Números y álgebra
Diploma del Bachillerato Internacional · IB Diploma · Mathematics: Analysis and Approaches · HL Tema 1 7:13 Narración en inglés · Subtítulos en inglés + 中文 quemados en pantalla
Capítulos
Transcripción
These are selected topics, not the whole syllabus, and every example is fictional.
这些是精选主题，不是完整大纲，所有例子都是虚构的。
The highest common factor takes the smallest shared prime powers; the lowest common multiple takes the largest.
最大公因数取共有质因数的最小幂；最小公倍数取所有质因数的最大幂。
Index laws add powers only for products; standard form keeps one digit before the point.
指数法则只对乘积把指数相加；科学记数法小数点前只有一位数。
Arithmetic sequences add a common difference, geometric ones multiply by a common ratio.
等差数列加公差，等比数列乘公比。
A percentage change is one multiplier, and you reverse it by dividing.
百分比变化就是乘一个乘数，反推时用除法。
A complex number is a point; its modulus is its distance from 0.
复数是平面上的点；模是它到原点的距离。
A binomial expansion is finite for a positive whole power and needs |x| < 1 otherwise.
正整数次幂的二项展开是有限的；其他情况需要 |x| < 1。
A Taylor polynomial copies a function near its centre.
泰勒多项式在展开中心附近模仿原函数。
A supplier packs 72 pencils and 90 pens into identical bags with nothing left over.
供应商把 72 支铅笔和 90 支钢笔装进相同的袋子，不能剩。
Split each number into primes: 72 is 2 cubed times 3 squared, and 90 is 2 times 3 squared times 5.
把每个数分解质因数：72 = 2³ × 3²，90 = 2 × 3² × 5。
The primes they share sit in the overlap: 2, 3 and 3, so the HCF is 18.
共有的质因数在重叠部分：2、3、3，所以最大公因数是 18。
That makes 18 bags of 4 pencils and 5 pens.
可装 18 袋，每袋 4 支铅笔和 5 支钢笔。
Multiply every prime in the diagram for the LCM: 360.
把图中所有质因数相乘得最小公倍数：360。
Now 84 pencils and 126 pens.
再看 84 支铅笔和 126 支钢笔。
84 is 2 squared times 3 times 7, and 126 is 2 times 3 squared times 7.
84 = 2² × 3 × 7，126 = 2 × 3² × 7。
The shared part is 2 times 3 times 7, which is 42, so 42 bags of 2 pencils and 3 pens.
共有部分是 2 × 3 × 7 = 42，所以装 42 袋，每袋 2 支铅笔和 3 支钢笔。
The LCM, 252, answers a different question: when repeated packs first match.
最小公倍数 252 回答的是另一个问题：重复包装何时第一次对齐。
And an HCF can never be bigger than the smaller number.
最大公因数永远不会大于较小的那个数。
A microscope records 0.000072 metres.
显微镜测得 0.000072 米。
Standard form needs one non-zero digit before the point, so move the point until it sits after the 7: five places right.
科学记数法要求小数点前只有一位非零数字，所以把小数点移到 7 的后面：向右移五位。
Moving right made the number bigger, so balance it with 10 to the minus 5: 7.2 times 10 to the minus 5.
向右移使数变大，所以乘 10⁻⁵ 来平衡：7.2 × 10⁻⁵。
16 to the three quarters: take the fourth root first, 2, then cube it: 8.
16 的 ¾ 次方：先开四次方得 2，再立方：8。
Root 72 is root 36 times root 2, so 6 root 2, and 1 over root 2 rationalises to root 2 over 2.
√72 = √36 × √2 = 6√2，1/√2 有理化为 √2/2。
But 2 squared plus 2 cubed is 12, not 2 to the 5, which is 32: powers add only when the terms multiply.
但 2² + 2³ = 12，不是 2⁵ = 32：只有相乘时指数才相加。
Start at 5 and add 3 each time: 5, 8, 11.
从 5 开始每次加 3：5、8、11。
Plotted against n, equal steps of 3 put the terms on a straight line.
以 n 为横轴作图，相等的步长 3 使各项落在一条直线上。
The eighth term adds seven steps, not eight: 5 plus 7 times 3 is 26.
第 8 项加了 7 步而不是 8 步：5 + 7 × 3 = 26。
The sum of the first eight terms is 8 over 2 times 5 plus 26, which is 124.
前八项和是 8/2 × (5 + 26) = 124。
Now multiply instead: 12, 6, 3, and so on, with ratio a half.
改为相乘：12、6、3……公比 ½。
The running totals are 12, 18, 21, 22.5: each step closes half the remaining gap to 24.
累计和是 12、18、21、22.5：每一步都把到 24 的差距缩小一半。
So the sum to infinity is 12 over 1 minus a half, which is 24.
所以无穷和是 12 ÷ (1 − ½) = 24。
With a ratio of 2 the terms grow without limit and there is no sum to infinity.
公比为 2 时各项无限增大，没有无穷和。
A coat is reduced by 20 percent to 240 yuan.
一件外套降价 20% 后是 240 元。
The discount is of the original price, so the sale price is 0.8 times P.
折扣按原价计算，所以售价是 0.8P。
Divide: P is 240 over 0.8, which is 300.
相除：P = 240 ÷ 0.8 = 300。
Now raise the sale price by 20 percent: 240 times 1.2 is 288, not 300, because the second 20 percent is of the smaller amount.
再把售价提高 20%：240 × 1.2 = 288，不是 300，因为第二个 20% 是按较小的数计算的。
Proportion starts with a constant k found from a known pair.
比例先由已知的一对数求常数 k。
Suppose y is 6 when x is 2.
设 x = 2 时 y = 6。
If y is directly proportional to x, then y equals k x and k is 3: the line y equals 3 x.
若 y 与 x 成正比，则 y = kx，k = 3：直线 y = 3x。
If y is inversely proportional, then y equals k over x and k is 12: the curve y equals 12 over x.
若成反比，则 y = k/x，k = 12：曲线 y = 12/x。
Doubling x doubles one and halves the other.
x 加倍时，一个加倍，另一个减半。
Plot z equals 3 plus 4 i: 3 along the real axis and 4 up.
画出 z = 3 + 4i：沿实轴 3，向上 4。
Its distance from 0 is the modulus, root of 9 plus 16, which is 5, not 7.
它到原点的距离就是模，√(9 + 16) = 5，不是 7。
The conjugate 3 minus 4 i is its reflection in the real axis, with the same modulus.
共轭复数 3 − 4i 是它关于实轴的对称点，模相同。
And z times its conjugate is 9 plus 16, which is 25: a real number.
z 乘以它的共轭是 9 + 16 = 25：一个实数。
Solve x squared minus 6 x plus 13 equals zero.
解 x² − 6x + 13 = 0。
The discriminant is 36 minus 52, which is minus 16, so the square root is 4 i.
判别式 36 − 52 = −16，平方根是 4i。
The roots are 3 plus or minus 2 i: a conjugate pair, as always with real coefficients.
根是 3 ± 2i：一对共轭复数，实系数方程总是如此。
To divide by 3 plus 4 i, multiply top and bottom by its conjugate: 3 minus 4 i over 25.
除以 3 + 4i 时，分子分母同乘它的共轭：(3 − 4i)/25。
Expand 1 plus 2 x to the 5.
展开 (1 + 2x)⁵。
Pascal's row for 5 is 1, 5, 10, 10, 5, 1.
帕斯卡三角形第 5 行是 1、5、10、10、5、1。
Each term also carries 2 to the k, so multiply: 1, 10, 40, 80, 80, 32.
每一项还带有 2ᵏ，相乘得：1、10、40、80、80、32。
A coefficient is the number in front; the whole term includes the power of x.
系数是前面的数；整项还包括 x 的幂。
For a power that is not a positive whole number, the expansion never stops.
指数不是正整数时，展开式永不终止。
1 plus x to the minus 1 begins 1 minus x plus x squared.
(1 + x)⁻¹ 开头是 1 − x + x²。
Near zero the curves agree: at x equals 0.1 the approximation gives 0.91 against 0.909.
在 0 附近两条曲线吻合：x = 0.1 时近似值 0.91，精确值 0.909。
Towards x equals minus 1 they part, and beyond it the series is not valid at all.
靠近 x = −1 时两者分开，超出范围级数根本不成立。
Near x equals 0, replace e to the x by a polynomial with the same value, slope and curvature: 1 plus x plus x squared over 2.
在 x = 0 附近，用一个函数值、斜率和弯曲程度都相同的多项式代替 eˣ：1 + x + x²/2。
The factorial sits in the denominator.
阶乘在分母上。
At 0.1 it gives 1.105, against the true 1.10517.
在 0.1 处得 1.105，真值 1.10517。
Further from the centre the two curves drift apart: an approximation, not the function.
离中心越远，两条曲线越分开：这是近似，不是原函数。
Check yourself.
自我检测。
The HCF of 20 and 28 is 4, and their LCM is 140.
20 和 28 的最大公因数是 4，最小公倍数是 140。
With first term 9 and difference 2, the 6th term is 9 plus 5 twos: 19.
首项 9、公差 2，第 6 项是 9 + 5 × 2 = 19。
400 down 10 percent and up 10 percent ends at 396, not 400.
400 先降 10% 再升 10%，结果是 396，不是 400。
The modulus of 8 plus 15 i is root of 64 plus 225: 17, not 23.
8 + 15i 的模是 √(64 + 225) = 17，不是 23。
The squares 1 to 25 add to 5 times 6 times 11 over 6: 55.
1² 到 5² 的和是 5 × 6 × 11 ÷ 6 = 55。