数学其实很简单，5岁也能懂
字幕摘录
| 时间 | 英文 | 中文 |
|---|---|---|
| 0:00 | You sat through years of math class. | 你坐了好几年的数学课 |
| 0:02 | You learned to follow steps, you passed tests, or you didn't, but | 你学会了遵循步骤, 你通过了测试, 或者你没有,但 |
| 0:06 | either way you walked out of school without knowing what any of it actually was. | 无论是哪种方式,你走出学校 并不知道它实际上是什么。 |
| 0:11 | Nobody told you what math is for, why each piece exists, or how it all connects. | 没有人告诉你什么是数学, 为什么每个作品都存在, 或者它是如何连接的。 |
| 0:18 | This video fixes that, not a lesson on one topic, the whole picture, in order in plain language. | 这段视频用简单的语言来修正,不是关于一个话题的教训,而是整个画面。 |
| 0:25 | By the end, you will know what every branch of math actually does and | 到最后,你会知道每个数学分支都做什么 |
| 0:29 | why | 为什么 |
| 0:30 | humans invented it. | 是人类发明的 |
| 0:32 | 始从 the. | |
| 0:34 | Math is a language. | 数学是一种语言。 |
展开字幕全文（244 条）
| 序号 | 英文 | 中文 |
|---|---|---|
| 1 | You sat through years of math class. | 你坐了好几年的数学课 |
| 2 | You learned to follow steps, you passed tests, or you didn't, but | 你学会了遵循步骤, 你通过了测试, 或者你没有,但 |
| 3 | either way you walked out of school without knowing what any of it actually was. | 无论是哪种方式,你走出学校 并不知道它实际上是什么。 |
| 4 | Nobody told you what math is for, why each piece exists, or how it all connects. | 没有人告诉你什么是数学, 为什么每个作品都存在, 或者它是如何连接的。 |
| 5 | This video fixes that, not a lesson on one topic, the whole picture, in order in plain language. | 这段视频用简单的语言来修正,不是关于一个话题的教训,而是整个画面。 |
| 6 | By the end, you will know what every branch of math actually does and | 到最后,你会知道每个数学分支都做什么 |
| 7 | why | 为什么 |
| 8 | humans invented it. | 是人类发明的 |
| 9 | 始从 the. | |
| 10 | Math is a language. | 数学是一种语言。 |
| 11 | It is the language humans invented to describe quantity, structure, and change. | 它是人类发明的语言,用来描述数量,结构和变化. |
| 12 | Every part of math you ever encountered was invented to solve a specific problem that words alone could not solve. | 你遇到的数学的每一部分都是被发明出来的,用来解决一个单词无法解决的具体问题. |
| 13 | That is the key idea. | 这是关键的想法。 |
| 14 | Every piece of math was an answer to a real question. | 每一个数学部分都是对一个真实问题的答案. |
| 15 | When you understand the question first, the math stops being random and starts making sense. | 当你首先理解问题时,数学就停止了随机性,开始讲道理. |
| 16 | The foundation of all math is numbers. | 所有数学的基础都是数字. |
| 17 | A number is a symbol that represents a quantity. | 数字是代表数量的一个符号。 |
| 18 | That's it. | 就这样 |
| 19 | Humans needed to count things. | 人类需要数东西。 |
| 20 | Sheep, | 羊头 |
| 21 | days, | 数日, |
| 22 | units of grain, so they invented symbols to represent amounts. | 所以他们发明了符号来表示数量。 |
| 23 | The counting numbers 1, 2, 3, and so on came first, then came 0, which was actually a major invention. | 计数数字1,2,3等等首先出现,然后出现0,这其实是一个重大发明. |
| 24 | 0 means the absence of quantity, and without it, | 0表示没有数量,没有数量, |
| 25 | calculation and record-keeping become far more complicated. | 计算和保存记录变得更加复杂。 |
| 26 | Then, | 接下来 |
| 27 | negative numbers. | 负数。 |
| 28 | If you earn $500 and know $700, the result is negative $200. | 如果你赚500美元,知道700美元,结果就是负200美元。 |
| 29 | You are below 0. | 你们在0以下 |
| 30 | Negative numbers represent quantities on the other side of nothing. | 负数代表无物的另一边的数量。 |
| 31 | Fractions and decimals came next. | 接下来是小数和小数 |
| 32 | Half a bag of concrete, | 半袋混凝土, |
| 33 | 2.5 meters of pipe. | 管长2.5米. |
| 34 | These describe quantities that fall between whole numbers. | 这描述了整个数字之间的数量。 |
| 35 | The key point is that our entire number system is a toolkit built piece by piece to describe more precise | 关键是,我们整个数字系统 是一个逐块构建的工具包 来描述更精确 |
| 36 | real-world situations. | 现实世界 |
| 37 | Now, here's what's actually happening when you do arithmetic. | 现在,这就是你做算术时实际发生的情况。 |
| 38 | Addition, subtraction, | 增减 |
| 39 | multiplication, and division are the four basic operations. | 乘法,而师是四大基本操作. |
| 40 | They are tools for combining and separating quantities. | 它们是合并和分离数量的工具。 |
| 41 | Multiplication is just fast addition. | 乘法只是快速的添加. |
| 42 | 6 times 4 means add 6 four times. | 6乘以4表示增加6乘以4倍. |
| 43 | Division is just the reverse. | 组织只是相反。 |
| 44 | Split 24 items into groups of six. | 将24件物品分为6组。 |
| 45 | You get four groups. | 你得到四组。 |
| 46 | The order in which you do these operations matters, | 您进行这些操作的顺序很重要, |
| 47 | which is why rules like order of operations exist. | 所以才有像行动秩序这样的规则 |
| 48 | If you calculate materials for a building project and apply the | 如果计算建筑项目的材料并应用 |
| 49 | operations in the wrong sequence, your material estimate is wrong. | 操作顺序错误,您的物质估计错误。 |
| 50 | The consequence is either waste or shortage on site. | 后果是废物或现场短缺。 |
| 51 | From arithmetic, we move to algebra. | 从算术,我们移动到代数。 |
| 52 | Algebra is arithmetic with unknowns. | 代数为未知数的算术. |
| 53 | Instead of knowing every number in a calculation, | 而不是知道每个数字的计算, |
| 54 | you know some of them and you have to find the rest. | 你认识其中一些人,你必须找到其他的 |
| 55 | We use letters, usually X or Y, to represent the unknown values. | 我们使用字母,通常是X或Y,来表示未知值. |
| 56 | An equation says that two expressions are equal. | 一个方程说两个表达法是平等的. |
| 57 | Solving the equation means rearranging it to find what the unknown | 解方程意味着重排它 找到未知的 |
| 58 | must be. | 必须是。 |
| 59 | If you know that materials cost $45 per unit and your total budget is | 如果你知道材料每单位花费45美元,你的总预算是 |
| 60 | $2,700, you write 45 times X equals | 2700美元,你写45倍X等于 |
| 61 | 2,700 and solve for X. | 2700和解决X. |
| 62 | The answer is 60 units. | 答案是60个单位。 |
| 63 | Algebra lets you work backwards from a result to find an input. | 代数让您从结果向后工作,以找到输入. |
| 64 | Every pricing model, every construction estimate, | 每个定价模式,每个建筑估计, |
| 65 | every financial projection you have ever encountered, use this logic. | 所有你遇到过的财务预测 都用这个逻辑 |
| 66 | The key point is | 关键是 |
| 67 | that algebra turns word problems from real life into solvable equations. | 代数将字问题从现实生活中变成可溶解的方程. |
| 68 | This is the important part. | 这一点很重要。 |
| 69 | Algebra has different types of equations and the type tells you what kind of relationship you're dealing with. | 代数有不同的方程类型,类型告诉你你正在处理什么样的关系. |
| 70 | A linear equation produces a straight line | 线性方程产生直线 |
| 71 | relationship. | 关系。 |
| 72 | Double the input, double the output, two workers, twice the output. | 双倍输入,双倍输出,双工人,双倍输出. |
| 73 | Simple. | 简单 |
| 74 | A | 页:1 |
| 75 | quadratic equation has one variable raised to the power of two and it describes situations where the relationship curves. | 四极方程有一个变量提升到两个的功率,它描述了关系曲线发生的情况. |
| 76 | The path a thrown object follows, the way profit rises and then falls as prices go up. | 抛物所走的道路,利润上升,然后随着物价上涨而下降。 |
| 77 | A polynomial equation can have the variable raised even higher and describes relationships with multiple turning points. | 一个多名方程可以使变量提高甚至更高,并描述与多个转折点的关系. |
| 78 | An exponential equation has the variable in the exponent and it describes growth or decline that accelerates over time. | 指数方程在指数中具有变量,它描述了随时间推移而加速的增长或下降. |
| 79 | Money compounding in an investment account, a | 投资账户中的复合货币,a |
| 80 | shape of the equation tells you the shape of the real relationship you are modeling. | 方程的形状可以告诉你 你建模中真实关系的形状。 |
| 81 | Now, here's why this matters to you. | 现在,这是为什么 这对你很重要。 |
| 82 | Every branch of math from this point on is built on top of what we just covered. | 数学的每个分支都建立在 我们刚刚覆盖的之上。 |
| 83 | They all use numbers, operations, and equations. | 他们都使用数字、操作和方程式。 |
| 84 | What changes is what they're being applied to. | 变化是它们被应用到的。 |
| 85 | Geometry is math applied to shape and space. | 几何是应用在形状和空间上的数学. |
| 86 | Humans needed to measure land, build structures, and navigate. | 人类需要测量土地、建筑结构和导航。 |
| 87 | Geometry is what they developed to do it. | 几何学是他们发展起来的 |
| 88 | Every shape has properties. | 每个形状都有属性. |
| 89 | A rectangle has a length and a width. | 矩形有长度和宽度. |
| 90 | Multiply them to get area, which tells you how much surface it covers. | 乘以获取区域, 它告诉你它覆盖了多少表面。 |
| 91 | A room that is four meters by five meters has | 一个四米五米的房间 |
| 92 | 20 square meters of floor. | 面积20平方米. |
| 93 | If you order flooring tiles without | 如果你点的地板没有 |
| 94 | calculating area first, you either buy too little and stop the job midway or you buy too much and waste money. | 先计算面积 要么买太少 中途停止工作 要么买太多 浪费钱 |
| 95 | Volume extends this idea into three dimensions. | Volume将这个想法扩展到三个维度. |
| 96 | A box, a cylinder, a sphere. | 一个盒子,一个圆柱,一个球体。 |
| 97 | Volume tells you how much the shape can hold. | 音量告诉你这个形状能维持多少. |
| 98 | A concrete column on a job site has a | 一个工地上的混凝土柱子有一个 |
| 99 | specific volume, and that volume tells you exactly how much concrete to order. | 具体音量, 并且该音量 确切地告诉你多少 具体需要订购。 |
| 100 | Geometry also gives us angle relationships. | 几何学也给了我们角度的关系. |
| 101 | A right angle is 90 degrees. | 右角为90度. |
| 102 | Two lines that meet at 90 degrees are perpendicular. | 两条在90度交汇的线是垂直的. |
| 103 | These relationships matter in construction, engineering, and any situation where fit and alignment have to be precise. | 这些关系在建筑、工程以及任何适合和调整的情况中都很重要。 |
| 104 | The key point is that geometry turns physical space into numbers you can calculate with. | 关键点是几何学将物理空间转化为数字,你可以用数字计算. |
| 105 | Connected to geometry is the Pythagorean theorem. | 连接到几何的是毕达哥里安定理. |
| 106 | In any right triangle, | 在任何正确的三角形中, |
| 107 | the square of the longest side equals the sum of the squares of the other two sides. | 的平面与其他两边的平面之和。 |
| 108 | This is written as a squared plus b squared equals | 写成平方加b平方等于 |
| 109 | squared. | 方块。 |
| 110 | If you need to know whether a corner of a building is truly square, | 如果你需要知道大楼的角落是否是真正的方形, |
| 111 | you measure three units along one wall, four units along the other, and check that the diagonal is | 你沿着一面墙测量三个单位, 沿着另一面测量四个单位, |
| 112 | exactly five units. | 五个单位 |
| 113 | If it is not, the corner is not square and everything built from it will be off. | 如果不是,角不是正方形的,从角上建造的一切都将关闭。 |
| 114 | The key point is that one theorem lets you calculate the length of any side of a right triangle if you know the other two. | 关键点是,一个定理允许你计算一个右三角形任意一侧的长度,如果你知道另外两个边. |
| 115 | Trigonometry extends this into angles. | 三角学将它延伸成角度. |
| 116 | It is the math of the relationship between a triangle's angles and its side lengths. | 它是三角形角度与其侧长的关系的数学. |
| 117 | Three functions, sine, cosine, and tangent, describe these ratios. | 三个函数(正弦、余弦和正弦)说明了这些比率。 |
| 118 | Sine is the ratio of the opposite side to the hypotenuse. | 弦是反面与下垂的比例. |
| 119 | Cosine is the ratio of the adjacent side to the hypotenuse. | 余弦(cosine)是相邻侧面与下垂的比例. |
| 120 | Tangent is opposite over adjacent. | 直径与相邻处相对. |
| 121 | If you know an angle and one side length, you can calculate every other | 如果你知道一个角度和一个边长,你可以计算每个边长 |
| 122 | measurement in the triangle. | 三角形测量。 |
| 123 | Surveyors use this to measure distances that cannot be walked. | 测量人员用这个测量无法行走的距离. |
| 124 | Engineers use it to calculate forces on diagonal supports. | 工程师们用它来计算对角支撑上的力量. |
| 125 | GPS satellites use it to calculate your position on Earth. | GPS卫星用它来计算你在地球上的位置. |
| 126 | The key point is that trigonometry lets you extract unknown distances and angles from known ones | 关键点是三角测量法允许您从已知的距离和角度提取未知的距离和角度 |
| 127 | using the properties of triangles. | 使用三角形的属性。 |
| 128 | Let me show you exactly how this works. | 让我告诉你这到底是怎么回事。 |
| 129 | Like, say you are installing a ramp. | 比如说你在安装斜坡 |
| 130 | You know the ramp is six meters long and must meet the ground at an eight degree angle. | 你知道坡道有6米长,必须八度角度与地面相遇. |
| 131 | You need to know how high the far end of the ramp rises. | 你需要知道坡道的远端有多高 |
| 132 | You use the sine function. | 您使用正弦函数 。 |
| 133 | Sine of eight degrees multiplied by six gives you approximately | 八度线乘六度 给你大概 |
| 134 | 0.83 meters. | 0.83米长. |
| 135 | That is the rise. | 这是上升。 |
| 136 | If you skip this calculation and guess, you either build a ramp | 如果你跳过这个计算和猜测, 你要么建立一个斜坡 |
| 137 | that is too steep for compliance or too shallow to reach the doorway. | 这太过陡峭,无法遵守 或太浅 无法到达大门。 |
| 138 | One formula using trigonometry gives you the exact answer | 一个使用三角测量的公式给出了准确的答案 |
| 139 | before a single piece of material is cut. | 在一块材料被切开之前 |
| 140 | Now we move to statistics. | 现在我们转向统计。 |
| 141 | Statistics is the math of data. | 统计是数据的数学. |
| 142 | When you have a large number of measurements or observations and you need to make sense of them, you use statistics. | 当您有大量的测量或观测数据,并且需要理解这些数据时,您会使用统计数据. |
| 143 | The mean is the average. | 其义为平均. |
| 144 | Add all values, divide by the count. | 添加所有值,除以数。 |
| 145 | If a company tracks monthly sales over a year and wants a single number that represents typical performance, the mean | 如果一个公司跟踪一年中的月销售量 想要一个代表典型业绩的单数 |
| 146 | gives it. | 给它。 |
| 147 | The median is the middle value when all numbers are arranged in order. | 中间值是排列所有数字时的中间值。 |
| 148 | In real estate, median home prices are used instead of | 在房地产中,使用房屋价格中位数,而不是 |
| 149 | averages because one extremely expensive house can distort an average and make a neighborhood look more expensive than it is for most buyers. | 平均而言,因为一个极其昂贵的房屋可能扭曲一个平均,使一个街区看起来比大多数买家更昂贵。 |
| 150 | The standard deviation tells you how spread out the values are. | 标准偏差告诉你这些值是怎样分布的. |
| 151 | A low standard deviation means values cluster tightly around the average. | 低标准偏差是指紧绕平均值的值组. |
| 152 | A high standard deviation means they are spread wide. | 高标准偏差意味着它们分布很广. |
| 153 | If you run a production line and your measurements have a high standard deviation, | 如果你运行生产线 和你的测量有很高的标准差, |
| 154 | your output is inconsistent, which signals a quality problem. | 您的输出不一致,这表示质量问题。 |
| 155 | The key point is that statistics converts raw data into reliable summaries that support decisions. | 关键是统计将原始数据转换成支持决策的可靠摘要. |
| 156 | Closely related is probability. | 密切相关的是概率。 |
| 157 | Probability is the math of likelihood. | 概率是可能性的数学. |
| 158 | Any time the future is uncertain and you need to reason about outcomes, you use probability. | 每当未来不确定,你需要 理性的结果,你使用概率。 |
| 159 | The probability of any event is expressed as a number between 0 and 1. | 任何事件的概率以0到1之间的数字表示. |
| 160 | 0 means it cannot happen. | 0表示不可能发生. |
| 161 | 1 means it will definitely happen. | 1表示它一定会发生. |
| 162 | 0.5 means it is equally likely to happen or not. | 0.5表示它同样有可能发生或不发生. |
| 163 | Insurance companies calculate the probability of various claims and price their policies to remain profitable. | 保险公司计算各种索赔的概率并定价其保单以保持盈利. |
| 164 | A structural engineer calculates the probability that a bridge experiences a load beyond its design spec within its expected lifetime. | 结构工程师计算桥梁在其预期寿命内承受超出其设计规格的负荷的概率. |
| 165 | And if that probability is too high, the design must change. | 如果概率过高,设计必须改变。 |
| 166 | The key point is that | 关键在于 |
| 167 | probability gives uncertain future events a numerical value that can be reasoned about and planned around. | 概率给未来不确定的事件一个数字值,可以围绕它进行推理和规划. |
| 168 | Now, here's what's actually happening when you bring algebra and geometry together. | 现在,当你把代数和几何组合起来时, 会发生什么。 |
| 169 | Coordinate geometry places shapes on a numbered grid. | 坐标几何将形状置于一个编号的网格上. |
| 170 | Every point on the grid has an X coordinate and a Y coordinate. | 网格上的每一点都有一个X坐标和一个Y坐标. |
| 171 | This means you can describe any shape with equations instead of drawings. | 这意味着你可以用方程式而不是图画来描述任何形状. |
| 172 | A straight line on the grid is described by a linear equation. | 网格上的直线用线性方程来描述. |
| 173 | A curve is described by a quadratic or polynomial equation. | 曲线由四极方程或多极方程描述. |
| 174 | This connection between algebra and geometry means that anything you can draw, you can also describe with an equation. | 代数和几何之间的这种联系意味着你可以画的任何东西,也可以用一个方程来描述. |
| 175 | And anything described by an equation can be drawn as a shape. | 而任何用方程描述的都可以被画成形状. |
| 176 | Engineers use this to model roads, bridges, and flight paths. | 工程师们用它模拟道路,桥梁和飞行路径. |
| 177 | Software uses this to render every line and curve on your screen. | 软件用它来渲染屏幕上的每条线和曲线. |
| 178 | The key point is that coordinate geometry unites the language of algebra with the visual language of shapes. | 关键点是协调几何将代数语言与形状的视觉语言统一起来. |
| 179 | This is the important part. | 这一点很重要。 |
| 180 | Calculus is built on everything we have covered. | 计算是建立在我们覆盖的一切之上的。 |
| 181 | It is the math of change. | 这是变化的数学。 |
| 182 | Specifically, it answers two questions. | 具体而言,它回答了两个问题。 |
| 183 | How fast is something changing at any given moment? | 任何时刻的变化都有多快? |
| 184 | And how much total change has happened over a period of time? | 在一段时间内发生了多少变化? |
| 185 | The first question is answered by derivatives. | 第一个问题由衍生物回答. |
| 186 | A derivative tells you the rate of change at a specific point on a function. | 一个衍生词会告诉你某个函数的特定点的变化率. |
| 187 | If a curve on a graph shows how the temperature of a piece of metal changes over time, | 如果一个图上的曲线显示 金属碎片的温度会随着时间变化 |
| 188 | the derivative at any point on that curve tells you exactly how fast the temperature is rising or falling at that exact moment. | 该曲线上的任何一点上的衍生物 都确切地告诉你 温度上升或下降的速度 |
| 189 | Engineers use derivatives to find the point where a structure bears maximum stress. | 工程师们利用衍生物寻找一个结构承受最大压力的点. |
| 190 | Financial analysts use them to find the point where a function is at its peak, which might be the price that maximizes profit. | 金融分析家利用它们来找出一个功能处于顶峰的点,这可能是利润最大化的价格. |
| 191 | The key point is that a derivative converts a function describing a total quantity into a function describing the rate of change of that quantity. | 关键点是,一个衍生物将描述总量的函数转换成描述该数量变化速率的函数. |
| 192 | The second question is answered by integrals. | 第二个问题由组成部分回答。 |
| 193 | An integral works in reverse. | 一个整体的作品反之。 |
| 194 | It adds up an infinite number of infinitely thin slices to find a total. | 它加起来是无限数量的无限薄片,可以找到一个总数. |
| 195 | If you know the rate at which water flows into a tank at every moment, | 如果你知道水流在水箱里的速度 |
| 196 | the integral tells you the total volume that entered the tank over any period of time. | 该组件告诉你在任何时间内进入坦克的总体积。 |
| 197 | If you know the force applied to an object at every point along its travel, the integral tells you the total energy transferred. | 如果你知道一个物体在旅行中的每一点 被施加的力 组成部分会告诉你所转移的总能量 |
| 198 | The key point is that an integral converts a function describing a rate into a total accumulated quantity. | 关键点是将描述一个率的函数转换为累积总量。 |
| 199 | Common mistakes. | 常见的错误。 |
| 200 | The most common is treating memorized steps as understanding. | 最常见的是将记性步骤视为理解。 |
| 201 | You learn to cross multiply | 你学会了交叉 |
| 202 | fractions without knowing why it works. | 分数不知何故. |
| 203 | You apply the quadratic formula without knowing what a quadratic relationship means in the real world. | 你应用四极公式,却不知道四极关系在现实世界意味着什么. |
| 204 | When you skip the understanding and go straight to the steps, you can follow examples, | 当你跳过理解,直接走到阶梯, 你可以效仿例子, |
| 205 | but cannot adapt when the situation changes. | 但不能在情况发生变化时适应。 |
| 206 | The fix is to always ask what the result actually means before moving on. | 问题在于,在继续前进之前,总是要询问结果的实际含义。 |
| 207 | The second mistake is treating each branch of math as separate. | 第二个错误是把数学的每个分支分别对待. |
| 208 | In practice, most real problems require several branches at once. | 实际上,大多数真正的问题需要几个分支同时出现. |
| 209 | An engineering problem might need geometry to define the shape, | 一个工程问题可能需要几何来定义形状, |
| 210 | algebra to set up relationships, trigonometry to handle angles, and calculus to find the optimal design. | 代数设置关系,三角测量处理角度,微积分寻找最佳设计. |
| 211 | The branches are connected. | 诸枝相连. |
| 212 | The third mistake is skipping units. | 第三个错误是跳过单位。 |
| 213 | A calculation without units is almost meaningless. | 没有单位的计算几乎毫无意义. |
| 214 | 10 meters per second is different from 10 meters per hour. | 10米每秒与10米每小时不同. |
| 215 | Always attach units to your numbers and track them through the calculation. | 总是在您的数字中附加单位并通过计算跟踪它们。 |
| 216 | The unit of the answer tells you whether you solved the right problem. | 答案的单位告诉你你是否解决了正确的问题. |
| 217 | Where does this actually matter? | 这到底有什么关系? |
| 218 | If you are managing a construction project, geometry determines material quantities, algebra lets you work backwards from budget to quantity, | 如果你正在管理一个建筑项目,几何决定 物质数量,代数 让你从预算倒向数量, |
| 219 | and statistics helps you evaluate contractor bids and identify outliers. | 和统计可以帮助您评价承包商的投标 和确定外部。 |
| 220 | If you work in finance, algebra builds pricing models, | 如果你在金融领域工作,代数构建了定价模式, |
| 221 | exponential functions model compound growth. | 指数函数模型复合增长。 |
| 222 | Statistics summarizes portfolio performance and probability quantifies risk. | 统计数据总结了投资组合的业绩,概率将风险量化。 |
| 223 | If you are in technology, every algorithm you use was designed using these tools. | 如果你在科技领域,你使用的每一个算法都是使用这些工具设计的. |
| 224 | Data compression, encryption, image rendering, and machine learning all reduce to applied math. | 数据压缩,加密,图像渲染,以及机器学习都减少为应用数学. |
| 225 | The consequence in each case is the same. | 每种情况的后果都是一样的。 |
| 226 | The person who understands the math can check the output, spot the error, and adapt. | 理解数学的人可以检查输出,发现错误,并适应. |
| 227 | The person who only knows the steps is dependent on the tool working correctly and | 只知道步骤的人依赖于正确操作的工具, |
| 228 | has no way to verify it. | 无法证实 |
| 229 | Here is what you can do now that you could not do before. | 这是你现在可以做的,你以前不能做的。 |
| 230 | You can look at any mathematical formula and identify which branch it belongs to and what kind of problem it is solving. | 您可以查看任何数学公式,确定它属于哪个分支,以及它正在解决什么样的问题. |
| 231 | You can recognize when a relationship is linear versus exponential and understand why it matters. | 你可以识别当一个关系是线性对指数, 并理解它为什么重要。 |
| 232 | You can read a statistical summary and assess whether it is telling you the full story or hiding something. | 你可以读一个统计摘要 并评估它是否告诉你整个故事 或者隐藏什么 |
| 233 | You can look at a calculus expression and understand whether it is asking about a rate of change or a total accumulation. | 您可以查看一个微积分表达式, 并了解它是在问变化率还是总积累。 |
| 234 | You now know that every piece of math was invented for a reason. | 现在你知道每个数学作品都是有原因的 |
| 235 | That the reason is always a real problem and that once you know the problem the math that solves it | 原因总是 一个真正的问题 一旦你知道问题 数学解决它 |
| 236 | stops being abstract and starts being useful. | 停止抽象,开始有用。 |
| 237 | You were not bad at math. | 你算术还不错 |
| 238 | You were taught formulas before you were shown what they were for. | 你被教过公式 之前,你被显示它们是什么。 |
| 239 | Now you have been shown what they are for. | 现在,你被展示了他们的目的。 |
| 240 | That is what you should have been given the first time. | 这是你第一次应该得到的 |
| 241 | Before you go do like hundreds of others and make sure to grab the | 在你走之前 像上百个其他人一样 确保抓住 |
| 242 | Simply scan the QR code or click on the link in the description box. | 简单扫描 QR 代码或点击描述框中的链接 。 |
| 243 | You do not want to miss out on this one. | 你不想错过这一次。 |
| 244 | Trust me. | 相信我,你还好吧? |
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