Pure Mathematics 3
A-Level Mathematics Topic 3 13:35 English narration · English + 中文 subtitles burned in
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For centuries, mathematicians hit a wall: what is the square root of minus one?
几个世纪以来，数学家撞上了一堵墙：负一的平方根是什么？
No ordinary number, squared, gives a negative.
没有哪个普通的数，平方后会得到负数。
They called it impossible, and imaginary.
他们称它为不可能，称它为虚的。
But then they dared to give it a name — the letter i — and to treat it as a genuine number.
但后来，他们大胆地给了它一个名字——字母 i—— 并把它当作一个真正的数来对待。
That single leap turned the number line into a whole flat plane, and opened one of the three new worlds of Pure Mathematics, part three.
就是这一跃，把数轴变成了一整个平面， 也开启了纯数学第三部分里三个新世界之一。
Pure three completes the A-Level core.
纯数学第三部分，为 A-Level 的核心画上句号。
It sharpens the algebra and calculus you already have, then opens three genuinely new worlds: vectors, which give direction to numbers; differential equations, which describe change itself; and complex numbers, that new plane.
它先磨利你已经掌握的代数与微积分， 再开启三个真正全新的世界：向量，给数字以方向；微分方程，描述变化本身； 还有复数，那个新平面。
Let's explore them.
让我们去探索它们。
We begin with two new algebra tools.
我们从两件新的代数工具开始。
The first is partial fractions: a single awkward fraction, with a factorised bottom, splits into a sum of simple pieces — each one easy to integrate or expand.
第一件是部分分式：一个别扭的单一分式，只要分母是分解好的， 就能拆成几个简单分式的和——每一个都容易积分或展开。
The second extends the binomial expansion.
第二件把二项展开式加以推广。
It no longer needs a whole-number power: it works for a fraction, or a negative power too, giving an infinite series — valid, though, only when the size of x is less than one.
它不再需要整数次幂：分数次幂、甚至负次幂它也管用，给出一个无穷级数—— 不过，只有当 x 的大小小于一时才成立。
Look at the pattern.
看这个模式。
One fraction with a factorised bottom becomes two simpler fractions — a constant over each linear factor.
一个分母已分解的分式，拆成两个更简单的分式——每个一次因式上面一个常数。
If a factor is squared, you also need a term with that square underneath.
如果某个因式是平方的，还要再加一项，分母是那个平方。
Always check first: the top must be lower degree than the bottom.
先检查：分子的次数必须低于分母。
If it is top-heavy, divide the polynomials first.
如果分子次数更高，就先做多项式除法。
You get a quotient plus a proper remainder fraction, and only that remainder splits into partial fractions.
得到一个商，再加上一个真分式余数， 只有那个余数才拆成部分分式。
Worked example.
例题。
Express x plus four, over x plus one times x minus two, in partial fractions.
把 x 加四，除以 x 加一乘 x 减二，写成部分分式。
Write it as A over x plus one, plus B over x minus two.
写成 A 在 x 加一上， 加上 B 在 x 减二上。
So x plus four equals A times x minus two, plus B times x plus one.
于是 x 加四等于 A 乘 x 减二，加上 B 乘 x 加一。
Put x equal to two: six equals three B, so B is two.
令 x 等于二：六等于三 B，所以 B 是二。
Put x equal to minus one: three equals minus three A, so A is minus one.
令 x 等于负一：三等于负三 A，所以 A 是负一。
Hence the fraction is two over x minus two, minus one over x plus one.
因此这个分式是二在 x 减二上，减去一在 x 加一上。
The binomial expansion also works when the power is a fraction or is negative, as long as the size of x is less than one.
当幂是分数或负数时，二项展开式同样成立，只要 x 的大小小于一。
The series is one, plus n times x, plus n times n minus one over two factorial times x squared, and so on for ever.
级数是一，加上 n 乘 x，再加上 n 乘 n 减一除以二阶乘再乘 x 的平方，并一直写下去。
For example, the square root of one plus x expands as one, plus half x, minus one eighth x squared, and more terms.
例如，一加 x 的平方根展开为一，加上二分之一 x，减去八分之一 x 平方，还有更多项。
Use it to approximate roots and rational powers when x is small.
当 x 较小时，用它来近似根式和有理次幂。
Three Pure two skills stay with you here.
纯数学第二部分的三项本领，在这里仍然要用。
First, the laws of logarithms with e to the x and natural log — same product, quotient and power rules as before.
第一，对数法则，配合 e 的 x 次方与自然对数—— 乘积、商、幂的规则和以前一样。
Second, trigonometry.
第二，三角。
You meet three reciprocal trig functions: secant is one over cosine, cosecant is one over sine, and cotangent is cosine over sine.
你要掌握三个倒数三角函数： 正割是余弦的倒数，余割是正弦的倒数，余切是余弦比正弦。
Memory aid: match the third letter — sec goes with cosine.
记忆口诀：看第三个字母——sec 对上 cosine。
From sine squared plus cosine squared equals one come two Pythagorean identities: sec squared is one plus tan squared, and cosec squared is one plus cot squared.
从正弦平方加余弦平方等于一， 得到两个毕达哥拉斯恒等式：正割平方等于一加正切平方，余割平方等于一加余切平方。
Keep the compound-angle and double-angle formulae, and the R-form of a sine plus b cosine.
复角与倍角公式，以及 a 正弦加 b 余弦的 R 形式，也都要保留。
Differentiation keeps the product, quotient and chain rules from Pure two, and the parametric and implicit methods.
微分保留纯数学第二部分的乘积法则、商法则和链式法则，以及参数与隐函数方法。
One new derivative is added: the derivative of inverse tangent of x is one over one plus x squared.
新加一个导数：反正切 x 的导数，是一比一加 x 的平方。
Worked example: differentiate inverse tangent of three x.
例题：求反正切三 x 的导数。
Use the chain rule with inverse tangent of three x: the outer derivative gives one over one plus three x all squared, then multiply by three.
用链式法则：外层给出一比一加三 x 的平方，再乘以三。
So the answer is three over one plus nine x squared.
所以答案是三比一加九 x 的平方。
Integration gains real power here.
积分在这里获得了真正的威力。
The star is integration by parts, which unpicks a product of two functions using this formula.
主角是分部积分，它用这个公式拆开两个函数的乘积。
Then watch for two patterns.
然后留意两种模式。
When the top of a fraction is the derivative of the bottom, the integral is simply a logarithm.
当分式的分子恰好是分母的导数时，积分就是一个对数。
And a fresh standard result — one over x squared plus a squared — integrates to an inverse tangent.
还有一个新的标准结果——一比 x 的平方加 a 的平方——积分得到一个反正切。
Choosing the right technique from the shape of the integrand is the whole skill.
根据被积式的形状选对方法，正是全部的本领所在。
Worked example.
例题。
Find the integral of x cosine x.
求 x 余弦 x 的积分。
Use integration by parts: take u as x, and d v by d x as cosine x.
用分部积分：取 u 为 x，取 d v 比 d x 为余弦 x。
Then d u by d x is one, and v is sine x.
于是 d u 比 d x 是一，v 是正弦 x。
Plug into the formula: u v minus the integral of v d u.
代入公式：u v 减去 v d u 的积分。
That is x sine x, minus the integral of sine x.
就是 x 正弦 x，减去正弦 x 的积分。
The integral of sine is minus cosine, so the minus times minus flips the sign.
正弦的积分是负余弦，负负得正。
The answer is x sine x plus cosine x, plus a constant.
答案是 x 正弦 x 加余弦 x，再加上常数。
Integration by substitution is the other big tool.
换元积分是另一件大工具。
A given change of variable turns a hard integral into an easy one: rewrite every piece in terms of the new letter, including d x, then integrate and return.
给定的变量替换，能把难积分变成简单的： 把每一部分都用新字母改写，包括 d x，再积分，最后换回去。
And remember partial fractions for integration: split a rational function first, then integrate each simple piece as a logarithm.
还有积分里的部分分式：先把有理函数拆开，再把每一项积成对数。
Match the method to the shape — parts, substitution, or partial fractions — before you write a line of algebra.
动笔之前，先按形状选方法——分部、换元，还是部分分式。
Now the first new world: vectors.
现在进入第一个新世界：向量。
Forces like wind and water have both size and direction — so they are vectors.
风与水这样的力，既有大小也有方向——所以它们是向量。
Write a vector as a column, or as x i plus y j plus z k, or as the arrow from A to B.
把向量写成列向量，或写成 x i 加 y j 加 z k，或写成从 A 到 B 的箭头。
Multiplying by a scalar — a plain number — stretches or shrinks it, and can reverse its direction if the number is negative.
乘以一个标量——一个普通的数——会把它拉长或缩短；若这个数是负的，方向还会反转。
A vector carries both a size and a direction, like a wind or a force.
向量同时带有大小和方向，就像风或力。
Its magnitude — its length — comes from the square root of the squares of its components.
它的模长——它的长度——由各分量平方和的平方根给出。
Shrink a vector down to length one, and you have a unit vector, pure direction.
把一个向量缩短到长度为一， 你就得到一个单位向量，纯粹的方向。
And a whole straight line in space is captured in a single, elegant equation: start at a point, then add any number of copies of a direction vector.
而空间中一整条直线，可以用一个简洁的方程抓住： 从一个点出发，再加上任意多个方向向量的副本。
Start at the point a, then add t copies of the direction b to reach any point on the line.
从点 a 出发，再加上 t 份方向 b，就能到达直线上任意一点。
The vector equation is r equals a plus t b.
向量方程是 r 等于 a 加 t b。
A position vector gives a point's place from the origin; a displacement from A to B is b minus a.
位置向量给出一点相对原点的位置；从 A 到 B 的位移是 b 减 a。
Two lines may be parallel if their directions are scalar multiples.
两条直线若方向成倍数关系，就平行。
They may intersect at one point.
它们也可能相交于一点。
Or they may be skew lines — not parallel, and never meeting, because they live in different planes in three dimensions.
或者是异面直线——既不平行，也永不相交，因为它们在三维空间的不同平面里。
Vectors bring a beautiful new operation.
向量带来一种优美的新运算。
Take two vectors, with an angle between them.
取两个向量，它们之间有一个夹角。
The scalar product — the dot product — multiplies matching components and adds them up.
数量积——也就是点积——把对应的分量相乘再相加。
Remarkably, that same number also equals the two lengths multiplied, times the cosine of the angle.
奇妙的是，这个同样的数， 还等于两个长度相乘，再乘以夹角的余弦。
So the dot product hands you the angle between any two vectors.
所以点积直接把任意两个向量的夹角交给你。
And when it comes out zero, the vectors are perpendicular.
而当它算出来为零时，这两个向量就相互垂直。
Worked example.
例题。
Find the angle between a equals i plus two j plus two k, and b equals two i plus two j plus k.
求 a 等于 i 加二 j 加二 k，与 b 等于二 i 加二 j 加 k 之间的夹角。
The dot product is one times two, plus two times two, plus two times one — that is eight.
点积是一乘二，加二乘二，加二乘一——等于八。
Each magnitude is three.
两个模长都是三。
So cosine of theta is eight over three times three, which is eight ninths.
所以夹角的余弦是八比三乘三，也就是九分之八。
That gives theta about twenty-seven point three degrees.
夹角大约二十七点三度。
The picture shows how the scalar product picks out how far one vector reaches along the other.
图中显示，数量积如何取出一个向量在另一个方向上的投影长度。
The second new world: differential equations.
第二个新世界：微分方程。
These link a quantity to its own rate of change.
它把一个量和它自身的变化率联系起来。
For a simple first-order equation whose variables are separable, the trick is to separate the variables — all the y's on one side, all the x's on the other — then integrate both sides.
对一个简单的一阶方程，诀窍是分离变量——所有的 y 放一边，所有的 x 放另一边—— 然后两边同时积分。
That gives a general solution, a whole family of curves, with a constant inside.
这给出一个通解，一整族曲线，里面带着一个常数。
A single known initial condition then fixes that constant, and selects the one particular curve you want.
而一个已知的条件，就能定住这个常数，从中挑出你想要的那条特解曲线。
Look at the family.
看这一族曲线。
Solving d y by d x equals x y gave y equals A e to the x squared over two.
解 d y 比 d x 等于 x y，得到 y 等于 A 乘 e 的 x 平方除以二次方。
Different values of A draw different curves.
不同的 A 画出不同的曲线。
The condition y equals one when x equals zero pins A to one, and highlights the single curve through that point — the particular solution.
条件 y 等于一当 x 等于零，把 A 钉为定值一， 并突出经过那一点的那条特解。
Always separate, integrate, then use the given condition to fix the constant.
永远先分离、再积分，最后用已知条件定常数。
Now the third world: complex numbers.
现在进入第三个新世界：复数。
Every one has the Cartesian form x plus i y — a real part and an imaginary part — and it lives as a point on the Argand diagram.
每个复数都有 x 加 i y 的形式——一个实部和一个虚部—— 它作为一个点，住在阿干图上。
Its modulus is its distance from the origin, and its argument is the angle it makes.
它的模是它到原点的距离，它的辐角是它所成的角。
The conjugate simply flips the sign of the imaginary part, a reflection across the real axis.
共轭只是把虚部的符号翻转，相当于对实轴作一次反射。
And to divide two complex numbers, you multiply the top and the bottom by the conjugate of the bottom, clearing the i from below.
而要把两个复数相除， 就把分子和分母同乘以分母的共轭，把下面的 i 清除掉。
On the Argand diagram the modulus of z is the distance from the origin, the argument is the angle from the positive real axis, and the conjugate is the reflection in the real axis.
在阿干图上，z 的模是到原点的距离，辐角是从正实轴量起的角，共轭是关于实轴的反射。
Two complex numbers are equal only when real parts match and imaginary parts match.
两个复数相等，当且仅当实部相同且虚部相同。
For a polynomial with real coefficients, any non-real roots come in conjugate pairs.
系数都是实数的多项式， 任何非实根都以共轭对出现。
The square roots of a complex number come from solving w squared equals z — two answers, opposite each other on the diagram.
一个复数的平方根，来自解 w 的平方等于 z—— 两个答案，在图上彼此相对。
Worked example.
例题。
Write three plus i, over one minus i, in the form x plus i y.
把三加 i 除以一减 i，写成 x 加 i y 的形式。
Multiply top and bottom by one plus i, the conjugate of the bottom.
分子分母同乘一加 i，也就是分母的共轭。
Expand the top: three plus three i plus i plus i squared.
展开分子：三加三 i 加 i 加 i 的平方。
That is two plus four i, because i squared is minus one.
因为 i 的平方是负一，得到二加四 i。
The bottom becomes two.
分母变成二。
Divide: one plus two i.
相除：一加二 i。
Always clear the denominator with the conjugate before you simplify.
化简之前，永远先用共轭清掉分母。
Complex numbers show their true beauty in polar form: a modulus r and an angle theta, written compactly as r e to the i theta.
复数在极坐标形式下，才显出它真正的美：一个模 r 和一个角 theta， 紧凑地写成 r e 的 i theta 次方。
Multiplication then becomes stunningly simple — the moduli just multiply, and the arguments simply add.
相乘于是变得惊人地简单——模直接相乘，辐角直接相加。
And here is the jewel: multiplying by i adds a right angle to the argument.
而这里就是那颗明珠：乘以 i，就给辐角加上一个直角。
So multiplying by i is nothing more than a quarter-turn in the plane.
所以乘以 i，不过就是在平面上转四分之一圈。
An equation or inequality in z describes a locus — one of the loci on the Argand diagram, a path or region.
z 的方程或不等式，在阿干图上描述一条轨迹——一条路径或一个区域。
The modulus of z minus a equals r is a circle of radius r centred at a: every point a fixed distance from a.
z 减 a 的模等于 r，是以 a 为圆心、半径 r 的圆：所有到 a 距离固定的点。
The modulus of z minus a equals the modulus of z minus b is the perpendicular bisector of the segment joining a and b.
z 减 a 的模等于 z 减 b 的模，是连接 a 与 b 线段的垂直平分线。
And the argument of z minus a equals theta is a half-line starting at a, at that fixed angle.
而 z 减 a 的辐角等于 theta，是从 a 出发、固定角度的一条半直线。
Sketch the locus before you algebra it.
先画轨迹，再动手代数。
Subtopic three point six is the same numerical skill from Pure two.
子主题三点六，就是纯数学第二部分的同一套数值本领。
When an equation cannot be solved exactly, trap the root with a sign change: evaluate the function at two ends of an interval; if the signs flip, a root lies between.
当方程无法精确求解时， 用变号锁住根：在区间两端计算函数值；若符号翻转，中间就有根。
Then close in with an iterative formula — x sub n plus one equals F of x sub n.
再用迭代公式逼近——x 下标 n 加一等于 F 作用于 x 下标 n。
The sequence converges when successive values get closer together.
当连续的值越来越靠近时，序列收敛。
If they drift apart, try a different rearrangement or a better start.
若它们越漂越远，就换一种整理方式，或换更好的初值。
One powerful method rides the tangent line, sliding to the axis again and again, each step landing nearer the answer, until it stops moving.
有一种强大的方法沿着切线滑动，一次又一次滑到横轴，每一步都落得离答案更近， 直到它不再移动。
Watch the iteration close in.
看迭代怎样收拢。
Start near the root, draw the tangent, read where it meets the axis, and repeat.
从靠近根的地方出发，画切线，读它与横轴的交点，再重复。
That is Newton-Raphson — fast when it works, and worth repeating from Pure two.
这就是牛顿—拉弗森法——奏效时很快，也值得从纯数学第二部分再讲一遍。
Before you go, four ways to keep your marks.
结束之前，四个保住分数的办法。
First, always split a rational function into partial fractions before you integrate or expand it.
第一，凡是有理函数，先分解成部分分式，再去积分或展开。
Second, use the dot product for the angle between vectors, and to test for perpendicularity.
第二，用点积来求向量之间的夹角，并检验是否垂直。
Third, choose your integration technique — parts, substitution, or partial fractions — from the shape of the integrand.
第三，根据被积式的形状， 选对你的积分方法——分部、换元，还是部分分式。
Fourth, give a complex number in exactly the form asked for, and show it on an Argand diagram.
第四，复数要按题目要求的形式作答， 并把它画在阿干图上。