中学是一个重要的年龄段，因为学生有足够的数学能力来解决高级问题，但他们还没有真正决定他们在生活中想做些什么。
Middle school is an important age because students have enough math capability to solve advanced problems, but they haven't really decided what they want to do with their lives.
他们想知道这些家长是怎样教出这么多的数学天才和音乐神童，这些家庭内部到底是什么样，他们自己是否也能做到。
They wonder what these parents do to produce so many math whizzes and music prodigies, what it's like inside the family, and whether they could do it too.
制造是如此容易，这就解释了为什么在软件开发中没有那么高的数学严密性。
Ease of manufacturing explains why we don't have much mathematical rigor in software development.
说，通过表明不确定性始终存在在数学中，，这个新成果是建立在这个基础上，而不管涉及的是什么材料。
The new work, says Szymczak, builds on that foundation by showing that the instability always exists in the math, no matter what materials are involved.
对程序员来说，最有效的离散数学的分支是概率理论。这是你在学校学完基本算术后的紧接着的课。你会问，什么是概率理论呢？
For programmers, the most useful branch of discrete math is probability theory. It's the first thing they should teach you after arithmetic, in grade school.
掌握数学的第二步是要认识到数学其实没有什么特别了不起的。
The second step in coming to grips with math is to realize that math is not particularly difficult.
但是这个例子这说明了，为什么得到的具体答案是数学解释而不是物理解释。
But the explanation is to why it comes up with that particular answer is a mathematical explanation not a physical one.
我要说的是，简单的数学运算阐明了，为什么任何大型的补偿抵消方案是不公正的、自相矛盾的、并最终是不可能的。
What I am about to explain is the simple mathematical reason why any large-scale programme of offsets is unjust, contradictory and ultimately impossible.
如此震动心灵的故事，解释了为什么这本薄薄的小书恰是对某些著作的一剂对症解药，因为多数学者研究档案后往往拿出令人望而却步的成果。
Such moving stories explain why this slim book is just the right antidote to the often daunting studies most scholars produce after working in the archives.
从另一方面说，如果数学是被发明的，为什么2+2不等于5呢？
On the other hand, if math is invented, then why can't a mathematician legitimately invent that 2 + 2 = 5?
Szymczak说，通过表明不确定性始终存在在数学中，，这个新成果是建立在这个基础上，而不管涉及的是什么材料。
The new work says Szymczak builds on that foundation by showing that the instability always exists in the math no matter what materials are involved.
也就是在这个月份的一天晚上，我，路德维格·霍勒斯·霍利，正坐在剑桥大学我的房间里埋头攻读数学著作，具体读的是本什么书，我记不大清楚了。
It was in this month something over twenty years ago that I, Ludwing Horace Holly , sat one night in my rooms at Cambridge, grinding away at some mathematical work, I forget what.
正确学习数学的方法是广度优先，而非深度优先。你需要生存在空间里，学习事物的名字，区分出什么是什么。
The right way to learn math is breadth-first, not depth-first. You need to survey the space, learn the names of things, figure out what's what.
听并圈出。陈小姐是一个数学/英语老师。格雷先生是一个音乐/语文老师。 再听一次并回答。 陈小姐是什么样的？ 格雷先生是什么样的？
Miss Chen is a maths / an English teacher. Mr Grey is a music / Chinese teacher. Listen again and answer. What is Miss Chen like? What is Mr Grey like?
这也说明了为什么一个人能成为钢琴天才或者数学高手是一个相当复杂的问题。
And that means determining why a person is a talented pianist or smart in mathematics is very tricky stuff.
本文给出什么是现代流变力学的基本概念和原理的见解，并把这些概念直接写成数学语言。
In this paper gives the view of what the basic concepts and principle of modern rheological mechanics are and put these concepts directly into mathematical language.
因为他们是密切相关的数学，我学到了什么。
For they are closely related to mathematics which I learned nothing about before.
它说没有什么以外的其他标志开始的权力，除非是做一些严重的数学。
It says nothing other than an indication of starting power unless one is up to doing some serious math.
正确学习数学的方法是广度优先，而非深度优先。你要考察的是整个数学世界，学习每个概念的名字，区分出什么是什么。
The right way to learn math is breadth-first not depth-first. You need to survey the space learn the names of things figure out what's what.
一如数学领域中的歌德巴赫猜想，“法律是什么”这个问题是法学中最著名的、尚未得到解决的问题之一。
Like the Goldbach conjecture in mathematics, what is law is one of the most famous still unsolved problems in jurisprudence.
无论它们是数学的、物理的、生物的、医学的、艺术的或音乐的，虚的对象有什么样的实体呢？
What entity do virtual objects have, whether they be mathematical, physical, biological, medical, artistic or musical?
那么，到底是什么原因导致大多数学习者在口语课堂上消极参与、不肯开口?我们有必要研究是哪些因素阻碍或促进了口语学习。
What cause learners' inactive participation or silence in the oral class?It is of necessity to investigate factors which may obstruct or enhance oral English acquisition.
我为什么说是可怕的是，因为我真正地没有善于数学和物理术语。
Why I say that's awful is because I really do not good at the mathematic and physics terminology.
因为他们是密切相关的，我什么都没学到关于数学之前。
For they are closely related to mathematics which I learned nothing about before.
是什么让数学时，以便有效地进入科学是一个神秘的谜，本书要达到不超过阐明有多深这个奥秘的。
What makes mathematics so effective when it enters science is a mystery of mysteries, and the present book wants to achieve no more than to explicate how deep this mystery is.
正确学习数学的方法是广度优先，而非深度优先。你要考察的是整个数学世界，学习每个概念的名字，区分出什么是什么。
The right way to learn math is breadth-first, not depth-first. You need to survey the space, learn the names of things, figure out what's what.
正确学习数学的方法是广度优先，而非深度优先。你要考察的是整个数学世界，学习每个概念的名字，区分出什么是什么。
The right way to learn math is breadth-first, not depth-first. You need to survey the space, learn the names of things, figure out what's what.
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