Summary of common mathematical formulas

First, the basic algebraic formula

1. Square difference formula: (a+b) × (a-b) = A2-B2.

2. Complete square formula: (AB) 2 = A2AB+B2

Complete cubic formula: (a b) 3 = (a b) (a2ab+B2)

3. Multiplication with the same base number: am× an = am+n (m both m and n are positive integers, a≠0).

Same base powers division: am ÷ an = am-n (m both m and n are positive integers, a≠0).

a0= 1(a≠0)

A-p = (a ≠ 0, p is a positive integer)

4. Arithmetic series:

( 1)sn = = na 1+n(n- 1)d；

(2)an = a 1+(n- 1)d；

(3)n =+ 1；

(4) If a, a and b are arithmetic progression, then: 2a = a+b;

(5) If m+n=k+i, then: AM+An = AK+AI;

(where: n is the number of terms, a 1 is the first term, an is the last term, d is the tolerance, and sn is the sum of the first n terms in arithmetic progression).

5. Geometric series:

( 1)an = a 1q- 1；

(2) Serial number = (q 1)

(3) If A, G and B are geometric series, then G2 = AB.

(4) If m+n=k+i, then: am? an=ak？ ai；

(5)am-an=(m-n)d

(6) =q(m-n)

(where: n is the number of terms, a 1 is the first term, an is the last term, q is the common ratio, and sn is the sum of the first n terms in the geometric series).

6. The root formula of unary quadratic equation: ax2+bx+c=a(x-x 1)(x-x2).

Where: x1=; x2= (b2-4ac 0)

Relationship between root and coefficient: x 1+x2=-, x 1? x2=

Second, the basic geometric formula

1. triangle: three points that are not on a straight line can form a triangle; The sum of the internal angles of the triangle is equal to180; Any two triangles.

The sum of sides is greater than the third side, and the difference between any two sides is less than the third side;

(1) angle bisector: the bisector of an angle of a triangle intersects the opposite side of the angle, and the line segment between the intersection of the vertex and the angle is called the bisector of the angle of the triangle.

(2) The midline of a triangle: the line segment connecting a vertex of the triangle with the midpoint of its opposite side is called the midline of the triangle.

(3) Height of the triangle: The vertical line from the vertex of the triangle to its opposite side is called the height of the triangle.

(4) The midline of the triangle: the line segment connecting the midpoints of the two sides of the triangle is called the midline of the triangle.

(5) Inner heart: the intersection of bisectors of angles is called inner heart; The distance from the center to the three sides of the triangle is equal.

Center of gravity: the intersection of the center lines is called the center of gravity; The distance from the center of gravity to the midpoint of each side is equal to one third of the center line here.

Vertical line: the intersection of high lines is called vertical line; The vertex of a triangle must be perpendicular to the opposite side.

Exterior center: The intersection of the perpendicular lines of the three sides of a triangle is called the exterior center of the triangle. The distance from the outer center to the three vertices of the triangle is equal.

Right triangle: A triangle with an angle of 90 degrees is a right triangle.

Properties of right triangle:

(1) The two acute angles of a right triangle are complementary;

(2) The median line on the hypotenuse of the right triangle is equal to half of the hypotenuse;

(3) In a right triangle, if there is an acute angle equal to 30, then the right side it faces is equal to half of the hypotenuse;

(4) In a right triangle, if a right-angled side is equal to half of the hypotenuse, the acute angle of this right-angled side is 30;

(5) In a right triangle, C2 = A2+B2 (where: A and B are the lengths of two right angles and C is the length of the hypotenuse);

(6) The radius of the circumscribed circle of the right triangle is also the center line on the hypotenuse;

Determination of right triangle;

(1) has an angle of 90;

(2) The median line of one side is equal to half the length of the side;

(3) If C2 = A2+B2, a triangle with side lengths A, B and C is a right triangle;

2. Area formula:

Square = side length × side length;

Rectangular = length × width;

Triangle = x base x height;

Trapezoid =;

Circle = R2

Parallelogram = base × height

Department = R2

Cube = 6× side length× side length

Cuboid = 2× (length× width+width× height+length× height);

Cylinder = 2π R2+2π RH;

Surface area of ball = 4r2

3. Volume formula

Cube = side length × side length × side length;

Cuboid = length× width× height;

Cylinder = bottom area × height = sh = π r2h

Cone = π r2h

Ball =

4. Formulas related to circles

Let the radius of the circle be r and the distance from the point to the center of the circle be d, then there are:

(1)d￠r: the point is in the circle (that is, the inside of the circle is the set of points whose distance from the center of the circle is less than the radius);

(2) d = r: the point is on the circle (that is, the upper part of the circle is the set of points whose distance from the center of the circle is equal to the radius);

(3) d-r: the point is outside the circle (that is, outside the circle is the set of points whose distance from the center of the circle is greater than the radius);

The nature and judgment of the positional relationship between line and circle;

If the radius ⊙O is r and the distance from the center of o to the straight line is d, then:

(1) straight line intersection ⊙ o: dr;

(2) The line is tangent to ⊙O: d = r；;

(3) the line is separated from ⊙O: d r;

The nature and judgment of the positional relationship between circles;

Let the radii of two circles be r and r respectively and the center distance be d, then:

(1) Two circles are separated from each other:

(2) circumscribe two circles:

(3) Two circles intersect: ();

(4) Two circles are inscribed: ();

(5) Two circles contain: ().

The formula of the circumference of a circle: c = 2π r = π d (where r is the radius of the circle, d is the diameter of the circle, and π ≈ 3.1415926 ≈);

Calculation formula of arc length corresponding to central angle: =;

Sector area: (1)S sector = π R2; ②S fan = r；

If the radius of the cone bottom is R and the length of the generatrix is L, then its lateral area: S side = π r;

Volume of cone: v = sh = π r2h.

Third, other common sense.

The mantissas of 1.2x, 3X, 7X, 8X all change periodically with 4; The mantissas of 4X and 9X both change with a period of 2;

In addition, the mantissas of 5X and 6X are always 5 and 6, where x is a natural number.

2. For any two numbers A and B, if A-B > 0, then A > B；; If a < b；; < 0, then a < b；; If a-b = 0, then a = b.

When a and b are any two positive numbers, if a/b > 1, then a > b；; If a/b < 1, then a < b；; If a/b = 1, then a = b.

When a and b are arbitrary negative numbers, if a/b > 1, then a < b；; If a/b < 1, then a > b；; If a/b = 1, then a = b.

For any two numbers A and B, when it is difficult to directly use the difference method or the commercial law is relatively large, we usually choose the middle value C. If

A > c, c > b, then we say a > B.

3. Engineering problems:

Workload = working efficiency × working time; Work efficiency = workload ÷ working hours;

Working hours = workload ÷ working efficiency; Total workload = sum of each workload;

Note: When solving practical problems, the total permanent workload is 1.

4. The phalanx problem:

(1) solid square: the total number of people in the square = (the number of people on the outermost side) 2.

Number of people on the outermost layer = (number of people on each side of the outermost layer-1) × 4

(2) Hollow phalanx: number of people in hollow phalanx = (number of people on each side of outermost layer) 2- (number of people on each side of outermost layer -2× number of layers) 2

= (number of people on each side of outermost layer-number of layers) × number of layers× 4 = number of people in hollow square.

Example: There is a three-story hollow square with 10 people on the outermost layer. How many people are there in the whole square?

Solution: (10-3) × 3× 4 = 84 (person)

5. Profit problem:

(1) Profit = selling price (selling price)-cost;

Profit rate =-1;

Sales price = cost ×( 1+ profit rate); Cost =

(2) Simple interest problem

Interest = principal × interest rate × term;

Sum of principal and interest = principal+interest = principal ×( 1+ interest rate× term);

Principal = principal and interest and present value (1+ interest rate × term).

Annual interest rate ÷ 12= monthly interest rate;

Monthly interest rate × 12= annual interest rate.

Example: someone deposits 2400 yuan with a term of 3 years, and the monthly interest rate 10.2 ‰ (that is, the monthly interest rate 1.02%). What is the total principal and interest when it expires in three years? "

Solution: Use the monthly interest rate. 3 years = 65438+2 months ×3=36 months

2400× (1+10.2 %× 36) = 2400×1.3672 = 3281.28 (yuan)

6. permutation number formula: p = n (n-1) (n-2) ... (n-m+1), (m≤n)

Combination number formula: c = p ÷ p = (regulation = 1).

"envelope error" problem: D 1=0 = 0, D2 = 1, D3 = 2, D4 = 9, D5 = 44, D6 = 265,

7. Age problem: the key is that the age difference remains unchanged;

After several years, age = age difference ÷ multiple difference-young age.

A few years ago, age = young age-age difference ÷ multiple difference

8. Date: 366 days in leap year and 365 days in average year, in which: 1, 3, 5, 7, 8, 10,1February are all 3 1 day, 4, 6, 9,/kloc-0.

9. Planting trees

(1) Linear tree planting: number of trees = total length interval+1.

(2) Ring planting: number of trees = total length interval

(3) Planting trees between buildings: number of trees = total length interval-1

(4) Rope cutting problem: fold it in half for n times, cut the M knife from above, and cut it into (2n× m+ 1) sections.

10. Chicken and rabbit are in the same cage:

Number of chickens = (number of rabbit feet × total number of heads-total number of feet) ÷ (number of rabbit feet-number of chicken feet)

(Usually, "per" quantity is regarded as "number of feet")

Gain and loss (the generalization of the problem of chickens and rabbits in the same cage);

Number of nonconforming products = (65438+ 0 points for nonconforming products × total number of products-actual total score) ÷ (points for each nonconforming product+points for each nonconforming product)

= total number of products-(points deducted for each unqualified product × total number of products+actual total score) ÷ (points deducted for each unqualified product+points deducted for each unqualified product)

Example: "the workers who produce light bulbs in the light bulb factory are paid according to their scores." Each qualified product will get 4 points, while each unqualified product will not be scored, and 15 points will be deducted. A worker produced 1000 light bulbs with a total score of 3525. How many of them are unqualified? "

Solution: (4×1000-3525) ÷ (4+15) = 475 ÷19 = 25 (pieces)

1 1. Profit and loss problem:

(1) One profit and one loss: (profit+loss) ÷ (difference between two distributions per person) = number of people.

(2) There are profits both times: (big profits-small profits) ÷ (the difference between each person's two distributions) = number of people.

(3) Two losses: (big loss-small loss) ÷ (the difference between two distributions per person) = number of people.

(4) One loss, just once: loss ÷ (difference between two distributions per person) = number of people.

(5) One profit, just one: surplus ÷ (the difference between two distributions per person) = number of people.

Example: "Children are divided into peaches, each person 10 peaches, 9 less, 8 more, and 7 peaches each. Q: How many children and peaches are there? "

Solution (7+9) ÷ (10-8) =16 ÷ 2 = 8 (a) ..............................................................................................................

10×8-9=80-9=7 1 (pieces) ....................................................................................................................................

12. Travel problems:

(1) average speed: average speed =

(2) Meet and catch up:

Encounter (deviation): distance/speed and = time.

Catch-up: distance/speed difference = time

(3) running water:

Downstream speed = ship speed+current speed;

Current speed = ship speed-current speed.

When two ships are sailing in opposite directions, the speed of ship A+the speed of ship B = the still water speed of ship A+the still water speed of ship B..

When two ships sail in the same direction, the still water speed of the rear (front) ship-the still water speed of the front (rear) ship = the speed at which the distance between the two ships decreases (widens).

(4) The train crosses the bridge:

Time for the train to get on the bridge completely = (bridge length-train length) ÷ train speed.

Time taken for the train to get off the bridge from the beginning = (bridge length+train length) ÷ train speed.

(5) Multiple meetings:

Go in the opposite direction, the first meeting is one kilometer away from A, and the second meeting is one kilometer away from B, so the two places are separated.

S = 3a-b (km)

(6) Clock problem:

The clock face is divided into 60 squares according to the minute hand. The rotation speed of the hour hand is the minute hand, which can be tracked every hour.

The hour hand and minute hand overlap 22 times a day and night, 44 times vertically, and do 180o22 times.

13. Exclusion principle:

A+B= +

A+B+C= + + + -

Where = e

14. Cattle eat grass:

Original grass quantity = (number of cattle-grass quantity growing every day) × days, where it is generally assumed that the grass quantity growing every day is X.

If in doubt, please consult the public education enterprises in China.