان در ۱۸۰ و تقسیم آن بر عدد پی، درجه به دست می‌آید.

زاویه در درجه = زاویه در رادیان . 

به عنوان مثال:

و بلعکس: با ضرب  در درجه، رادیان بدست می‌آید:

جدول زیر تبدیل چند زاویه پرکاربرد را نمایش می‌دهد:

درجه 30° 45° 60° 90° 180° 270° 360°
رادیان 0

Definition

Radian describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. More generally, the magnitude in radians of such a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, θ = s / r, where θ is the subtended angle in radians, s is arc length, and r is radius. Conversely, the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians; that is, s = .

Conversion between radians and degrees

As stated, one radian is equal to 180/π degrees. Thus, to convert from radians to degrees, multiply by 180/π.

For example:

Conversely, to convert from degrees to radians, multiply by π/180.

For example:

Radians can be converted to turns (complete revolutions) by dividing the number of radians by 2π.

Radian to degree conversion derivation

The length of circumference of a circle is given by , where  is the radius of the circle.

So the following equivalent relation is true:

 [Since a  sweep is needed to draw a full circle]

By the definition of radian, a full circle represents:

Combining both the above relations:

 

Conversion of common angles
Turns Radians Degrees Gradians (Gons)
0 0 0g
1/24 π/12 15° 16 2/3g
1/12 π/6 30° 33 1/3g
1/10 π/5 36° 40g
1/8 π/4 45° 50g
1/6 π/3 60° 66 2/3g
1/5 2π/5 72° 80g
1/4 π/2 90° 100g
1/3 2π/3 120° 133 1/3g
2/5 4π/5 144° 160g
1/2 π 180° 200g
3/4 3π/2 270° 300g
1 2π 360° 400g

Advantages of measuring in radians

Some common angles, measured in radians. All the large polygons in this diagram are regular polygons.

In calculus and most other branches of mathematics beyond practical geometry, angles are universally measured in radians. This is because radians have a mathematical “naturalness” that leads to a more elegant formulation of a number of important results.

Most notably, results in analysis involving trigonometric functions are simple and elegant when the functions’ arguments are expressed in radians. For example, the use of radians leads to the simple limit formula

which is the basis of many other identities in mathematics, including

Because of these and other properties, the trigonometric functions appear in solutions to mathematical problems that are not obviously related to the functions’ geometrical meanings (for example, the solutions to the differential equation , the evaluation of the integral , and so on). In all such cases it is found that the arguments to the functions are most naturally written in the form that corresponds, in geometrical contexts, to the radian measurement of angles.

The trigonometric functions also have simple and elegant series expansions when radians are used; for example, the following Taylor series for sin x :

If x were expressed in degrees then the series would contain messy factors involving powers of π/180: if x is the number of degrees, the number of radians is y = πx / 180, so

Mathematically important relationships between the sine and cosine functions and the exponential function (see, for example, Euler’s formula) are, again, elegant when the functions’ arguments are in radians and messy otherwise.

Dimensional analysis

Although the radian is a unit of measure, it is a dimensionless quantity. This can be seen from the definition given earlier: the angle subtended at the centre of a circle, measured in radians, is equal to the ratio of the length of the enclosed arc to the length of the circle’s radius. Since the units of measurement cancel, this ratio is dimensionless.

Although polar and spherical coordinates use radians to describe coordinates in two and three dimensions, the unit is derived from the radius coordinate, so the angle measure is still dimensionless.

sum of positive number and its reciprocal is at least two

جمع عددی مثبت با عکس خود همواره بزرگتر یا مساوی دو میباشد . این نامساوی معروف و پرکاربرد را به طرق مختلفی می توان اثبات کرد . اما درزیر یکی از این نوع اثبات های حالب را ببینید که اساس کارش ساده است . در مثلث قائمه وتر بزرگتر از اضلاع است و…

مساحت شولیکانو و مساحت سایر اشکال

Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the planeSurface area is its analog on the two-dimensional surfaceof a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).