Positive and negative numbers

The concepts of positive and negative numbers

A number (1) greater than 0, such as 3, 1.5,1/2,584, is called a positive number. The number of primary school students is positive except 0, and the positive number is greater than 0.

(2) Like -3,-1.5,-1/2, -584, etc. , supplement? -? Numbers with a sign (pronounced negative) are called negative numbers. Negative number is less than 0.

(3) Zero is neither positive nor negative, and zero is the boundary between positive and negative numbers.

note:

(1) For emphasis, positive numbers can sometimes be preceded by? +? (Pronunciation is positive), for example: 3, 1.5, or +3,+1.5.

(2) The concepts of positive numbers and negative numbers cannot be simply understood as: use? +? The number of the symbol is positive, use? -? The number of the symbol is negative.

For example, must -a be negative? The answer is not necessarily. Because the letter a can represent any number, if a represents a positive number, then -a is a negative number; If a stands for 0, -a is still 0; When a stands for negative number, -a is not negative (in this case, -a is positive).

Positive and negative numbers represent

Positive and negative numbers are generated according to actual needs. With the development of society, the natural numbers, fractions and decimals in primary school can no longer meet the actual needs. For example, some quantities with opposite meanings, such as income 200 yuan, expenditure 100 yuan, zero plus minus six, etc. , not only has the opposite meaning, but also represents a certain number. How to express them?

We define the quantity of one meaning as positive and the quantity of another opposite meaning as negative, thus producing positive numbers and negative numbers.

When positive and negative numbers are used to represent quantities with opposite meanings, which meaning is positive can be chosen at will, but it is customary to put it? Advance, rise, income, temperature above zero? Wait until the rules are correct, then let it go? Retrograde, descent, expenditure, sub-zero temperature? Negative and other regulations.

rational number

Some concepts of knowledge point 1 rational number

Rational Numbers: Integers and fractions are collectively called rational numbers.

Note: (1) Sometimes, for the need of research, an integer can also be regarded as a number with a denominator of 1. In this case, the score contains an integer. But the fraction in this lecture does not include the fraction with denominator of 1

(2) Because fractions can be interchanged with finite decimals and infinite circulating decimals, and all the above decimals can be expressed by fractions, we regard both finite decimals and infinite circulating decimals as fractions.

(3)? 0? That is, neither positive nor negative, but? 0? Is an integer.

Integers include positive integers, zero and negative integers. For example: 1, 2, 3, 0,-1, -2, -3 and so on.

Scores include positive and negative scores, such as: 1/2, 0.6,-1/2, -0.6 and so on.

Knowledge point 2 Classification of rational numbers

(1) According to the relationship between integer and fraction:

(2) According to the relationship between positive number, negative number and 0:

Note: Generally, positive numbers and 0 are called non-negative, negative numbers and 0 are called non-positive numbers, positive numbers and 0 are called non-negative integers (also called natural numbers), and negative numbers and 0 are called non-positive integers.

If the number is represented by letters, a >: 0 means that a is a positive number; A<0 means that a is a negative number; Answer? 0 means that a is non-negative; Answer? 0 means a is not positive.

Knowledge point 3 axis

The number axis is an important tool to understand the concept and operation of rational numbers, and the idea of combining numbers with figures representing numbers (such as the number axis) is an important idea in learning mathematics. As Professor Hua said:

Numbers and shapes are interdependent. How can we divide them into two sides?

There is little intuition in counting missing shapes, and it is difficult to be nuanced in counting several shapes.

The combination of numbers and shapes is good in all aspects, but everything is wrong when it is separated.

Don't forget, the unity of geometry and algebra is always linked and never separated!

The first marriage between numbers and axes established the corresponding relationship between numbers and points in a straight line, revealed the internal relationship between numbers and shapes, and became the basis of the combination of numbers and shapes.

1. Definition of the number axis: The straight line defining the origin, positive direction and unit length is called the number axis.

The definition of number axis contains three meanings:

(1) axis is a straight line, which can extend to both ends indefinitely;

(2) The number axis has three elements: origin, positive direction and unit length, which are indispensable;

(3) The selection of origin, positive orientation and determination of unit length are all based on actual needs? Rules? (usually with the right as the positive direction).

2. Number of axis drawings:

(1) Draw a straight line (usually a horizontal line).

(2) Select a point on a straight line as the origin, and use this point to represent zero (mark? 0? )。

(3) Determine the positive direction (generally, the right direction is positive), which is indicated by an arrow.

(4) Choose an appropriate length as the unit length, and take a point every other unit length from the origin to the right, which is expressed as 1, 2, 3 in turn; From the origin to the left, take a point every other unit length, which is expressed as-1, -2, -3 in turn.

note:

(1) The location of the origin and the size of the unit length can be appropriately selected according to the actual situation;

(2) When determining the unit length, according to the actual situation, sometimes one point can be taken from every two (or more) unit lengths, which is expressed as 2, 4, 6, from the origin to the right; From the origin to the left, expressed as -2, -4, -6, and;

3. The relationship between points on the number axis and rational numbers:

All rational numbers can be represented by points on the number axis. Positive rational numbers can be represented by points to the right of the origin, and negative rational numbers can be represented by points to the left of the origin and small origin.

4. Compare the size of rational numbers with the number axis:

Of the two numbers displayed on the axis, the number on the right is always greater than the number on the left. Positive numbers are all greater than 0; Negative numbers are all less than 0; Positive numbers are greater than all negative numbers.

Knowledge point 4 reciprocal

Definition of 1. Reciprocal

(1) Geometric definition of antipodal: The number represented by two points with the same distance from the origin on both sides of the number axis is called reciprocal antipodal. For example, 4 and -4 are reciprocal.

(2) Algebraic definition of antipodes: There are only two numbers with different signs (they are all the same except different signs), and we say that one of them is the antipodes of the other.

2. The nature of the reciprocal:

Any number has a reciprocal, and there is only one. The reciprocal of a positive number is negative, the reciprocal of a negative number is positive, and the reciprocal of 0 is 0.

0 is the only number whose opposite number is equal to itself. Conversely, if a=-a, then a must be 0.

3. The characteristics of reciprocal:

If a and b are reciprocal, then a+b=0 (or a=-b).

If a+b=0 (or a=-b), then a and b are reciprocal.

4. How to find the reciprocal of a number: (see book)

5. Simplification of multiple symbols

(1) Add a before the number? +? Number, still the same as the original number, such as +5=5, +(-5)=-5.

(2) add a before a number? -? The number becomes the reciprocal of the original number. For example, -(-3) is the inverse of -3, so -(-3)=3.

The concept of absolute value of knowledge point 5

1. Geometric definition of absolute value: the absolute value of a number A is the distance between the point representing the number A on the number axis and the origin, and the absolute value of the number A is recorded as? 丨丨 answer?

2. Algebraic definition of absolute value: the absolute value of a positive number is itself; The absolute value of a negative number is its reciprocal; The absolute value of 0 is 0.

Comparison of Rational Numbers in Knowledge Point 6

Positive numbers are all greater than 0, negative numbers are all less than 0, positive numbers are greater than all negative numbers, and two negative numbers are smaller in absolute value.

When using the number axis, the number on the right side of the number axis is always greater than the number on the left side.

Addition and subtraction of rational numbers

Addition of rational numbers

The operation of combining two rational numbers into one rational number is called the addition of rational numbers.

The two rational numbers added have the following situations:

(1) Both numbers are positive numbers;

(2) Both numbers are negative;

(3) The signs of two numbers are different, that is, one is positive and the other is negative;

(4) One is a positive number and the other is 0;

(5) One is negative and the other is 0;

(6) Both are 0.

Knowledge point 2 rational number addition rule

(1) Add two numbers with the same sign, take the same sign, and add the absolute values.

(2) Add two numbers with different absolute values, 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. Two opposite numbers add up to 0.

(3) When a number is added to 0, the number is still obtained.

Knowledge point 3 Arithmetic of rational number addition

(1) additive commutative law: a+b = b+a.

(2) Additive associative law: (a+b)+c=a+(b+c).

Knowledge point 4 rational number subtraction rule

Subtracting a number is equal to adding the reciprocal of this number, that is, a-b=a+(-b).

Knowledge point 5 rational number addition and subtraction mixed operation

1. The significance of unifying addition and subtraction of rational numbers into addition

The subtraction in the mixed operation of addition and subtraction of rational numbers can be converted into addition according to the subtraction rule of rational numbers.

In this way, the original mixed operation is unified into addition operation. The formula after unification into addition is the sum of several positive or negative numbers. Sometimes we call this formula algebraic sum.

2. The method of rational number addition and subtraction mixed operation

(1) The subtraction in rational number mixing operation is converted into addition by using the subtraction rule.

(2) Applying the law of addition, additive commutative law and the law of additive association.

Multiplication and division of rational numbers

Knowledge point 1 rational number multiplication rule

Multiply two numbers, the same sign is positive, the different sign is negative, and then multiply by the absolute value. Any number multiplied by 0 is 0.

The concept of reciprocal of knowledge point 2

Two numbers whose product is 1 are reciprocal.

Because of one? 1/a(a？ 0), so when A is a rational number other than 0, the reciprocal of A is 1/a, and if A and B are reciprocal, ab= 1.

Knowledge Point 3 Popularization of Rational Number Multiplication Rule

(1) Multiplies several numbers that are not equal to 0, and the sign of the product is determined by the number of negative factors. When there are odd negative factors, the product is negative; When there are even negative factors, the product is positive.

(2) When several numbers are multiplied, as long as one factor is 0, the product is 0.

Knowledge Point 4 Arithmetic of Rational Number Multiplication

(1) Multiplication commutation law: ab=ba.

(2) multiplicative associative law: (ab)c=a(bc).

(3) Distribution law: a(b+c)=ab+ac.

Knowledge point 5 rational number division rule

(1) divided by a number is equal to multiplying the reciprocal of this number. Answer? b=a？ 1/b(b？ 0)。

(2) Divide two numbers, the one with the same sign is positive, and the one with different signs is negative, and divide by the absolute value. Divide 0 by any number that is not equal to 0 to get 0.

Knowledge point 6 mixed operation of multiplication and division of rational numbers

In addition to transfer, determine the symbol.

Knowledge Point 7 Elementary Arithmetic of Rational Numbers

Multiply first, then divide, then add and subtract. If there are brackets, count them first. In the operation at the same level, the order from left to right should be followed.

Power of rational number

The meaning of knowledge point 1 rational number power

Knowledge point 2 Properties of rational number multiplication operation

The power of any positive number is positive; The odd power of a negative number is negative and the even power of a negative number is positive. Any degree of 0 is 0.

Knowledge point 3 Operation sequence of rational number mixed operation

Calculate the power first, then multiply and divide, and finally add and subtract. If there are brackets, count them first.

Knowledge point 4 scientific counting method

Knowledge point 5 the significance of learning divisor

In production practice and real life, there are not only a large number of accurate figures, but also a large number of approximate figures. Approximation is a number close to reality.

There are two reasons for the approximate figure: First, it is sometimes impossible to get a completely accurate figure, for example, the radius of the sun is about 696,000 kilometers; Second, sometimes it doesn't have to be completely accurate, such as buying 10 Jin of rice, sometimes it may be a little more, sometimes it may be a little less.

Knowledge point 6 significant figures

The rounded approximation, from the first non-zero number on the left to the most accurate number, is called the significant number of this number.

Method Skill 1: In a formula that only contains multiplication and division, it can be determined by? Negative? The number of symbols determines the symbol of the result. ? Negative? When there are odd numbers, the result is negative; ? Negative? When the number of symbols is even, the result is positive.

Method tip 2: Fraction, decimal multiplication and division mixed operation, usually turning decimal into fraction and treating fraction as false fraction. In the multiplication and division of Huasong product form, the sign of product sum must be determined first. In the case of braces, the general method of removing brackets is from the inside out, that is, removing small, medium and curly braces in turn, or from the outside in. When performing mixed operations, we should pay attention to two points: one is the operation order, and the other is the operation symbol.

Method Tip 3: Using the arithmetic rules of rational numbers flexibly and changing the operation order by adding or subtracting brackets appropriately can often simplify the operation. Rounding, grouping, splitting, elimination, consistent decomposition and overall processing are commonly used methods and skills in rational number operation.