Number axis

A specific geometry

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This entry is made by Compilation and application of scientific encyclopedia entries of "Science Popularization China" to examine.

Number axis, a specific geometric figure. A straight line is composed of countless points aggregate There are countless real numbers, including positive real numbers, zero and negative real numbers. Because of their commonness, real numbers are represented by countless points on a straight line. At this time, a straight line specifying the origin, positive direction and unit length is used to represent the real number. When the right side is specified as a positive direction, the number represented by the upper point on the right is always greater than the number represented by the upper point on the left. Positive numbers are greater than zero, and zero is greater than negative numbers.

Chinese name: Number axis
Foreign name: number axis
Classification: mathematics

Inverse number: Only two numbers with different symbols, the others are the same
absolute value: Distance from point to origin
Role: Compare Size
inventor: Descartes

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brief introduction

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A straight line is a collection of countless points. Real numbers include positive real numbers, zero, and negative real numbers. Because of their commonness, real numbers are represented by countless points on a straight line. At this time, a straight line specifying the origin, positive direction and unit length is used to represent the real number. When the right side is specified as a positive direction, the number represented by the upper point on the right is always greater than that represented by the upper point on the left. Positive numbers are greater than zero, and zero is greater than negative numbers^[1] 。

concept

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In mathematics, you can use a straight line The point on represents the number. This line is called the number line. It meets the following requirements:

(1) Take any point on the straight line to represent 0. This point is called origin （origin）；

(2) Generally, it is specified that the straight line from the origin to the right (or up) is a positive direction, and from the origin to the left (or down) is a negative direction;

(3) Select the appropriate length as Unit length , on the straight line, from the origin to the right, take a point every other unit length, which successively represents 1 (1 unit length to the right), 2 (2 units length to the right), 3 (3 units length to the right); From the origin to the left, use a similar method to represent - 1 (length of 1 unit to the left), - 2 (length of 2 units to the left), - 3 (length of 3 units to the left)

On the number axis, in addition to the number 0 to be represented by the origin, it is necessary to represent any rational number that is not zero. According to the positive and negative signs of this number, determine which side of the number axis it is on (usually the positive number is on the right side of the origin, and the negative number is on the left side of the origin). Then, in the corresponding direction, determine how many units of length it is from the origin, and draw the corresponding points. In addition, a point mark 1 on the number axis means that the line segment from the origin to the point contains 1 unit length, and the specific length is unlimited. In addition, a unit length on the number axis does not necessarily mean a grid. For example, a grid can also be marked with 5, which can be considered as a reduction of the coordinate system due to some need. A grid of this mark 5 actually contains 5 unit lengths, but the coordinate system is reduced due to some need to better represent it.

effect

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1. The number axis can visually show numbers. The points on the horizontal number axis correspond to real numbers one by one, that is, each real number can be represented by a point on the number axis

2. Compare the size of real numbers. Take 0 as the center. The number on the right is larger than the number on the left.

3、 imaginary number It can also be represented by the vertical number axis perpendicular to the horizontal number axis and at the same origin, which forms the horizontal number axis Complex plane 。

4. It can be formed by two mutually perpendicular number axes with the same origin Rectangular coordinate system ； It can be formed by three mutually perpendicular number axes with the same origin Space rectangular coordinate system To determine the position of the object.

The number axis has the completeness of numbers. It can not only represent rational numbers and irrational numbers (collectively called real numbers), but also represent imaginary numbers. At the same time, it can establish a coordinate system, forming a relatively tight number system

Geometric meaning

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The number line is a specific geometric figure; The origin, positive direction and unit length are the three elements of the number axis.

1) Starting from the origin, in the positive direction radial （ Positive halfshaft ）Point correspondence on Positive number , rays in the opposite direction（ Negative halfshaft ）Point correspondence on negative , the origin corresponds to zero.

2) For the two numbers on the number axis, the number in the positive direction is always larger than the number on the other side.

3) Positive numbers are greater than 0, negative numbers are less than 0, and positive numbers are greater than all negative numbers.

Note: The unit length refers to taking an appropriate length as the unit length, for example, taking 2m as the unit length "1", then 4m means two unit lengths. Length unit It refers to meters, centimeters, millimeters and other units representing length.

The two should not be confused.

The points on the number axis correspond to the numbers one by one. (Any number, including imaginary number, can be represented by a point on the number axis.)

The positive direction of the number axis is generally right, but it does not rule out the possibility of left, and the closer the number is to the positive direction, the greater the number is, and the opposite is the farther away from the positive direction, the smaller the number is.

When drawing a number axis, you should first draw a horizontal line and a positive direction, then draw a zero, and then draw the unit length according to the meaning of the question.

usage

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Mathematically, the number axis is a one-dimensional graph, and integers are evenly distributed on a line as special points. The number axis is a straight line that specifies the origin, direction and unit length. The origin, direction and unit length are the three elements of the number axis. It is often used to help teach simple addition or subtraction (especially when there are negative numbers in the operation).

In most cases, the number line is represented as horizontal (which is not necessary, of course). It is divided into two symmetrical parts by the origin 0. Usually positive numbers are to the right of 0 and negative numbers are to the left of 0. All real numbers correspond to points on number axis one by one.

Related concepts

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Inverse number

Only two numbers with different symbols are called Opposite to each other One of them is called the opposite of the other.

(a ≠ 0) The opposite number of a is - a, and the opposite number of 0 is 0.

absolute value

The distance from the point representing a number on the number axis to the origin is called the absolute value of the number

The absolute value of a positive number is itself, and the absolute value of a negative number is its Inverse number 。 The absolute value of 0 is 0.

Formula | a |=?

If a is greater than 0, the absolute value of a is also equal to a;

If a is equal to 0, then the absolute value of a is equal to 0;

If a is less than 0, the absolute value of a is equal to the opposite number of a.

Property: absolute value is nonnegative

Rational number comparison size:

All positive numbers are greater than 0, 0 is greater than all negative numbers, and positive numbers are greater than all negative numbers.

Note: The number on the right side of the number axis is always larger than the number on the left side. Compared with the two negative numbers, the one with a larger absolute value is smaller.

computing time

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The number axis is used to represent the global Meridian , two of this line segment Endpoint Represents 18 0 ° longitude The midpoint of the line segment represents the 0 ° longitude, so that the longitude positions of all places in the world can be represented on this line segment. The direction of the arrow represents the direction of the earth's rotation. Therefore, the longitude from 0 ° to 180 ° is the east longitude, and the time zone on the far right is East 12th District , the earliest time; From 0 ° meridian westward to 180 ° meridian is Western Meridian , the leftmost time zone is West 12th District The time is the latest. There is a difference of 24 hours between the eastern and western districts.

On this number axis, the more to the right, the earlier the time, and the larger the value, which is consistent with the mathematical meaning of the number axis. Therefore, if the time of place B in Figure 1 is known, the time of place A is required. Place A is on the right side of place B. Add, that is, time of place A is equal to time of place B plus time of place A and place B time difference ； On the contrary, the time of place B is required, and place B is on the left of place A. By subtraction, it can be recorded as "right plus left minus". At the same time, since the direction of the number axis represents the direction of the earth from west to east rotation In this sense, the direction can also be recorded as "East plus West minus". In this way, it is not easy to make mistakes by combining the choice of addition and subtraction and the time sooner or later with the mathematical meaning of the number axis. In addition, use the two Endpoint To represent 18 and normalize 0 ° longitude, which can avoid crossing the date line, thus simplifying the calculation.

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