The distance between the point corresponding to a number on the number axis and the origin (zero point) is called the absolute value of the number. Absolute value can only be non-negative. Algebraic definition: | a | = a(a >；; 0)| a | =-a(a & lt； 0) |a|=0(a=0) means that the absolute value of a positive number is itself, and the absolute value of a negative number is its reciprocal (note: reciprocal is a sign transformation).

Geometric meaning

On the number axis, the distance from the origin of a number is called the absolute value of the number. For example, it is the distance between the point represented on the exponential axis and the origin. This distance is 5, so the absolute value of is 5.

Algebraic meaning

The absolute values of positive numbers and 0 are themselves, the absolute values of negative numbers are their opposites, and the absolute values of 0 are equal to the absolute values of two numbers whose opposites are 0. The absolute value of a is represented by |a |, which is read as "the absolute value of a".

Edit this application.

The absolute value of a positive number is itself. The absolute value of a negative number is its reciprocal. , the absolute value is non-negative, ≥0. The absolute value of 0 is still zero. The absolute value of a special zero is both his own and his opposite number. Write |0|=0 |3|=3 =|-3|=3 When a≥0, |a|=a is A.

Edit the related properties of this paragraph.

No matter the algebraic or geometric meaning of absolute value, it reveals the following properties of absolute value: (1) The absolute value of any rational number is a number greater than or equal to 0, which is the nonnegativity of absolute value. (2) There is only one number whose absolute value is equal to 0, which is 0. (3) There are two numbers whose absolute values are equal to the same positive number, and the two numbers are opposite. (4) The absolute values of two opposite numbers are equal. Absolute value equations sum inequality: (1) | a | *| b | = | ab | (2) | a |/| b | = | a/b | (b ≠ 0) (3) a 2 = | a | 2 This property is generally used in a unary with absolute value. = | x+y | & lt； =|x|+|y| It can be concluded that | x |-| y | < = | x-y | < =|x|+|y|, because | x|-|-y| < = | x+(-y) | < =|x|+|-y|

Edit the absolute inequality of this paragraph.

(1) To solve the absolute value inequality, we must try to remove the absolute value symbol in the formula and convert it into a general algebraic type to solve it. (2) There are two main methods to prove absolute inequality: a) removing the absolute symbol and transforming it into general inequality proof: method of substitution, discussion method and flat method; B) Using the inequality: | a |-| b | ≦| a+b ||| | a |+| b |, the formula within the absolute value is split and combined in this way, and the items are added and subtracted, so that the formula to be proved is related to the known formula.

Edit this paragraph about the dispute of absolute value.

If we take 1 km south as+1 and 1 km north as-1, and find the absolute value of-1, the result is 1 km south? ! Obviously, there is something wrong here. The problem is that both positive and negative numbers are relative numbers, not absolute numbers, so the absolute value of relative numbers should be unsigned numbers, not positive numbers. Therefore, the unsigned number is not just a zero, there should be other unsigned numbers! So there is |-1 | =|+1| =1,where1is not a positive number, but an unsigned number like 0!

Possible calculation methods of unsigned numbers;

If three women are -3 and four men are +4, how many people are there in a * * *? The calculation method is to add the absolute values of two numbers, which is seven people. If you ask the difference between men and women, the calculation method is to add the relative numbers, which is+1. If south 1 km is taken as+1, and 2 km north is taken as -2, how many kilometers has a * * * traveled? The calculation method is to add the absolute values of two numbers, which is 3 kilometers. If you are asked how many kilometers you have traveled, the calculation method is the sum of relative numbers, which is-1. If 10 degree to zero degree is+10, and negative 5 degree is -5, ask: a * * *, how many degrees is the difference? The calculation method is to add the absolute values of two numbers, that is, 15 degrees. If you ask how many degrees the sum of temperatures is, the calculation method is to add the relative numbers, which is +5. If the title doesn't say anything positive, for example, the postman delivers letters to the south 10 meters and then to the north 5 meters. Before doing the problem, he must write: remember what is positive, generally don't write again, because it is either positive or negative, only one is known. So the concept of absolute value is also controversial. Some people don't think the absolute value must be positive. This shows that mathematics is also developing constantly. Moreover, the mathematics we see is only a stage in the historical process, and it has not affected normal learning.

The absolute value is an unsigned number.

When yin and yang are in balance, things show neither yin nor yang, that is, the state of zero (zero does not mean anything, but actually represents balance, (-1)+(+ 1)=0, which is not balance! )。 So the so-called (-1)+(+3)=+2 means the imbalance of Yin and Yang. There are two more yang than yin, so it is +2. And the so-called (+1)+(-3)=-2, the reason is the same, but at this time, Yin is the majority, and Yin is two more than Yang. Men, women and men are the same. Three men (+3) and two women (-2) are unbalanced, so there is (+3)+(-2)=+ 1, and men have one more than women. So is the charge. If we rub the glass rod with silk, the charge on the glass rod will be unbalanced and the glass rod will be charged. For example, (0)-(-2)=+2, that is, Yin is reduced under balance, and the result is Yang, here is +2. So what is the absolute value? The absolute value is an unsigned number. For example, three people, we don't talk about men and women, only about people, so what symbol should we use to represent them? Obviously it can't be expressed by symbols. Here 3 can only be an unsigned number, if we record it as 3 (note that 3 here is different from +3, +3 is a signed number, and 3 is an unsigned number). In this way, when we ask, how many people are there in three men (assuming +3) plus three women (assuming -3) and one * * *, we must add them up in absolute terms, that is, |+3 |-3 | = 6, that is, six people. This is an unsigned number. According to the previous mathematical concept, it is wrong for us to understand 6 as a positive number because it becomes six people.