Rational Numbers: Integers and fractions are collectively called rational numbers.
Number axis: The straight line defining the positive direction, origin and unit length is called number axis.
Inverse number: two numbers distributed on both sides of the origin and at the same distance from the origin are called inverse numbers.
Absolute value: the distance from a point on the number axis to the origin is called the absolute value of the rational number represented by this point. (absolute value)
Solution of absolute value of rational number: the absolute value of positive number is itself; The absolute value of a negative number is its reciprocal; The absolute value of 0 is still 0.
Rational number addition rule:
1. Add two numbers with the same sign, and add the absolute values of the two numbers.
2. Add two numbers with different signs, take the sign of the addend with larger absolute value, and subtract the one with smaller absolute value from the one with larger absolute value; The sum of two opposite numbers is 0.
Add 3.0 to any rational number and you still get this rational number.
4. additive commutative law and the law of association are still valid in the addition of rational numbers.
Additive commutative law: A+B = B+A.
Additive associative law: (a+b)+c=a+(b+c)
Algebraic sum: The sum of several rational numbers with the plus sign omitted is called the algebraic sum of these numbers. (algebraic sum)
Support removal rules:
1. When there is a "+"in front of the bracket, the sign of the number in the bracket remains unchanged after removing the bracket and the "+"in front of it.
2. When there is a "-"sign in front of the bracket, when the bracket and the "-"sign in front of it are removed, the sign of the number in the bracket will change.
Parenthesis rule:
1. When the brackets with "+"in front of them are filled in, the signs of the numbers in the brackets remain unchanged.
2. When the brackets with "-"in front are filled in, the symbols of the numbers in the brackets should be changed.
Rational number multiplication (multiplication) rule:
1. Two numbers with the same sign are multiplied as positive, two numbers with different signs are multiplied as negative, and the absolute value is multiplied.
2. Multiply any rational number by 0 to get 0.
Multiplicative commutative law: ab=ba
Law of multiplicative association: (ab)c=a(bc)
Multiplication and distribution law: a(b+c)=ab+ac.
Rational number division rule:
1. Multiply two numbers with the same sign to get a positive number, divide two numbers with different signs to get a negative number, and then divide by the absolute value.
2.0 is not divisible, and dividing 0 by any number other than 0 will get 0.
When a number is divided by a number that is not 0, it is equal to multiplying the reciprocal of this number. (Countdown)
Power: the operation of multiplying several identical factors is called power, and the result of power is called power. (Power) If there are n A's multiplied, it can be written as an, where an is called n power, also called n power .. A is called the base of a power, and A can take any rational number; N is called the exponent of power and can be any positive integer.
Approximation: A number similar to an exact value is called an approximation of the exact value. (approximate value)
Significant digit: All digits from the first non-zero digit on the left to the nearest digit are called significant digits of this approximation. (significant figure, significant figure)
Algebraic expression: representing letters and letters; A formula for multiplying or adding numbers and letters, a single number or letter, is called an algebraic formula. (algebraic expression)
Algebraic value: the value of algebra is calculated by replacing the letters in algebra with numerical values according to the original operational relationship of algebra.
Monomial: An algebraic expression consisting of the product of letters and numbers, or a single letter and number is called a monomial. (single item)
Coefficient: the numerical factor in a single item is called the coefficient of the item. (coefficient)
Number of times of a monomial: The sum of the indices of all the letters in the monomial is called the number of times of the monomial. (degree)
Polynomial: An algebraic expression consisting of the sum of several monomials is called a polynomial.
Term: each monomial that constitutes a polynomial is called a polynomial term ($ TERM)
Constant term: the term without letters in polynomial is called constant term.
Degree of Polynomial: The degree of the highest term in a polynomial is called the degree of the polynomial.
Algebraic expression: monomials and polynomials are collectively called algebraic expressions. (Integral expression)
Equation: An equation with an "=" sign indicating equality is called an equation.
Equation: An equation with an unknown number is called an equation.
Solution of the equation: the value of the unknown that can make the left and right sides of the equation equal is called the solution of the equation.
The root of an equation: the solution of an equation with only one unknown number, also called the root of the equation.
Solving an equation: The process of finding the solution of an equation is called solving an equation.
Equality is the basic nature;
The addition (or subtraction) of the same number or algebraic expression on both sides of the 1. equation still holds.
2. Multiply (or divide) both sides of the equation by the same number (divisor is not 0), and the equation still holds.
One-dimensional linear equation: An equation with only one unknown number and the degree of the unknown number is 1 is called one-dimensional linear equation.
Moving term: a symbol that moves any term on the left or right side of the equation to the opposite side and changes its nature. This deformation is called transposition of terms.
The main steps of solving application problems with column equations;
1. Read the question carefully, understand the meaning of the question, find out the quantitative relationship in the question, and find out the equivalent relationship between them.
2. Set an unknown number, and use the algebraic expression containing the unknown number to represent the equivalence relation involved in the topic.
3. List the equations according to the equation relationship.
4. Find the solutions of the listed equations.
5. Check whether the solution of the equation conforms to the practical significance of the problem.
6. write the answer.
Savings equivalence relation: after-tax interest = principal × deposit × interest rate ×( 1-20%)
Work equivalence relation: total work = work efficiency × working time = total work ÷ working time.
Straight line: A line that does not bend and can extend indefinitely is called a straight line. (straight line)
The nature of a straight line: only one straight line passes through two points.
Ray: A point on a straight line and the part beside it are called rays. This point is called the endpoint of the ray. (end point)
Line segment: two points on a straight line and the part between them are called line segments, and these two points are called the endpoints of line segments. (extreme point)
Distance: The length of the line segment connecting two points is called the distance between two points.
Distance attribute: between two points, the line segment is the shortest.
Definition of midpoint of line segment: If point C is a point on line segment AB and AC=AB is satisfied, point C is called the midpoint of line segment AB. (midpoint, midpoint)
Angle: The figure formed by two rays drawn from a point is called an angle and the vertex of the angle, and these two rays are called the edges of the angle. Angle can also be regarded as a graph formed by the end position and the start position of rotation when the light rotates around its endpoint. The starting position of rotation is called the starting edge of angle, and the ending position of rotation is called the ending edge of angle.
Classification criteria of angles:
1. When the starting edge and the ending edge of the angle are on the same straight line. This angle is called a right angle. (Angle of straight line)
2. The angle formed by overlapping two sides of an angle after rotation is called fillet. (rounded corners)
3. Half of a right angle is called a right angle. (right angle)
4. An angle smaller than right angle is called acute angle. (acute angle)
5. The angle greater than right angle and less than right angle is called obtuse angle.
Angular bisector: A ray passing through the vertex of an angle divides an angle into two equal angles, so this ray is called the angular bisector of this angle. (Angular bisector)
Intersection line: two straight lines with only one common point are called intersection lines, and this common point is called intersection point. There is only one intersection point when two straight lines intersect.
Vertical line:
1. Among the four angles formed by the intersection of two straight lines, if one of them is equal to 90, then the two straight lines are perpendicular to each other, and the verticality is indicated by the symbol "⊥", and the intersection of the two straight lines is called the foot of the vertical line.
2. There is one and only one straight line perpendicular to the known straight line.
Distance from point to straight line: the line segment from a point outside the straight line to the vertical line is called the vertical line segment. The length of a vertical section starting from a point outside a straight line is called the distance from the point to the straight line.
Parallel lines: Two straight lines that do not intersect the same plane are called parallel lines.
(2) Archimedes
King Shiloh of Syracuse asked the goldsmith to make a crown out of pure gold. Because it is suspected that there is silver mixed in it, Archimedes is invited to identify it. When he entered the bathtub to take a bath, the water overflowed outside the bathtub, so he realized that although the weight of objects made of different materials was the same, the discharged water would be different because of their different volumes. According to this truth, it can be judged whether the crown is adulterated.
Zu Chongzhi and his son Zuxuan (also a famous mathematician in China) solved the calculation of the volume of a sphere with ingenious methods. They adopted a principle at that time: "If the power supply potential is the same, the products should not be different." That is, two entities located between two parallel planes are cut by any plane parallel to these two planes. If the areas of two sections are always equal, the volumes of two solids are also equal. This principle is in the west.
19101012. Hua was born in Jintan County, Jiangsu Province. He comes from a poor family and is determined to study hard. When I was in middle school, in a math class, the teacher gave the students a famous question: "There is a number, three places, and two;" 5 digits, 5 numbers, and the remaining 3; There are seven places in seven, and there are two left. What's the number of this? " While everyone was thinking, Hua stood up and said, "23". His answer surprised the teacher and won his praise. From then on, he fell in love with mathematics.
After finishing the first grade of junior high school, Hua dropped out of school because of his poor family, so he had to stand in front of the counter for his parents, but he still insisted on teaching himself mathematics. Through his unremitting efforts, his paper "Why can't the solution of Su Jiaju's algebraic quintic equation be established" was discovered by Professor Xiong Qinglai, head of the Department of Mathematics of Tsinghua University, and invited him to Tsinghua University; Hua was hired as a university teacher, which is unprecedented in the history of Tsinghua University.
1In the summer of 936, Hua, an outstanding mathematician, was a visiting scholar at Cambridge University in England for two years. At this time, the news of the Anti-Japanese War spread all over Britain, and he returned to the motherland with strong patriotic enthusiasm to give lectures for The National SouthWest Associated University.
China attaches great importance to the direct application of mathematical methods in industrial and agricultural production. He often goes deep into factory guidance, popularizes the application of mathematics and writes popular science books.
Hua has also set a shining example for young people to become self-taught. He is a self-taught mathematician without a college degree. He said: "I am not afraid of difficulties and study hard. This is my main experience in learning mathematics well." "The so-called genius is to rely on unremitting efforts.
(3)(4) preview before class, find difficulties, attend classes and grasp the key points.
Review after class, find rules and practice typical exercises repeatedly.
(5) Too much, or your own good.