Integers can be regarded as fractions with the denominator 1 Positive integer, 0' negative integer' is positive fraction, and negative fraction written as fraction is called rational number. Take any point on a straight line to represent the number 0, and this point is called the origin.
Only two numbers with different negative signs are called opposites. The distance between the point representing the number A on the number axis and the origin is called the absolute value of the number A, and the absolute value of the positive number recorded as IaI is itself. The absolute value of a negative number is its reciprocal; The absolute value of 0 is 0 (1). Positive numbers are greater than 0, 0 is greater than negative numbers, and positive numbers are greater than negative numbers. (2) Two negative numbers, the larger one has the smaller absolute value.
Rational number addition rule: 1. Add the same sign, take the same negative sign, and then add the absolute value. 2. Add two numbers with different signs and different absolute values, take the negative sign of the addend with larger absolute value, and subtract the one with smaller absolute value with the larger absolute value. Two numbers with opposite numbers add up to 0. 3. Add a number and add 0 to get this number.
Rule of rational number subtraction: subtracting a number is equal to adding the reciprocal of this number.
Rational number multiplication rule: two numbers are multiplied, the positive sign and negative sign of the same sign, and the absolute value is multiplied. Multiply any number by 0 to get 0.
Division of rational numbers: dividing by a number that is not 0 is equal to multiplying the reciprocal of this number.
A single number or letter is called a monomial, and the number factor in the monomial is also called the product of this monomial. In a monomial, the sum of the exponents of all the letters is called the number of times of the monomial. The sum of several monomials is called a polynomial, and the term of each monomial without letters is called the degree of the highest term in a constant polynomial, which is called the degree of this polynomial.
1. Pique formula S=a+ 1/2b- 1.
2. One of the equal sum series: 5+6*(n- 1)
Geometric formulas and theorems (junior high school)
1 There is only one straight line at two points.
The line segment between two points is the shortest.
The complementary angles of the same angle or equal angle are equal.
The complementary angles of the same angle or the same angle are equal.
One and only one straight line is perpendicular to the known straight line.
Of all the line segments connecting a point outside the straight line with points on the straight line, the vertical line segment is the shortest.
7 Parallel axiom passes through a point outside a straight line, and there is only one straight line parallel to this straight line.
If both lines are parallel to the third line, the two lines are also parallel to each other.
The same angle is equal and two straight lines are parallel.
The internal dislocation angles of 10 are equal, and the two straight lines are parallel.
1 1 are complementary and two straight lines are parallel.
12 Two straight lines are parallel and have the same angle.
13 two straight lines are parallel, and the internal dislocation angles are equal.
14 Two straight lines are parallel and complementary.
Theorem 15 The sum of two sides of a triangle is greater than the third side.
16 infers that the difference between two sides of a triangle is smaller than the third side.
The sum of the internal angles of 17 triangle is equal to 180.
18 infers that the two acute angles of 1 right triangle are complementary.
19 Inference 2 An outer angle of a triangle is equal to the sum of two non-adjacent inner angles.
Inference 3 The outer angle of a triangle is greater than any inner angle that is not adjacent to it.
2 1 congruent triangles has equal sides and angles.
Axiom of Angular (SAS) has two triangles with equal angles.
The Axiom of 23 Angles (ASA) has the congruence of two triangles, which have two angles and their sides correspond to each other.
The inference (AAS) has two angles, and the opposite side of one angle corresponds to the congruence of two triangles.
The axiom of 25 sides (SSS) has two triangles with equal sides.
Axiom of hypotenuse and right angle (HL) Two right angle triangles with hypotenuse and right angle are congruent.
Theorem 1 The distance between a point on the bisector of an angle and both sides of the angle is equal.
Theorem 2 is a point with equal distance on both sides of an angle, which is on the bisector of this angle.
The bisector of an angle 29 is the set of all points with equal distance to both sides of the angle.
The nature theorem of isosceles triangle 30 The two base angles of isosceles triangle are equal (that is, equilateral and equiangular).
3 1 Inference 1 The bisector of the vertices of an isosceles triangle bisects the base and is perpendicular to the base.
The bisector of the top angle, the median line on the bottom edge and the height on the bottom edge of the isosceles triangle coincide with each other.
Inference 3 All angles of an equilateral triangle are equal, and each angle is equal to 60.
34 Judgment Theorem of an isosceles triangle If a triangle has two equal angles, then the opposite sides of the two angles are also equal (equal angles and equal sides).
Inference 1 A triangle with three equal angles is an equilateral triangle.
Inference 2 An isosceles triangle with an angle equal to 60 is an equilateral triangle.
In a right triangle, if an acute angle is equal to 30, the right side it faces is equal to half of the hypotenuse.
The center line of the hypotenuse of a right triangle is equal to half of the hypotenuse.
Theorem 39 The distance between the point on the vertical line of a line segment and the two endpoints of the line segment is equal.
The inverse theorem and the point where the two endpoints of a line segment are equidistant are on the middle vertical line of this line segment.
The perpendicular bisector of the 4 1 line segment can be regarded as the set of all points with equal distance from both ends of the line segment.
Theorem 42 1 Two graphs symmetric about a line are conformal.
Theorem 2: If two figures are symmetrical about a straight line, then the symmetry axis is the perpendicular to the straight line connecting the corresponding points.
Theorem 3 Two graphs are symmetrical about a straight line. If their corresponding line segments or extension lines intersect, then the intersection point is on the axis of symmetry.
45 Inverse Theorem If the straight line connecting the corresponding points of two graphs is bisected vertically by the same straight line, then the two graphs are symmetrical about this straight line.
46 Pythagorean Theorem The sum of squares of two right angles A and B of a right triangle is equal to the square of the hypotenuse C, that is, A 2+B 2 = C 2.
47 Inverse Theorem of Pythagorean Theorem If the three sides of a triangle A, B and C are related in length A 2+B 2 = C 2, then the triangle is a right triangle.
The sum of the quadrilateral internal angles of Theorem 48 is equal to 360.
The sum of the external angles of the quadrilateral is equal to 360.
The theorem of the sum of internal angles of 50 polygons is that the sum of internal angles of n polygons is equal to (n-2) × 180.
5 1 It is inferred that the sum of the external angles of any polygon is equal to 360.
52 parallelogram property theorem 1 parallelogram diagonal equality
53 parallelogram property theorem 2 The opposite sides of parallelogram are equal
It is inferred that the parallel segments sandwiched between two parallel lines are equal.
55 parallelogram property theorem 3 diagonal bisection of parallelogram.
56 parallelogram decision theorem 1 Two groups of parallelograms with equal diagonals are parallelograms.
57 parallelogram decision theorem 2 Two groups of parallelograms with equal opposite sides are parallelograms.
58 parallelogram decision theorem 3 A quadrilateral whose diagonal is bisected is a parallelogram.
59 parallelogram decision theorem 4 A group of parallelograms with equal opposite sides are parallelograms.
60 Rectangle Property Theorem 1 All four corners of a rectangle are right angles.
6 1 rectangle property theorem 2 The diagonals of rectangles are equal
62 Rectangular Decision Theorem 1 A quadrilateral with three right angles is a rectangle.
63 Rectangular Decision Theorem 2 Parallelograms with equal diagonals are rectangles
64 diamond property theorem 1 all four sides of the diamond are equal.
65 Diamond Property Theorem 2 Diagonal lines of diamonds are perpendicular to each other, and each diagonal line bisects a set of diagonal lines.
66 Diamond area = half of diagonal product, that is, S=(a×b)÷2.
67 diamond decision theorem 1 A quadrilateral with four equilateral sides is a diamond.
68 Diamond Decision Theorem 2 Parallelograms whose diagonals are perpendicular to each other are diamonds.
69 Theorem of Square Properties 1 All four corners of a square are right angles and all four sides are equal.
Theorem of 70 Square Properties 2 Two diagonal lines of a square are equal and bisected vertically, and each diagonal line bisects a set of diagonal lines.
Theorem 7 1 1 is congruent with respect to two centrosymmetric graphs.
Theorem 2 About two graphs with central symmetry, the connecting lines of symmetric points both pass through the symmetric center and are equally divided by the symmetric center.
Inverse Theorem If a straight line connecting the corresponding points of two graphs passes through a point and is bisected by the point, then the two graphs are symmetrical about the point.
The property theorem of isosceles trapezoid is that two angles of isosceles trapezoid on the same base are equal.
The two diagonals of an isosceles trapezoid are equal.
76 isosceles trapezoid decision theorem A trapezoid with two equal angles on the same base is an isosceles trapezoid.
A trapezoid with equal diagonal lines is an isosceles trapezoid.
Theorem of bisecting line segments by parallel lines If a group of parallel lines are tangent to a straight line.
Equal, then the line segments cut on other straight lines are also equal.
79 Inference 1 A straight line passing through the midpoint of one waist of a trapezoid and parallel to the bottom will bisect the other waist.
Inference 2 A straight line passing through the midpoint of one side of a triangle and parallel to the other side will be equally divided.
Trilaterality
The median line theorem of 8 1 triangle The median line of a triangle is parallel to and equal to the third side.
Half of
The trapezoid midline theorem is parallel to the two bases and equal to half of the sum of the two bases L = (a+b) ÷ 2s = l× h.
Basic properties of ratio 83 (1) If a:b=c:d, then ad=bc.
If ad=bc, then a: b = c: d.
84 (2) Combinatorial Properties If A/B = C/D, then (A B)/B = (C D)/D.
85 (3) Isometric Property If A/B = C/D = … = M/N (B+D+…+N ≠ 0), then
(a+c+…+m)/(b+d+…+n)=a/b
86 parallel lines are divided into segments and the theorem of proportion. Three parallel lines cut two straight lines, and the corresponding results are obtained.
The line segments are proportional.
It is inferred that the line parallel to one side of the triangle cuts the other two sides (or the extension lines of both sides), and the corresponding line segments are proportional.
Theorem 88 If the corresponding line segments obtained by cutting two sides (or extension lines of two sides) of a triangle are proportional, then this straight line is parallel to the third side of the triangle.
A straight line parallel to one side of a triangle and intersecting with the other two sides, the three sides of the cut triangle are directly proportional to the three sides of the original triangle.
Theorem 90 A straight line parallel to one side of a triangle intersects the other two sides (or extension lines of both sides), and the triangle formed is similar to the original triangle.
9 1 similar triangles's decision theorem 1 Two angles are equal and two triangles are similar (ASA)
Two right triangles divided by the height on the hypotenuse are similar to the original triangle.
Decision Theorem 2: Two sides are proportional and the included angle is equal, and two triangles are similar (SAS).
Decision Theorem 3 Three sides are proportional and two triangles are similar (SSS)
Theorem 95 If the hypotenuse of a right triangle and one right-angled side and another right-angled side
The hypotenuse of an angle is proportional to a right-angled side, so two right-angled triangles are similar.
96 Property Theorem 1 similar triangles has a high ratio, and the ratio corresponding to the center line is flat with the corresponding angle.
The ratio of dividing lines is equal to the similarity ratio.
97 Property Theorem 2 The ratio of similar triangles perimeter is equal to similarity ratio.
98 Property Theorem 3 The ratio of similar triangles area is equal to the square of similarity ratio.
The locus from 107 to a point with equal distance on both sides of a known angle is the bisector of this angle.
The trajectory from 108 to the equidistant point of two parallel lines is a straight line parallel and equidistant to these two parallel lines.
139 every inner angle of a regular n-polygon is equal to (n-2) ×180/n.
140 Theorem Radius and apothem Divides a regular N-polygon into 2n congruent right triangles.
14 1 the area of the regular n polygon Sn = PNRN/2 P represents the perimeter of the regular n polygon.
142 The area of a regular triangle √ 3a/4a indicates the side length.
143 if there are k positive n corners around a vertex, then the sum of these angles should be
360, so k× (n-2) 180/n = 360 is changed to (n-2)(k-2)=4.
145 sector area formula: s sector =n r 2/360 = LR/2.
Practical tools: common mathematical formulas
Formula classification formula expression
Multiplication and factorization A2-B2 = (a+b) (a-b) A3+B3 = (a+b) (A2-AB+B2) A3-B3 = (A-B (A2+AB+B2))
Trigonometric inequality | A+B |≤| A |+B||||| A-B|≤| A |+B || A |≤ B < = > -b≤a≤b
|a-b|≥|a|-|b| -|a|≤a≤|a|
The solution of the unary quadratic equation -b+√(b2-4ac)/2a -b-√(b2-4ac)/2a
The relationship between root and coefficient x1+x2 =-b/ax1* x2 = c/a Note: Vieta theorem.
discriminant
B2-4ac=0 Note: This equation has two equal real roots.
B2-4ac >0 Note: The equation has two unequal real roots.
B2-4ac & lt； Note: The equation has no real root, but a complex number of the yoke.
The sum of the first n terms of some series
1+2+3+4+5+6+7+8+9+…+n = n(n+ 1)/2 1+3+5+7+9+ 1 1+ 13+ 15+…+(2n- 1)= N2
2+4+6+8+ 10+ 12+ 14+…+(2n)= n(n+ 1) 12+22+32+42+52+62+72+82+…+N2 = n(n+ 1)(2n+ 1)/6
13+23+33+43+53+63+…n3 = N2(n+ 1)2/4 1 * 2+2 * 3+3 * 4+4 * 5+5 * 6+6 * 7+…+n(n+ 1)= n(n+ 1)(n+2)/3
Lateral area of cylinder S=c*h=2pi*h lateral area of cone s =1/2 * c * l = pi * r * l.
Cylinder volume formula V=s*h cylinder V=pi*r2h