Of all the mathematics in Newton's time, Newton's work was more than half. Indeed, Newton has made great achievements in astronomy and physics. In mathematics, he has made creative achievements and contributions from binomial theorem to calculus, from algebra and number theory to classical geometry and analytic geometry, finite difference, curve classification, calculation method and approximation theory, and even probability theory.
Discover binomial theorem
1665, Newton, who was only 22 years old, discovered the binomial theorem, which is an essential step for the all-round development of calculus. The binomial theorem holds that energy is discovered by direct calculation.
Binomial series expansion is a powerful tool to study series theory, function theory, mathematical analysis and equation theory. Today, we will find that this method is only applicable to the case where n is a positive integer. When n is a positive integer of 1, 2, 3, ..., the series ends at n+ 1 If n is not a positive integer, the series will not end, and this method is not applicable. But you know, Leibniz introduced the word function in 1694. In the early stage of calculus, it is the most effective method to treat transcendental function with the level of transcendental function.
Create calculus
Newton's most outstanding achievement in mathematics was the creation of calculus. His outstanding achievement is to unify all kinds of special skills to solve infinitesimal problems since ancient Greece into two general algorithms-differential and integral, and establish the reciprocal relationship between these two operations. For example, area calculation can be regarded as the inverse process of finding tangent.
At that time, Leibniz had just put forward the research report of calculus, which triggered the debate on the patent right of calculus invention until Leibniz's death. Later generations have decided that the difference product was invented by them at the same time.
In the method of calculus, Newton's extremely important contribution is that he not only clearly saw, but also greatly used the methodology provided by algebra, which is much superior to geometry. He replaced the geometric methods of cavalieri, Gregory, Huygens and Barrow with algebraic method, and completed the algebra of integral. Since then, mathematics has gradually shifted from the subject of feeling to the subject of thinking.
In the early days of micro-products, they were used by people with ulterior motives because they did not establish a solid theoretical foundation. This led to the famous second mathematical crisis. This problem was not solved until the limit theory was established in19th century.
Introducing polar coordinates to develop cubic curve theory
Newton made a profound contribution to analytic geometry. He is the founder of polar coordinates. The first one studied the high-order plane curve extensively. Newton proved how to put the general cubic equation
In his book Cubic Curve, Newton listed 72 out of 78 possible forms of cubic curve. These are the most attractive; The most difficult thing is: just as all curves can be projected as the center of a circle; All cubic curves can be used as curves.
The center of the projection. This theorem was a mystery until 1973 was proved.
Newton's cubic curve laid the foundation for studying higher plane straight lines, and expounded the importance of asymptotes, nodes and points. Newton's work on cubic curves inspired many other research work on higher plane curves.
Advanced equation theory, open variational method
Newton also made a classical contribution to algebra, and his generalized arithmetic greatly promoted the theory of equations. He found that the imaginary roots of real polynomials must appear in pairs, and found the upper bound law of polynomial roots. He expressed the sum formula of the roots of polynomials by using the coefficients of polynomials, and gave a generalization of Cartesian sign rule that limits the number of imaginary roots of real polynomials.
Newton also designed a method to find the logarithm of the approximate values of the real roots of numerical equations and transcendental equations. The modification of this method is now called Newton method.
Newton also made great discoveries in the field of mechanics, which is a science to explain the motion of objects. The first law of motion was discovered by Galileo. This law shows that if an object is at rest or moving in a straight line at a constant speed, it will remain at rest or continue to move in a straight line at a constant speed as long as there is no external force. This law, also known as the law of inertia, describes a property of force: force can make an object move from rest to motion, from motion to rest, and can also make an object change from one form of motion to another. This is the so-called Newton's first law. The most important problem in mechanics is how objects move under similar circumstances. Newton's second law solved this problem; This law is considered to be the most important basic law in classical physics. Newton's second law quantitatively describes that force can change the motion of an object. Indicates the time change rate of speed (i.e. acceleration A is directly proportional to force F, but inversely proportional to the mass of the object, i.e. a=F/m or F = Ma). The greater the force, the greater the acceleration; The greater the mass, the smaller the acceleration. Both force and acceleration have magnitude and direction. Acceleration is caused by force, and the direction is the same as force; If several forces act on an object, the resultant force will produce acceleration. The second law is the most important, and all the basic equations of power can be derived from it by calculus.
In addition, Newton formulated the third law based on these two laws. Newton's third law points out that the interaction between two objects is always equal in size and opposite in direction. For two objects in direct contact, this law is easier to understand. The downward pressure of the book on the sub-table is equal to the upward support of the table on the book, that is, the action is equal to the reaction. So is gravity. The force that an airplane in flight pulls up the earth is numerically equal to the force that the earth pulls down the airplane. Newton's laws of motion are widely used in science and dynamics.
Elementary plane geometry
self-evident truth
1 Any two different points determine a straight line through them.
Let AB be a given line segment and OX be a known ray, then there is only one point C on the ray OX, so the line segment OC=AB.
Geometry can be moved without changing its shape and size.
4 Parallelism axiom: At most, a straight line parallel to the known straight line can be drawn through a point outside the known straight line.
5 Archimedes' axiom: given the line segment AB & gtCD, when the latter is used to measure the former, it will always surpass the former after being measured several times, or there must be a positive integer N, so that (n- 1)CD≤AB≤Ncd.
Biaxial symmetry and central symmetry
Axisymmetry of 1: Fold in half along a straight line, and the parts on both sides of the straight line are completely coincident. This straight line is called symmetry axis, and the points that can overlap together are called symmetry points. If this is a graph, it is called an axisymmetric graph. (e.g. isosceles triangle)
Property: The perpendicular to the symmetry point is the axis of symmetry.
2 Center symmetry: two figures can overlap each other when they rotate around a center180. This point is called the center of symmetry, and the point that can overlap is called the symmetrical point. If this is a graph, it is called a centrosymmetric graph. (e.g. parallelogram)
Property: The midpoint of the symmetry point is the center of symmetry.
Three basic concepts
Angle of bisector of middle vertical line and 1 line segment
Properties of the perpendicular line in (1):
Any point on the vertical line of 1 is equal to the distance between the two ends of the line segment.
All points equidistant from both ends of the line segment are on the middle vertical line.
(2) The nature of the angular bisector:
Any point on the bisector of the angle1is equidistant from both sides of the same angle.
All points equidistant on both sides of an angle are on the bisector of the angle.
2 perspective
(1) Angle of view of a line segment: When two rays from a point pass through both ends of a known line segment, the angle formed by the two rays is called the angle of view of the point to the known line segment.
(2) Perspective from point to circle: two tangents drawn from a point outside the circle (regarded as rays), and the included angle between these two tangents is called perspective from point to circle.
Three congruent triangles.
1 decision theorem: s.a.s.a.s.a.s.a (big corner)
S.s.a: Two triangles must be congruent, if two sides and the diagonals of their big sides correspond equally.
Certificate: A/Sina = A 1/Sina 1, b/sinb = b1∈ (0/sinb1), if both a and A 1 are big faces, then a = a/.
Max (b, b 1) ≥ 90 is a small contradiction with b, b 1, so B=B 1.
Note: Small corners are invalid.
2 congruent right triangle:
(1) right-angled edge
(2) Right angle hypotenuse
(3) Right-angled edge, adjacent angle or relatively acute angle
(4) acute bevel edge
Four parallel lines
Existence theorem of 1: On a plane, two lines perpendicular to a known line are parallel to each other.
2 Decision Theorem: Two known straight lines are cut by a third straight line. Two known straight lines are parallel to each other if one of the following conditions holds:
1 isosceles angle is equal.
2 The internal dislocation angles are equal.
3. Complement the internal angle on the same side.
3 property theorem: if two lines are cut by a third line, then it is formed.
1 isosceles angle is equal.
2 The internal dislocation angles are equal.
3. Complement the internal angle on the same side.
Inference: (1) If two straight lines are perpendicular to one of the two parallel lines, they are also perpendicular to the other.
(2) The perpendicular lines of intersecting lines also intersect.
4 parallel cutting theorem:
(1) Two straight lines are cut by a set of parallel lines. If the segments cut on one line are equal, the segments cut on another line are equal.
If two straight lines are cut into equal segments by a set of cutting lines, and two cutting lines are parallel, all cutting lines are parallel to each other. (Note that it is not the inverse theorem of 1)
(2) Angle parallel cutting theorem: both sides of an angle are cut by parallel lines. If the segments cut on one side are equal, the segments cut on the other side are also equal.
Inverse theorem of angle parallel cutting theorem: two sides of an angle are cut into equal segments by a set of cutting lines, then all cutting lines are parallel to each other.
(3) On the parallel cutting theorem of proportion:
1 Two straight lines are cut by a straight line parallel to the third side, and the cut line segments must be proportional.
If two straight lines are cut by a set of sectioning lines in proportion, and two of the sectioning lines are parallel, all the sectioning lines are parallel to each other.
The two sides of a triangle are cut by a set of parallel lines, and the cut line segments must be proportional.
Inverse theorem: If two sides of a triangle are directly proportional to the line segment cut by a straight line, then the straight line is parallel to the third side.
(4) the median line theorem
The median line of any triangle of 1 is parallel to the third side, which is equal to half of this side.
The center line of the trapezoid is parallel to the bottom, which is equal to half of the sum of the two bottoms.
Five figures
(1) triangle
1 external angle theorem: each external angle of a triangle is greater than any internal diagonal.
2 isosceles triangle: four lines are combined into one.
3 triangle inequality theorem:
(1) big side to big angle, big angle to big side.
(2) In a triangle, any one side is less than the sum of the other two sides and greater than their difference.
Inference: For any three points A, B and C, there is always ∣AB-AC∣≤BC≤AB+AC.
(3) If two triangles have two equal sides, then
The angle of 1 is large, and the opposite side is large.
The third side is bigger and the diagonal is bigger.
4 Five Hearts
(1) outer circle center: the intersection point of three perpendicular lines is also the center of the circumscribed circle.
(2) Center of gravity: the intersection point of the median lines of three sides.
(3) vertical center: the intersection of three high lines (and three vertices form a vertical center group)
(4) Inner heart: The intersection of the bisectors of the three inner angles is also the center of the inscribed circle.
(5) Paracenter: The intersection of three bisectors of an inner corner and an outer corner of two other inner corners has three points, which are also the center of the tangent circle.
Theorem of bisector of inner angle and outer angle: Let the bisector of triangle and its outer angle intersect with the opposite side and its extension line, then the intersection points are divided into inner side and outer side respectively, and the score ratio is equal to the ratio of two adjacent sides. (Inverse theorem exists)
6 regular triangle: PA≤PB+PC, when P is located on the arc BC opposite to point A in its circumscribed circle, take the equal sign.
(2) parallelogram
1 Definition: Two pairs of quadrangles with parallel opposite sides.
2 property theorem:
1 Two pairs of opposite sides are equal.
Two pairs of diagonal lines are equal.
The three diagonals are equally divided.
3 Decision Theorem: A quadrilateral with one of the following conditions must be a parallelogram.
1 Two pairs of opposite sides are equal.
Two pairs of diagonal lines are equal.
The three diagonals are equally divided.
A pair of opposite sides are parallel and equal.
4 rectangle: equilateral parallelogram (two diagonal lines are equal, and the line connecting the midpoint of the opposite side is the symmetry axis)
Diamond: equilateral parallelogram (two diagonal lines are equally divided and diagonal lines are symmetrical)
Square: A quadrilateral (4 axes of symmetry) that is both rectangular and rhombic.
③ trapezoid
1 Definition: A pair of quadrilaterals with parallel opposite sides.
2 isosceles trapezoid: the two waists are equal, the two bottom angles are equal, the diagonal lines are equal, and the connecting line between the two bottom midpoints is the symmetry axis.
(4) Polygon
1 sum of internal angles: (n-2) * 180, sum of external angles: 360.
Regular polygon: A polygon with equal sides and angles.
(5) circle
1 symmetry: take the center of the circle as the symmetry center and any diameter as the symmetry axis.
2 inequality theorem: arc, chord, central angle, distance between chord centers l = rθ = (n180) * 2π r.
3 tangent theorem
(1) The tangent of the circle is perpendicular to the radius of the tangent point.
(2) The straight line passing through the outer end of the radius of a circle and perpendicular to the radius is the tangent of the circle.
(3) The length of two tangents drawn from a point outside the circle is equal, and the ray drawn from this point to the center of the circle bisects the perspective of this point to the circle.
(4) Common tangent theorem: the two outer tangents of two circles are equal in length, and the two inner common tangents are also equal in length.
(5) Tangency theorem of two circles:
1 Tangent point of two circles is on the connecting line, otherwise, the same point on the connecting line of two circles must be tangent.
The necessary and sufficient condition of the circumscribed circle of 2 is OO'= R+R'+R', and the necessary and sufficient condition of the inscribed circle is oo' =∣r-r '∣.
4 Circumferential angle: the angle whose vertex is on the circle and whose two sides intersect with the circle.
(In a circle, the circumferential angle subtended by the same arc is equal to half the central angle subtended)
Chord angle: the angle at which one side intersects the circle and the other side is tangent to the circle at the vertex.
The tangent angle of a circle is equal to the circumferential angle of the arc it contains.
Angle inside the circle: the angle of the vertex inside the circle.
(The inner angle of a circle is equal to the sum of the arcs contained in the circumferential angle and the vertex angle it subtends.)
Outer angle of a circle: an angle whose vertex is outside the circle and whose two sides share a common point with the circle.
(The outer angle of a circle is equal to the difference between the circumferential angles of the two arcs it contains)
Summary: 1 is the same arc: the inner angle of the circle >; Circumferential angle = chord tangent angle > outer angle of circle
If an angle and two sides of a circle have a common point and are equal to the angle of the circle, then the vertex of the angle must be on the circle.
5 circle inscribed quadrilateral: diagonal complementarity. (Inverse theorem exists)
Circular circumscribed quadrilateral: opposite sides and equality. (Inverse theorem exists)
6 circle power theorem: given a circle o, if any secant passes through a and b through point p, then
P=PA*PB=∣PO2-R2∣, let p'= PO2-R2, and this value of p' is called the power of point P to circle O. Specifically, the power of a point outside the circle is positive, the power of a point inside the circle is negative, and the power of a point on the circle is zero.
7 four * * * circle judgment:
(1) diagonally complementary quadrilateral
(2) Two points point to a line segment from the same perspective.
(3) Power theorem: PA*PB=PC*PD.
Six similar triangles
1 Fundamental Theorem: A straight line parallel to one side of a triangle and intersecting with the other two sides, the cut triangle is similar to the original triangle.
2 Decision Theorem: Two triangles must be similar if they meet one of the following conditions:
(1) Two pairs of corresponding angles are equal (average).
(2) A pair of corresponding angles are equal and their sides are proportional.
(3) Three pairs of corresponding edges are proportional (s.s.s)
(4) Two pairs of corresponding sides are in proportion, and the diagonals of the main sides are equal (S.s.a)
3 The ratio of any pair of corresponding line segments (such as the corresponding height, median line and angular bisector) in similar triangles is equal to the similarity ratio.
Seven areas
S (parallelogram) =ah=absinα
S (rectangle) =ab
s(diamond)= ah = absinα=( 1/2)l 1 L2。
S (square) =a2= ( 1/2)l2
S (triangle) = (1/2) ah = (1/2) absinc.
S (circle) =πR2
S (sector) = (n/360) π R2 = (1/2) θ r 2.
s(bow)=( 1/2)R2(απ/ 180-sinα)
Beresinar formula: s (quadrilateral) = (1/4) [4e2f2-(A2-B2+C2-D2) 2]1/2.
Brahmagudda formula: s (quadrilateral inscribed with a circle) = [(s-a) (s-b) (s-c) (s-d)]1/2 (s is a semi-circle).
Helen formula: s (triangle) = [s(s-a)(s-b)(s-c)] 1/2.
Eight basic trajectories:
The locus of 1 equidistant from two known points is the middle vertical line connecting the two points.
Within a known angle, the locus of equidistant points on both sides is the bisector of this angle.
The locus of a point equidistant from two parallel known straight lines is a straight line, which is parallel to these two known straight lines and equidistant from them.
The locus of a point with a fixed distance from a known straight line is a pair of straight lines on both sides of the known straight line and parallel to the known straight line, where the distance from each known straight line is equal to the fixed length.
The distance from the fixed point is equal to the trajectory of the fixed length point, which is a circle with the fixed point as the center and the fixed length as the radius.
For a certain line segment, the locus of the point whose viewing angle is equal to a fixed angle is a double arc with the fixed line segment as the chord.
7 For a certain line segment, the locus of the point whose viewing angle is equal to the right angle is a circle with the diameter of the fixed line segment.
Nine special concepts
1 Euler line: the line of the outer center, center of gravity and vertical center of a triangle.
(The distance from the center of gravity to one side is equal to half the distance from the opposite vertex to the vertical center)
Newton line: the midpoint of three diagonals of a complete quadrilateral.
3 Mick point: the sides of a complete quadrilateral intersect into four triangles, and their circumscribed circles are * * * points.
4 Seymour pine thread:
(1) The necessary and sufficient condition of the orthogonal projection line of the points on three sides of a triangle or its extension line is that the points are on the circumscribed circle of the triangle. The straight line of orthographic projection is called Simpson line of triangle at a certain point.
(2) The orthogonal projection line of Mick's point of a quadrilateral. This line is called a completely quadrilateral Simpson line.
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