Number and algebra
International Baccalaureate · IB Diploma · Mathematics: Analysis and Approaches · SL Topic 1 4:38 English narration · English + 中文 subtitles burned in
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Transcript
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 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。
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。
0.00045 is 4.5 times 10 to the minus 4.
0.00045 = 4.5 × 10⁻⁴。
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。