The real number knowledge point of the first volume of eighth grade mathematics is 1.
1, the concept and classification of real numbers
② Irrational number
The decimal of infinite cycle is called irrational number.
When understanding irrational numbers, we should grasp the moment of "infinite non-circulation", which can be summarized into four categories:
Endless prescriptions, such as √7, 3 √2, etc. ;
Numbers with specific meanings, such as pi, or simplified numbers containing pi, such as pi/? +8 and so on;
Numbers with specific structures, such as 0.101001001….
Some trigonometric function values, such as sin60.
2. Reciprocal, reciprocal and absolute value of real numbers
① Countdown
A real number and its inverse are a pair of numbers (only two numbers with different signs are called inverse numbers, and the inverse of zero is zero). Seen from the number axis, the points corresponding to two opposite numbers are symmetrical about the origin. If a and b are opposites, then a+b=0, a=-b, and vice versa.
② Absolute value
On the number axis, the distance between the point corresponding to a number and the origin is called the absolute value of the number. The absolute value of |a|≥0 .0 is itself, and it can also be regarded as its inverse. If |a|=a, then a ≥ 0; If |a|=-a, then a≤0.
③ reciprocity
If A and B are reciprocal, there is ab= 1, and vice versa. The numbers whose reciprocal equals itself are 1 and-1. 0 has no reciprocal.
④ Number axis
The straight line that specifies the origin, positive direction and unit length is called the number axis (pay attention to the three elements specified above when drawing the number axis).
When solving problems, we should really master the idea of combining numbers with shapes, understand the one-to-one correspondence between real numbers and points on the number axis, and use them flexibly.
⑤ Estimate
3, square root, arithmetic square root and cubic root
① arithmetic square root
Generally speaking, if the square of a positive number X is equal to A, that is, x2=a, then this positive number X is called the arithmetic square root of A. In particular, the arithmetic square root of 0 is 0.
Property: Positive numbers and zeros have only one arithmetic square root, and the arithmetic square root of 0 is 0.
② Square root
Generally speaking, if the square of a number x is equal to a, that is, x2=a, then this number x is called the square root (or quadratic root) of a.
Property: a positive number has two square roots in opposite directions; The square root of zero is zero; Negative numbers have no square root.
The operation of finding the square root of a number is called square root. Pay attention to the double nonnegativity of √a: √ a ≥ 0; a≥0
③ Cubic root
Generally speaking, if the cube of a number X is equal to A, that is, x3=a, then this number X is called the cube root (or cube root) of A. ..
Representation: 3 √a A a.
Property: positive numbers have positive cubic roots; Negative numbers have negative cubic roots; The cube root of zero is zero.
Note: -3 √a=3 √-a, which means that the negative sign in the cube root symbol can be moved beyond the root sign.
The real number knowledge points in the first volume of mathematics in grade eight.
1, real number comparison
① Real number comparison size
Positive numbers are greater than zero, negative numbers are less than zero, and positive numbers are greater than all negative numbers;
The number represented by two points on the number axis is always larger on the right than on the left;
Two negative numbers, the larger one has the smaller absolute value.
② Several common real number comparison methods.
Number axis comparison: The number on the right is always greater than the number on the left of the two numbers represented on the number axis.
Difference comparison: let a and b be real numbers.
a-b & gt； 0? a & gtb；
a-b=0？ a = b；
a-b & lt； 0? a
Quotient comparison method: let a and b be two positive real numbers,
Absolute value comparison method: Let A and B be two negative real numbers, then ∣ A ∣ > ∣b∣？ A< B.<p = "">
Ping method: Let A and B be two negative real numbers, then A2 >；; b2？ a
2. Calculation of arithmetic square root (quadratic root)
(1) contains the quadratic root sign "√"; The root sign a must be non-negative.
② Nature:
(3) If the operation result contains the form of "√", the following requirements must be met:
The factor of the square root is an integer, and the factor is an algebraic expression.
The number of square roots does not contain factors or factors that can be completely opened.
3. Real number operation
① Six operations: addition, subtraction, multiplication, division, multiplication and extraction.
② Operation sequence of real numbers
First calculate the power sum root, then multiply and divide, and finally add and subtract. If there are brackets, count them first.
③ Operation law
Additive commutative law a+b= b+a
Additive associative law (a+b)+c= a+( b+c)
Multiplicative commutative law ab= ba
Multiplicative associative law (ab)c = a( bc)
Distribution law of multiplication to addition a( b+c )=ab+ac
How to learn primary school mathematics well
First, proper study methods and habits.
1, prepare well before class and take the initiative in class. The quality of preparation before class directly affects the effect of listening to lectures.
2. Listen carefully in class and take notes.
3. Review in time and turn knowledge into skills.
4. Seriously finish homework, form skills and skills, and improve the ability to analyze and solve problems.
5. Summarize in time and systematize what you have learned.
Therefore, we will continue to "preview before listening" in the future; Review first, then do your homework; A good habit of summarizing periodically and periodically.
Second, good learning motivation and interest.
Learning motivation is the direct driving force to promote your study. Hua said: "if you are interested, you will never get tired, you will never get tired." Therefore, you will find time to study. " I'm glad you like math class. I hope you can have more fun in math learning.
Third, strong will.
In the process of learning mathematics, you have encountered many difficulties, large and small. You can strengthen your confidence, face difficulties bravely and overcome difficulties, which requires a strong will. Facing difficulties with confidence and trying to overcome them is a sign of perseverance. You have this very valuable quality, when you encounter difficulties or setbacks in your study, you will not be discouraged; When you get good grades, you are not complacent, but good at summing up experiences and lessons, exploring learning rules and methods, and going forward bravely. Only in this way can we achieve good results.
Fourth, self-confidence and diligence
Mathematician Zhang Guanghou said: "There is no shortcut on the road of learning mathematics, let alone opportunism. Only by diligent study and perseverance can we achieve excellent results. " You know the truth that practice makes perfect. After repeated practice, you really achieved good results.
It is necessary to prepare for the exam calmly and calmly with a good attitude. Not impetuous before the exam can make you review with high speed and quality. In addition, facing the exam with a positive attitude can make you play to your normal level or even surpass it.
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