Real line, decimals and significant figures(实线,小数,和有效数字)

Sets and elements

A set is a collection of objects, referred to as elements. A set may be represented, for example, by a list of elements surrounded by curly brackets and separated by commas, or using set builder notation {…|…}, where the vertical line is an abbreviation for “such that”. For example, { x | x is a natural number less than 5 }
and {0, 1, 2, 3, 4} represent the same set, whose elements are precisely 0, 1, 2, 3 and 4.
(A natural number is a whole counting number, including zero. Note that some people do not count zero as a counting number, but we do in this course!)

Element symbol

The symbol \in is an abbreviation for “is an element of”, and \notin is an
abbreviation for “is not an element of”. For example, if

A={xx is a natural number less than 5}A=\{x | x \text { is a natural number less than } 5\}

then 2A2 \in A, but 5A5 \notin A.

Subset symbols

If A and B are sets and we write ABA \subseteq B or BAB \supseteq A, then we mean that every element of A is also an element of B, and say that A is a subset of B. For example
{1,2,3}{1,2,3,4}\{1,2,3\} \subseteq\{1,2,3,4\} and {1,2,3,4}{1,2,3}\{1,2,3,4\} \supseteq\{1,2,3\}, but {1,2,3,4}{1,2,3}\{1,2,3,4\} \nsubseteq\{1,2,3\}.

Equality of sets

If A and B are sets then A = B if and only if ABA \subseteq B and BAB \subseteq A, that is, A and B have precisely the same elements. Order and repetition are not important.
For example, {1, 2, 3, 4} = {4, 1, 3, 2} = {4, 1, 3, 1, 2, 3}.

Intersection, union and slash

If A and B are sets then put

  • AB={xxA and xB}A \cap B=\{x | x \in A \text { and } x \in B\}, called the intersection of A and B.
  • AB={xxA or xB}A \cup B=\{x | x \in A \text { or } x \in B\}, called the union of A and B.
  • A\B={xxA and xB}A \backslash B=\{x | x \in A \text { and } x \notin B\}, called A slash B, the result of removing from A all elements from B.

Natural numbers

The set N={0,1,2,3,}\mathbb{N}=\{0,1,2,3, \ldots\} of natural numbers forms a number system, closed under addition and multiplication, by which we mean that if m and n are natural numbers, then m+n and mn (the result of multiplying m by n) are also natural numbers.


The set Z={0,±1,±2,±3,}\mathbb{Z}=\{0, \pm 1, \pm 2, \pm 3, \ldots\} of integers forms a number system, closed under addition, subtraction and multiplication.


The set Q={a/ba,bZ,b0}\mathbb{Q}=\{a / b | a, b \in \mathbb{Z}, b \neq 0\} of fractions, also called rational numbers (derived from the word ratio), forms a number system, closed under addition, subtraction, multiplication and division by nonzero elements. To add and multiply rational numbers, use the rules

ab+cd=ad+bcbd and abcd=acbd\frac{a}{b}+\frac{c}{d}=\frac{a d+b c}{b d} \quad \text { and } \quad \frac{a}{b} \cdot \frac{c}{d}=\frac{a c}{b d}

Decimal expansion of a real number

A nonnegative real number α\alpha has a decimal expansion

α=bnbn1b2b1a1a2a3\alpha=b_{n} b_{n-1} \ldots b_{2} b_{1} \cdot a_{1} a_{2} a_{3} \ldots

where the bib_{i} and aja_{j} are digits from the set {0, 1, 2, . . . , 9}.

Recurring decimal expansions

A real number is rational if and only if it has a recurring
decimal expansion, which means the pattern of digits repeats forever from some point
onwards. For example,

67=0.857142857142=0.8˙57142˙,\frac{6}{7}=0.857142857142 \ldots=0 . \dot{8} 5714\dot{2} ,

where the two dots indicate the start and finish of the pattern of digits that gets repeated.

The real number line

The real numbers form a number system R\mathbb{R} that is closed under addition, subtraction, multiplication and division by nonzero elements. We visualise R\mathbb{R} as a continuous number line, called the real line, with zero in the middle, negative numbers to the left and positive numbers to the right.

The number π\pi

The real number π\pi = 3.14159… is the ratio of the perimeter of a circle to its diameter. The fact that π\pi is the same for all circles is an advanced argument involving limits (a concept explained in Module 3) that relies on the fact that circles can be approximated arbitrarily well using similar triangles and the fact that ratios of corresponding sides of similar triangles are equal.


A real number that is not rational (that is, cannot be expressed as a fraction
involving a 2\sqrt{2} , 3\sqrt{3} and π\pi are irrational, though the proof for π\pi is difficult. (A proof that 2\sqrt{2} is irrational is given in the notes
accompanying Module 1, Video 1.2.)

Significant figures

Real numbers may be approximated by rational numbers with finite decimal expansions. The number of digits counted to the right from the leftmost positive digit is called the number of significant figures. For example, real numbers represented by 26.103, 0.00304 and 0.003040 are quoted to 5, 3 and 4 significant figures respectively.

Scientific notation

A positive real number α\alpha is expressed in scientific notation if it has the form

α=ba1a2am×10k\alpha=b \cdot a_{1} a_{2} \ldots a_{m} \times 10^{k}

where m is nonnegative and k is an integer. For example, 193.034 and 0.003040 become 1.93034×1021.93034 \times 10^{2} and 3.040×1033.040 \times 10^{-3} respectively in scientific notation. The number of digits used in scientific notation is the number of significant figures being quoted, and this avoids ambiguity in the case of large whole numbers (with zeros as place-holders).

Accuracy rule for addition and subtraction

When adding or subtracting numbers, the final answer should be quoted to the least number of decimal places that occurs.

Accuracy rule for multiplication and division

When multiplying or dividing numbers, the final answer should be quoted to the least number of significant figures that occurs.

The Theorem of Pythagoras and properties of the square root of 2(毕达哥拉斯定理和2的平方根性质)

Square roots

If αR\alpha \in \mathbb{R} then α=β\sqrt{\alpha}=\beta if and only if β0\beta \geq 0 and β2=α\beta^{2}=\alpha. In particular, 0=0\sqrt{0}=0.
If α>0\sqrt{\alpha}>0 then we call α\sqrt{\alpha} the positive square root and α-\sqrt{\alpha} the negative square root of α\alpha.
For example, 2 is the positive square root of 4, and -2 is the negative squareroot of 4.
We also write α12\alpha^{\frac{1}{2}} for α\sqrt{\alpha}, and note the following laws hold, where α,β0\alpha,\beta \geq 0 and γ>0\gamma>0:

α12β12=αβ=αβ=(αβ)12\alpha^{\frac{1}{2}} \beta^{\frac{1}{2}}=\sqrt{\alpha} \sqrt{\beta}=\sqrt{\alpha \beta}=(\alpha \beta)^{\frac{1}{2}}



Negative real numbers do not have square roots within the real number system R\mathbb{R} (though they do have square roots in the larger complex number system C\mathbb{C}).


Any expression involving a square root symbol √ is called a surd expression.
For example,

2,23,2+3,42,12+5,a2+b2,a+b+22c\sqrt{2}, \quad 2 \sqrt{3}, \quad \sqrt{2}+\sqrt{3}, \quad 4-\sqrt{2}, \quad \frac{1}{2+\sqrt{5}}, \quad \sqrt{a^{2}+b^{2}}, \quad \frac{\sqrt{a+\sqrt{b+2}}}{2 \sqrt{c}}

are all surd expressions. (More generally, surd expressions can involve other types of roots, such as cube roots, or tenth roots, but simple square roots are the most common.)

The Distributive Law (expanding brackets)

If a,b,cRa, b, c \in \mathbb{R} then

(a+b)c=ac+bc and a(b+c)=ab+ac(a+b) c=a c+b c \quad \text { and } \quad a(b+c)=a b+a c

It follows that, if a,b,c,dRa, b, c, d \in \mathbb{R}, then

(a+b)(c+d)=ac+ad+bc+bd(a+b)(c+d)=a c+a d+b c+b d

Squares and differences of squares

As a consequence of distributivity, we have the following identities, for all a,bRa, b \in \mathbb{R}:

(a+b)2=a2+2ab+b2,(ab)2=a22ab+b2,a2b2=(a+b)(ab)(a+b)^{2}=a^{2}+2 a b+b^{2}, \quad(a-b)^{2}=a^{2}-2 a b+b^{2}, \quad a^{2}-b^{2}=(a+b)(a-b)

The last identity is called the difference of two squares formula, and it features in many
algebraic tricks or manipulations. For example,

77×83=(803)(80+3)=80232=64009=639177 \times 83=(80-3)(80+3)=80^{2}-3^{2}=6400-9=6391

and, for any positive reals a and b,


Theorem of Pythagoras

If a and b are the lengths of the shorter sides of a right angled triangle, and c is the length of the hypotenuse (the longest side), then


so that c=a2+b2,a=c2b2c=\sqrt{a^{2}+b^{2}}, a=\sqrt{c^{2}-b^{2}} and b=c2a2b=\sqrt{c^{2}-a^{2}}


Irrational surds

If nNn \in \mathbb{N} then n\sqrt{n} is rational if and only if nn is a perfect square, that is n=m2n=m^{2} for some mN.m \in \mathbb{N} . Thus, for example, 2,3,5\sqrt{2}, \sqrt{3}, \sqrt{5} and 6\sqrt{6} are irrational.

Algebraic expressions, surds and approximations

Differences of two squares formula

In any well-behaved arithmetic we have the identity


known as the difference of two squares formula. It is exploited, for example, in the next important idea (rationalising the denominator).

Rationalising the denominator

If aZ,bNa \in \mathbb{Z}, b \in \mathbb{N} and a±b0a \pm \sqrt{b} \neq 0 then

1a±b=(1a±b)(abab)=aba2b.\frac{1}{a \pm \sqrt{b}}=\left(\frac{1}{a \pm \sqrt{b}}\right)\left(\frac{a \mp \sqrt{b}}{a \mp \sqrt{b}}\right)=\frac{a \mp \sqrt{b}}{a^{2}-b} .

For example,



137=(137)(3+73+7)=3+797=3+72.\frac{1}{3-\sqrt{7}}=\left(\frac{1}{3-\sqrt{7}}\right)\left(\frac{3+\sqrt{7}}{3+\sqrt{7}}\right)=\frac{3+\sqrt{7}}{9-7}=\frac{3+\sqrt{7}}{2} .

It is a fact, discussed below, that if aZa \in \mathbb{Z} and bN,b \in \mathbb{N}, such that bb is not a perfect square, then a±ba \pm \sqrt{b} is irrational. The terminology rationalising the denominator comes about, as in the examples above, where the original fraction has an irrational number in the denominator. The fraction is then transformed by this technique, so that the denominator becomes rational (an integer, in fact, in each of these examples).

Continued fractions

An expression obtained by taking a number and adding a reciprocal, where the denominator itself is another expression involving a number added to a reciprocal, and allowing this process to repeat, is called a continued fraction.
If the entire expression is just a reciprocal, but the denominator can be expressed symbolically in terms of the entire expression, then the expression can be fed into itself indefinitely, to obtain an infire continued fraction. For example, the expression


leads to the infinite continued fraction


By truncating such an expression at some place, one obtains a finite continued fraction, which then evaluates to an ordinary fraction.

Equations and inequalities

Positive, negative, nonnegative and nonpositive real numbers

A real number xx is called positive if it appears to the right of zero on the real number line, in which case we write a>0,a>0, and negative if it appears to the left of zero, in which case write a<0.a<0 . We say that xx is nonnegative if it is positive or zero, in which case we write a0,a \geq 0, and nonpositive if it is negative or zero, in which case we write a0.a \leq 0 .

Multiplication and (signed) areas of rectangles

If aa and bb are real numbers then the product of aa and bb is the result of multiplying aa by bb , denoted typically by a×ba \times b or aba b (juxtaposition). The product aba b may be represented, geometrically, by the (signed) area of a rectangle with side lengths aa and bb . If both aa and bb are nonnegative then this is just the usuale nonnegative) area.

If one of aa and bb is positive and the other negative then the area aba b of the rectangle is regarded as negative.
If both aa and bb are negative then the area aba b is regarded as positive.
The formal rigorous explanation for this last fact is based on laws, or axioms, of arithmetic, and involves an advanced argument in abstract algebra that one typically sees in second or third year university mathematics.

Intuitively, one can just think of negatives of negatives being positive. It wouldn’t make sense for the negative of a negative number to remain negative. This is mirrored in common speech when one uses double negatives: for example, to say “I didn’t not say that!” is a complicated way of saying “I said that!” (though its use can help to emphasise the positive in a heated argument.) (The speech analogy also can break down, as some languages, including some colloquial English, use a double negative in certain turns of phrase to reinforce a single negative.)
For example, 2 × 2 = 4 and

2×(2)=(2)×2=42 \times(-2)=(-2) \times 2=-4


(2)×(2)=(2×(2))=(4)=4(-2) \times(-2)=-(2 \times(-2))=-(-4)=4

Factorisation of zero

If a and b are real numbers then the product ab is zero if and only if a = 0 or b = 0 (including the possibility that a = b = 0).
This follows from the representation of the product ab as the (signed) area of a rectangle with side length a and b. If the area vanishes, then at least one of the side lengths must vanish.
This fact about the factorisation of zero is very useful for solving equations. For example,
if x is a real number such that


then either x - 1 = 0 or x - 2 = 0, so that either x = 1 or x = 2.
This fact clearly generalises to more than two factors. For example, if


then either x - 4 = 0, x + 1 = 0 or x + 3 = 0, so that either x = 4, x = -1 or x = -3.


Let a,bR.a, b \in \mathbb{R} . We write a<ba<b if aa appears to the left of bb on the real number line, which is equivalent to the difference bab-a being positive (and the difference aba-b being negative).

In this case we say that aa is less than bb and bb is greater than a,a, and also write b>a.b>a .
We write aba \leq b if a=ba=b or a<b,a<b, and write aba \geq b if a=ba=b or a>ba>b .
For example,


Note that -10 < 5 even though you might think of -10 as a “larger” number than 5 in terms of “size”.
(Later, we introduce the magnitude or absolute value of a real number, which captures the idea of “size” precisely. The word “larger” is ambiguous, and has to be read in context in mathematics.)

Properties of inequalities

Let a,b,cRa, b, c \in \mathbb{R}

  • If a<ba<b and cdc \leq d then a+c<b+d,a+c<b+d, so that, in particular, a+c<b+ca+c<b+c
  • It follows also, from the previous part, that if a<ba<b then ac<bca-c<b-c .
  • If a<ba<b and c>0c>0 then ac<bca c<b c and ac<bc\frac{a}{c}<\frac{b}{c}
  • If a<ba<b and c<0c<0 then ac>bca c>b c and ac>bc,\frac{a}{c}>\frac{b}{c}, so that, in particular, a>b-a>-b .
  • If 0<a<b0<a<b then 1a>1b>0\frac{1}{a}>\frac{1}{b}>0 .

Variations of these properties hold using >,>, \leq and \geq .

Sign diagrams, solution sets and intervals

The sign of a real number

The sign of a real number x refers to whether x is positive or x is negative. We say that two numbers have the same sign if they are both positive or they are both negative. We say that two numbers have different signs if one is positive and the other is negative.
Consider real numbers a and b. The product ab is positive if and only if a and b have the same sign. The product ab is negative if and only if a and b have different signs.

Sign diagrams

A sign diagram is a table with two rows, the first row indicating values of
some variable, x say, and the second row indicating the sign (indicated by a plus sign + for positive and a minus sign - for negative) of some particular expression involving x. The line in the table between the rows represents the real line, and important points for x are marked off, typically corresponding to x for which the expression evaluates to zero (and this is usually obvious when the expression has been factorised).

For example, here is the sign diagram for the expression (x1)(x2),(x-1)(x-2), which is zero, when x=1x=1 and x=2,x=2, positive when x<1x<1 and when x>2,x>2, and negative when 1<x<21<x<2 :
Sign diagrams1

Here is the sign diagram for the expression x(x21)=x(x+1)(x1),x\left(x^{2}-1\right)=x(x+1)(x-1), which is zero when x=0,x=1x=0, x=1 and x=1,x=-1, positive when 1<x<0-1<x<0 and when x>1,x>1, and negative when x<1x<-1 and when 0<x<10<x<1 :
Sign diagrams2

One can also construct sign diagrams when an expression becomes undefined, especially
when fractions are involved where the variable appears in the denominator, in which case the letter “u” is used in the second row, as an abbreviation for “undefined”.

Here is the sign diagram for the expression x21x=(x+1)(x1)x,\frac{x^{2}-1}{x}=\frac{(x+1)(x-1)}{x}, which is identical to the previous example, except for noting that the expression is undefined for x=0x=0 :

Sign diagrams3

For another variation, here is the sign diagram for the expression xx21=x(x+1)(x1),\frac{x}{x^{2}-1}=\frac{x}{(x+1)(x-1)}, where now the expression is undefined for x=1x=-1 and x=1:x=1 :

Sign diagrams4

Empty set

The empty set is the set without any elements and denoted by .\emptyset . If AA is any set then A=AA \cup \emptyset =A, A=A \cap \emptyset = \emptyset and A\A=A \backslash A=\emptyset .

Solution sets

The solution set (within the real number system R\mathbb{R}) of an equation or an inequality involving a variable x is the set of all real numbers x that satisfy the given equation or inequality. If no real numbers satisfy the given equation or inequality then the solution set is empty.
For example, the solution set of the equation 2x + 1 = 7 is the set {3}, since x = 3 is the only solution.
The solution set of the equation (x - 1)(x - 2) = 0 is the set {1, 2}, since the solutions are precisely x = 1 and x = 2.
The solution set of the inequality 2x+1<72 x+1<7 is the set {xRx<3}\{x \in \mathbb{R} | x<3\}
The solution set of the inequality 2x+172 x+1 \geq 7 is the set {xRx3}\{x \in \mathbb{R} | x \geq 3\}
The solution set of the inequality (x1)(x2)<0(x-1)(x-2)<0 is the set {xR1<x<2}\{x \in \mathbb{R} | 1<x<2\}
The solution set of the inequality (x1)(x2)0(x-1)(x-2) \geq 0 is the set {xRx1 or x2}\{x \in \mathbb{R} | x \leq 1 \text { or } x \geq 2\}
The solution set of the equation x+1=x+2x+1=x+2 is the empty set ,\emptyset, since if the solution set contained some xx then, taking xx away from both sides of the equation would produce the absurdity 1=21=2
The solution set of the inequality x2+1<0x^{2}+1<0 is also the empty set ,\emptyset, since squares of real numbers are always nonnegative: if the solution set were to contain some xx then, taking 1 away from both sides, one would conclude that x2<1,x^{2}<-1, which is negative, contradicting that x2x^{2} is nonnegative.

Interval notation

 If a,bR and a<b then \text { If } a, b \in \mathbb{R} \text { and } a<b \text { then }

Interval notation1
Interval notation2

It is also common to write R=(,),\mathbb{R}=(-\infty, \infty), the interval that covers the entire real line.

Coordinate systems

Cartesian plane

The Cartesian plane or xy-plane consists of all ordered pairs (x, y) as x and y range over all reals numbers, denoted by

R2={(x,y)x,yR},\mathbb{R}^{2}=\{(x, y) | x, y \in \mathbb{R}\},

containing a horizontal xx-axis {(x,0)xR}\{(x, 0) | x \in \mathbb{R}\} and a vertical yy-axis {(0,y)yR}.\{(0, y) | y \in \mathbb{R}\} . The axes are perpendicular and intersect at the origin O=O(0,0).O=O(0,0) .
Cartesian plane
If P = P(a, b) is a point in the xy-plane, then we call a the x-coordinate and b the y-coordinate of P, obtained by projecting P to the closest points on the x-axis and y-axis respectively. Together, we call the pair (a, b) the Cartesian coordinates of P. In the above diagram, the coordinates of P are positive. In general the coordinates can be positive, negative, zero or any mixture of these.

Quadrants of the Cartesian plane

The Cartesian plane is divided by the coordinate axes into four quadrants.
 Quadrants of the Cartesian plane
The first quadrant consists of points with (+,+)(+,+) coordinates; the second with (,+)(-,+) coordinates; the third with (,)(-,-) coordinates; the fourth with (+,)(+,-) coordinates.

Distance between two points in the Cartesian plane

The distance dd between points P(x1,y1)P\left(x_{1}, y_{1}\right) and Q(x2,y2)Q\left(x_{2}, y_{2}\right) in the plane is given by the formula


that is, the square root of the sum of the squares of the differences in their coordinates. This follows from the Theorem of Pythagoras, by creating a right-angled triangle with shorter sides parallel to the xx and yy-axes.

In this diagram, the horizontal side of the triangle has length x2x1x_{2}-x_{1} and the vertical side has length y2y1.y_{2}-y_{1} . The length of the hypotenuse is the distance dd between PP and Q,Q, so that d2=(x2x1)2+(y2y1)2,d^{2}=\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}, by Pythagoras, and the above formula is immediate by taking positive square roots.

Note that the above diagram is particular in that it assumes the differences in coordinates are both positive. However, the mathematics works and the formula holds in all cases, including when these differences are zero or negative. This is because taking the square of a difference of two real numbers, inside the square root sign, produces the same outcome, regardless of the order in which the difference is taken.

Coordinates in space

To model positions in space mathematically, we think of the xy-plane now as ‘horizontal’ (including the y-axis, which we previously thought of as being
‘vertical’), and introduce a third z-axis, another copy of the real line, which we think of as the new ‘vertical’, which is perpendicular to the xy-plane. All axes pass through a common point, again called the origin, and denoted by O = O(0, 0, 0). Every point P in space has coordinates (a, b, c), for some real numbers a, b and c, which are formed by projecting P to the closest points on the x, y and z-axes respectively. In this way we identify space with the set of triples of real numbers, denoted by

R3={(a,b,c)a,b,cR}\mathbb{R}^{3}=\{(a, b, c) | a, b, c \in \mathbb{R}\}

The xx-axis is identified with the subset of triples {(x,0,0)xR},\{(x, 0,0) | x \in \mathbb{R}\}, the yy-axis with the subset {(0,y,0)yR}\{(0, y, 0) | y \in \mathbb{R}\} and the zz-axis with the subset {(0,0,z)zR}.\{(0,0, z) | z \in \mathbb{R}\} . In the following diagram, only the positive parts of the axes emanating out of the origin are drawn, and you have to imagine the negative halves of the axes extending behind the page (for the xx-axis), to the far left (for the yy-axis) and far below (for the zz-axis).
Coordinates in space
Imagine the point P hovering in space above the (horizontal) xy-plane. We project down (vertically) onto the xy-plane to get to some point Q, say, which has projections a and b on the x and y-axes respectively. The closest point on the z-axis to Q is the origin, with z-value 0, so the coordinates of Q become (a, b, 0). If we move from P directly across (horizontally) to the z-axis then we meet it at some point, say c. Then the triple of real numbers (a, b, c) that we get in this way become the coordinates of P.

Distance between two points in space

The formula for the distance between two points in
space generalises the earlier formula for the distance between two points in the xy-plane. As a special case consider the distance from P(a, b, c) to the origin O(0, 0, 0).
Distance between two points in space
Add a line segment joining O to P, whose length is the distance d from P to O. The points O, P and Q form a right-angled triangle with shorter side-lengths c (the closest point to P on the z-axis) and the distance from O to Q, which we denote by h.

By Pythagoras, d2=h2+c2.d^{2}=h^{2}+c^{2} . But, by Pythagoras again, h2=a2+b2,h^{2}=a^{2}+b^{2}, since hh is the length of the hypotenuse formed by the right-angled triangle using O,QO, Q and the point on the xx-axis labelled by aa , whose shorter side-lengths are now just aa and bb . Hence


Taking the square root, we deduce that the distance from P(a,b,c)P(a, b, c) to the origin O(0,0,0)O(0,0,0) is


This diagram assumed a,ba, b and cc are positive, but the mathematics works regardless of the signs of the coordinates of P.P .
More generally, the distance dd between points P(x1,y1,z1)P\left(x_{1}, y_{1}, z_{1}\right) and Q(x1,y2,z2)Q\left(x_{1}, y_{2}, z_{2}\right) is given by the formula


that is, the square root of the sum of the squares of the differences in their coordinates. This can be shown using a similar but more elaborate diagram to the above, or alternatively can be deduced from the special case, by translating both of the points parallel to themselves so that one of them goes to the origin. If PP is translated to O(0,0,0)O(0,0,0) then QQ is translated to the point with coordinates (x2x1,y2y1,z2z1),\left(x_{2}-x_{1}, y_{2}-y_{1}, z_{2}-z_{1}\right), and the general result now follows from the special case.

The midpoint between two points in the plane or in space

The midpoint of the line segment joining points P(x1, y1, z1) to Q(x2, y2, z2) in space has coordinates

(x1+x22,y1+y22,z1+z22)\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}, \frac{z_{1}+z_{2}}{2}\right)

that is, the triple formed by taking the averages of the respective coordinates of PP and Q. By ignoring the zz-coordinate in the previous formula, we see that the midpoint of the line segment joining points P(x1,y1)P\left(x_{1}, y_{1}\right) and Q(x2,y2)Q\left(x_{2}, y_{2}\right) in the plane has coordinates

(x1+x22,y1+y22)\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)

Distance and absolute value

Absolute value

If αR\alpha \in \mathbb{R} then the absolute value or magnitude of α,\alpha, denoted by α,|\alpha|, is the distance from α\alpha to 0, regarded as points on the real number line, so that

α={α if α0α if α<0|\alpha|=\left\{\begin{aligned} \alpha & \text { if } \alpha \geq 0 \\-\alpha & \text { if } \alpha<0 \end{aligned}\right.

Note that α|\alpha| just reproduces itself if α\alpha is positive or zero, but takes its negative (which turns out to be positive) if α\alpha is negative. For example, 2=2|2|=2 and


The absolute value is always nonnegative.

Useful laws, including the triangle inequality

The following laws hold, for all α,β,γR,\alpha, \beta, \gamma \in \mathbb{R}, with γ0\gamma \neq 0 :

α=α=α2,αβ=αβ,α/γ=α/γ,α+βα+β,|-\alpha|=|\alpha|=\sqrt{\alpha^{2}}, \quad|\alpha \beta|=|\alpha| \beta|, \quad| \alpha / \gamma|=| \alpha|/| \gamma|, \quad| \alpha+\beta| \leq| \alpha|+| \beta |,

the last of which is known as the triangle inequality. For example,


23=2×(3)=6=6 and 23=23=23.|2||-3|=|2 \times(-3)|=|-6|=6 \quad \text { and } \quad \frac{|2|}{|-3|}=\left|\frac{2}{-3}\right|=\frac{2}{3}.

Observe further that


consistent with the triangle inequality (using “less than” in this particular case, which is included in “less than or equals”). The reason for the terminology “triangle inequality” may seem cryptic, but is related to the fact that the shortest distance between any two points P and Q in the plane or in space is along the straight line joining them. One may think of this straight line as one edge of a triangle made with P and Q and a third point R typically not on this shortest straight line segment. In our context with absolute values, all points are constrained to lie in the real line, so the “triangles” PQR that arise turn out to be degenerate (having no area).

Distance between two points in the real line

Distance between two points in the real line: If α,βR\alpha, \beta \in \mathbb{R} then αβ|\alpha-\beta| is the distance between α\alpha and β,\beta, as points on the real number line.
Distance between two points
Note that the distance between α\alpha and β,-\beta, or between α-\alpha and β\beta is


Lines and circles in the plane

Lines in the plane

AA line in the xyx y -plane is the set of points (x,y)(x, y) satisfying an equation
either of the form y=mx+ky=m x+k (the nonvertical case), where mm is the slope and kk is the yy-intercept, or x=x=\ell (the vertical case), where \ell is the xx-intercept. All lines may be put in the form

ax+by=ca x+b y=c

for some constants a, b and c.

This equation is only interesting by assuming that at least one of aa or bb is nonzero. If a=0a=0 then this assumption forces bb to be nonzero, in which case the equation describes a horizontal line (y=cb).\left(y=\frac{c}{b}\right) . If b=0b=0 then this assumption forces aa to be nonzero, in which case the equation describes a vertical line (x=ca).\left(x=\frac{c}{a}\right) .

Slope of a line determined by two distinct points

A line is completely determined by two distinct points. Let P(x1, y1), Q(x2, y2) be distinct points in the plane. The slope of the line joining P to Q is

 vertical rise  horizontal run =y2y1x2x1=y1y2x1x2\frac{\text { vertical rise }}{\text { horizontal run }}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{y_{1}-y_{2}}{x_{1}-x_{2}}

if x1x2,x_{1} \neq x_{2}, and infinite if x1=x2x_{1}=x_{2} (when the line is vertical). An equation of the line through PP and QQ is


provided that x1x2x_{1} \neq x_{2}.

If the slope is positive then the line slopes upwards as one moves from left to right. If the slope is negative then the line slopes downwards as one moves from left to right. If the slope is zero then the line is horizontal.
If the slope is m, then one moves m units vertically (upwards if m is positive and downwards if m is negative) for each unit that one moves horizontally towards the right.

Parallel and perpendicular lines

Two nnonvertical lines in the xyx y -plane with slopes m1m_{1} and m2m_{2} respectively are parallel if m1=m2,m_{1}=m_{2}, and perpendicular if m1m2=1,m_{1} m_{2}=-1, in which case m1m_{1} and m2m_{2} become negative reciprocals of each other.

Equation of a circle

A circle in the xyx y -plane, centred at P(x0,y0)P\left(x_{0}, y_{0}\right) with radius rr is the set of points (x,y)(x, y) satisfying the equation

(xx0)2+(yy0)2=r2 . \left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}=r^{2} \text { . }

Any tangent to the circle is perpendicular to the radius joining the centre of the circle to the intersection point.


significant figures:有效数字
scientific notation:科学计数法
surd expression:不尽根式