Numbers and the number system
Pearson Edexcel · International GCSE · Mathematics A Topic 1 6:02 English narration · English + 中文 subtitles burned in
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Transcript
Four families carry topic 1.
第 1 主题有四类要点。
Every percentage change is a multiplier, and a reverse percentage divides by it.
每个百分比变化都是一个乘数，反向百分比就除以它。
A rounded value has bounds half a unit either side.
近似值的上下界各差半个单位。
Standard form keeps a between 1 and 10.
科学记数法中 a 在 1 到 10 之间。
And a ratio shares an amount by parts over the total.
按比分配就是用份数除以总份数再乘总量。
Order matters: 3 plus 4 times 2 squared is 19, because the index and the multiplication come before the addition.
运算顺序很重要：3 + 4 × 2² = 19，因为乘方和乘法先于加法。
Brackets change it: 3 plus 4, times 2, is 14.
括号会改变结果：(3 + 4) × 2 = 14。
Split 72 into 8 and 9, and keep splitting until every branch ends in a prime: 72 is 2 cubed times 3 squared.
把 72 分成 8 和 9，一直分下去，直到每一枝都是质数：72 = 2³ × 3²。
Fractions add over a common denominator.
分数相加要先通分。
One third is four twelfths and one quarter is three twelfths, so together they make seven twelfths.
三分之一等于十二分之四，四分之一等于十二分之三，所以合起来是十二分之七。
A fraction of an amount multiplies: three quarters of 48 is 36.
求一个量的几分之几用乘法：48 的四分之三 是 36。
To divide, multiply by the reciprocal.
除以一个分数，就乘以它的倒数。
For a recurring decimal, let x equal 0.306306 and so on; then 1000 x is 306.306 and so on.
对于循环小数，设 x 等于 0.306306……，那么 1000x 等于 306.306……。
Subtracting removes the tail: 999 x equals 306, so x is 306 over 999, which cancels to 34 over 111.
相减消去循环部分：999x 等于 306，所以 x 等于九百九十九分之三百零六，约分得一百一十一分之三十四。
Factor to primes first; the Venn diagram then reads off both answers.
先分解质因数，然后用韦恩图同时读出两个答案。
360 is 2 cubed times 3 squared times 5, and 84 is 2 squared times 3 times 7.
360 等于 2 的三次方乘 3 的平方乘 5，84 等于 2 的平方乘 3 乘 7。
The primes they share go in the overlap: 2, 2 and 3.
共有的质因数放在重叠 部分：2、2 和 3。
The rest of 360, which is 2, 3 and 5, goes on its own side, and 84 keeps its 7.
360 剩下的 2、3 和 5 放在它自己的一侧，84 留下 7。
The highest common factor is the overlap's product, 2 times 2 times 3, which is 12.
最大公因数是重叠部分的乘积，2 乘 2 乘 3，等于 12。
The lowest common multiple is the product of everything, 12 times 30 times 7, which is 2520.
最小公倍数 是所有数的乘积，12 乘 30 乘 7，等于 2520。
Take the largest square factor out: root 50 is root 25 times root 2, which is 5 root 2.
提出最大的平方因数：√50 = √25 × √2 = 5√2。
To rationalise 5 over 3 minus root 2, multiply top and bottom by the conjugate, 3 plus root 2: the denominator becomes 9 minus 2, which is 7.
要使 5/(3 − √2) 分母有理化，分子分母同乘共轭 3 + √2：分母变成 9 − 2 = 7。
For 6 over root 3, multiply by root 3 over root 3: 2 root 3.
对于 6/√3，乘以 √3/√3：得 2√3。
Fill every region, then make the regions add up to the total.
先填好每个区域，再检查各区域之和等于总数。
Of 30 students, 18 play cricket, 15 play tennis and 7 play both.
30 名学生中，18 人打板球，15 人打网球，7 人两样都打。
The overlap is filled first with 7.
先在重叠部分填 7。
Cricket only is 18 minus 7, which is 11; tennis only is 15 minus 7, which is 8.
只打板球的是 18 减 7，等于 11；只打网球的是 15 减 7，等于 8。
That accounts for 26 students, so 4 play neither, and they sit outside both circles.
这样共 26 人，所以有 4 人两样都不打，他们在两个圆之外。
Check: 11 plus 7 plus 8 plus 4 is 30.
检验：11 加 7 加 8 加 4 等于 30。
Every percentage change is a multiplier.
每一次百分比变化都是一个乘数。
A 12 percent rise multiplies by 1.12; a 20 percent fall multiplies by 0.8.
增加百分之十二就乘以 1.12；减少百分之二十就乘以 0.8。
A reverse percentage divides: 6225 after a 17 percent fall means 6225 divided by 0.83, which is 7500 rupees, never 6225 plus 17 percent.
逆向百分比用除法：减少百分之十七之后是 6225， 意味着 6225 除以 0.83，等于 7500 卢比，绝不是 6225 再加上百分之十七。
Compound change applies the multiplier once per year: 2000 at 3 percent grows to 2060, then 2121.80, then 2185.45, and after four years 2251.02.
复利变化每年乘一次乘数：2000 按百分之三增长到 2060，然后 2121.80，然后 2185.45，四年后是 2251.02。
Count the total parts before sharing, and find one unit before scaling.
分配之前先数总份数，按比例放大之前先求一份。
Simplify a ratio by dividing through: 45 to 60 is 3 to 4.
化简比时两边同除：45 比 60 等于 3 比 4。
To share 350 pounds in the ratio 2 to 5, there are 7 parts, so one part is 50 pounds; the shares are 100 and 250.
把 350 英镑按 2 比 5 分配，共 7 份，所以一份 是 50 英镑；两份分别是 100 和 250。
The unitary method finds one first: 7 pens cost 4.20, so one pen costs 0.60, and 12 pens cost 7.20.
单一法先求一个：7 支笔 4.20，所以一支笔 0.60，12 支笔 7.20。
Choose the extremes the question needs, then compute once.
先选出题目需要的极端值，再只计算一次。
A value carries bounds of plus or minus half its accuracy unit.
一个数值的上下界是加减精度单位的一半。
A speed of 19 kilometres per hour to the nearest whole lies from 18.5 up to 19.5, and a time of 1.5 hours to 1 decimal place lies from 1.45 up to 1.55.
速度为每小时 19 千米、精确到整数，范围是 18.5 到 19.5；时间 1.5 小时、精确到一位小数，范围是 1.45 到 1.55。
The greatest distance uses both upper bounds: 19.5 times 1.55 is 30.2 kilometres to 3 significant figures.
最大距离用两个上界：19.5 乘 1.55 等于 30.2 千米，保留三位有效数字。
To estimate, round to 1 significant figure first.
估算时先把每个数保留一位有效数字。
In standard form, 604000 is 6.04 times 10 to the 5.
用标准形式写，604000 等于 6.04 乘 10 的 5 次方。
A metre is 100 centimetres, but one square metre is not 100 square centimetres.
1 米是 100 厘米，但 1 平方米不是 100 平方厘米。
It holds 100 rows of 100 small squares: 10 000 square centimetres.
它包含 100 行、每行 100 个小方格：10 000 平方厘米。
Volume cubes the factor, so a litre is 1000 cubic centimetres.
体积要立方，所以 1 升是 1000 立方厘米。
And on the calculator, keep full precision and round only at the end.
用计算器时保持全部精度，最后才取近似。
Check yourself.
自我检测。
3 plus 4 times 2 squared is 19.
3 + 4 × 2² = 19。
Root 50 is 5 root 2.
√50 = 5√2。
A 12 000 pound car losing 15 percent a year is worth 12 000 times 0.85 cubed after 3 years: 7369 pounds 50.
价值 12 000 英镑的车每年贬值 15%，3 年后是 12 000 × 0.85³ = 7369.50 英镑。
350 pounds in the ratio 2 to 5 is 100 and 250.
350 英镑按 2:5 分是 100 和 250。
And 0.00007 is 7 times 10 to the minus 5.
0.00007 = 7 × 10⁻⁵。