Definition of absolute value
The distance between the point corresponding to a number on the number axis and the origin (O point) is called the absolute value of the number. Absolute value can only be non-negative.
Algebraic definition:
a = a(a & gt； 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, and the absolute values of negative numbers are their opposites.
The absolute values of two opposite numbers are equal.
What is the absolute value of a? Answer? Express delivery. Read? The absolute value of a? .
Application 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 any rational number is non-negative, that is, the absolute value of any rational number is? 0。
The absolute value of 0 is still 0.
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 a。
When a<0, a =-a.
There is a-b = b-a.
Comparing the sizes of two negative numbers, the absolute value is larger but smaller.
For example: if 2(x? 1)? 3 + 2(y？ 4) =0, then x = _ _ _, y = _ _ _. (It is an absolute value).
Answer:
2(X- 1)-3=0
X=5/2
2Y-8=0
Y=4
The absolute values of a pair of opposites are equal;
The absolute value of example +2 is equal to? The absolute value of 2 (because they are equal in unit length from the origin on the number axis)
Computer language implementation
In computer language, the binary number of a positive number is 0, and the binary number of a negative number is 1.
In a 32-bit system, the number of bytes is 4, and the expression for finding the absolute value is:
ABS(x)=(x & gt； & gt3 1)^ x-(x & gt； & gt3 1)
Code is usually implemented by macros:
# Define ABS (x) ((x) > & gt3 1)^(x))-((x)& gt； & gt3 1)
Note: "> > operators with" "and" > > move to the left, and ""XOR.
Some properties of absolute value
No matter the algebraic meaning or geometric meaning of absolute value, the following related properties of absolute value are revealed:
(1) The absolute value of any rational number is a number greater than or equal to 0, which is not negative.
(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 equality and inequality:
( 1) a * b = ab
(2) a / b = a/b (b？ 0)
(3)a^2=·^2
This property is generally used in quadratic equations with absolute values. For example, x^2-3 x +2=0 2 = 0 2 = 0 can be changed to
X 2-3 x+2 = 0, (x- 1) (x-2) = 0, x= 1 or 2, x=? 1 or? 2
(4)x-y & lt； = x+y & lt； = x + y
It can be concluded that X-Y
Absolute inequality
(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) Remove the sign of absolute value and turn it into a general inequality proof: method of substitution, discussion method and plate method;
B) Use the inequality: a-b ≦ a+b ≦ a+b, so that the formula in absolute value should be split and combined, and items should be added and subtracted to link the formula to be proved with the known formula.
Argument about 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!
Therefore,-1 =+1 = 1, where 1 is not a positive number, but an unsigned number similar to 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 the sum of 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, that 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 (say +3) plus three women (say -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.
Examples of application of absolute value
The absolute value of 0 is both its own number and the opposite number (the opposite number of 0 is itself, but (-0) does not exist), write |0|=0.
|3|=3 =|-3|
When a. 0，|a|=a
When a<0, |a|=-a
Existence |a-b|=|b-a|
Comparing the sizes of two negative numbers, the absolute value is larger but smaller.
For example, if | 2 (x- 1)-3 |+2 (y-4) | = 0, then x = _ _ _, and y = _ _ _ _.
Answer:
2(X- 1)-3=0, 2Y-8=0, because after removing the brackets, the number in brackets should be multiplied by two (|2(y-4)|).
The solution is X=5/2 and Y=4.
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