People's Education Press: Junior High School Mathematics, Grade 7, Semester 2: Crossing from Line to Plane

Imagine you're searching for a seat in a cinema. If there were only one row (one-dimensional), you'd need just one number; but in reality, cinemas have multiple rows and seats (two-dimensional), so you must have both the 'row number' and 'seat number'. If you're given 'Row 3, Seat 5' but sit in 'Row 5, Seat 3', it's clearly wrong—this is the precise definition of 'ordered' in mathematics and real life.

I. Logical Evolution from One Dimension to Two Dimensions

A point on a number line requires only one real number to be fixed in position, while a point in a plane exists across two mutually perpendicular dimensions. After establishing a Cartesian coordinate system, for any point $M$ in the coordinate plane, there is a unique pair of ordered real numbers $(x, y)$ corresponding to it; conversely, for any pair of ordered real numbers $(x, y)$, there is a unique point $M$ in the coordinate plane corresponding to it. Thisone-to-one correspondenceis the foundation of the idea of combining numbers with geometry.

Core Definition

Ordered Pair: A pair of two numbers arranged in order is called an ordered pair, denoted as $(a, b)$.

Note the Details

‘Ordered’ means $(x, y) \neq (y, x)$ (unless $x = y$). The order determines the directional property represented by the numbers (horizontal offset or vertical offset).

II. Bidirectional Mapping of One-to-One Correspondence

This mapping ensures that 'numbers' can precisely describe the position of 'shapes,' and 'shapes' can intuitively reflect the properties of 'numbers,' allowing geometric figures in the plane to be processed algebraically. We summarize this relationship as:

Using Numbers to Solve Shapes: Calculating the area, perimeter, or determining positional relationships of shapes using coordinates.
Using Shapes to Aid Numbers: Gaining intuitive understanding of function properties or equation solutions through visual observation.

🎯 Core Principle

A point $P$ on the plane $\longleftrightarrow$ an ordered pair $(x, y)$.

In the coordinate $(x, y)$, $x$ is the horizontal coordinate, and $y$ is the vertical coordinate.

QUESTION 1

In the ordered pair $(5, 2)$, what does the number $5$ represent?

Distance from the point to the $x$-axis

Horizontal coordinate (x)

Vertical coordinate (y)

Distance from the point to the origin

QUESTION 2

If a cinema seat is represented by the ordered pair $(a, b)$, where $a$ denotes the row number and $b$ the seat number, what does $(3, 6)$ represent?

Row 6, Seat 3

Row 3, Seat 6

Between Row 3 and Row 6

A total of 9 seats

QUESTION 3

The necessary and sufficient condition for the ordered pairs $(m, n)$ and $(n, m)$ to be equal is:

$m = 0$

$n = 0$

$m = n$

$m = -n$

QUESTION 4

Point $P$ lies on the $x$-axis and is 4 units away from the origin. What is the $y$-coordinate of point $P$?

$4$

$-4$

$0$

Cannot be determined

QUESTION 5

The ordered pair representing the origin $O$ is:

$(1, 1)$

$(0, 0)$

$(0, 1)$

$(1, 0)$

QUESTION 6

How many real numbers are needed to determine the position of a point in the plane?

Infinitely many

QUESTION 7

Given point $A(2, 3)$, if we swap its horizontal and vertical coordinates to obtain point $B$, what is the coordinate of point $B$?

$(2, 3)$

$(3, 2)$

$(-2, -3)$

$(3, -2)$

QUESTION 8

Which of the following statements is correct?

$(2, 3)$ and $(3, 2)$ represent the same point

Ordered pairs can only represent integers

Points in the coordinate plane correspond one-to-one with ordered real pairs

Only one number is needed to determine a point's position

QUESTION 9

If the ordered pair $(x-1, 5)$ represents a point on the $y$-axis, what is the value of $x$?

$0$

$1$

$5$

$-1$

QUESTION 10

[Short Answer] Plot the following sets of points in the same Cartesian coordinate system and connect them sequentially with line segments. (1) $(2, 0), (4, 0), (2, 2), (2, 0)$; (2) $(0, 2), (0, 4), (-2, 2), (0, 2)$. What did you discover?

Form two triangles of the same shape but different positions

Form two squares

Form two straight lines

Cannot form a figure