数学专业术语 定理证明过程中常见的短语和句子

数学常用短语与从句(Phrases or clauses frequently used in mathematics)常见的涉及运算的短语或句子1. Differentiate both sides of the equation and we get...Integrate both sides of the equation and we get...2. Differentiating both sides of the equation , we get...Integrating both sides of the equation , we get...3.Add (A) to (B) and we have…(其中(A) 、(B)为某些表达式，如不等式、等式、方程等，下同) 4．Subtract(B) from (A) and we have …5. Multiplying each term of t he equation by …, we obtain…6. Dividing the equation through by ..., we have ...7. (A) and (B) together give …(A) and (B) together yield …(A) and (B) together imply …8. Comparing (A) with (B), it is easy to see th at …9. Substituting (A) into (B), we obtain …10. Eliminating (the parameter) t from (A) and (B),we have …11. By introducing a new variable …, we can then rewrite (A) as followsBy introducing a new variable …, we can then rewrite (A) in the followi ng form12. By a simple calculation, we obtain from (A) …定理证明过程中常见的短语和句子1.下面的句型可用来表达“根据什么即可得到什么”的意思According to definition , it follows …According to hypothesis , it follows …According to assumptions, it follows …According to theorem(N), it fol lows …According to lemma (A) , it follows …According to corollary (B) , it follows …According to the remark , it follows …According to the fact that … , it follows …(可以把上面的“according to ”换成“ by” )Since …, it follows …2. 如果一个论断可以通过一些简单运算或简单推理而获得，由于这些运算或推理比较简单，读者可以自行推算，因而只需直接写出论断来，这时可用下面句型：(1) It is easy to see that …It is easy to show that …It is easy to prove that …It is easy to verify that …It is easy to check that …(2) It can easily be seen that …It can easily be sho wn that …It can easily be proved that …It can easily be verified that …It can easily be checked that …3.如果所要提及的结论比较显浅，或是众所周知，无需作进一步的证明，这时可用下面句型：(1) It is clear that …It is obvious that …It is evident that …It is well-known that …(2) Clearl y, …Obviously, …Evidently,…4.为了证明一个定理有时需要引进辅助函数，这时可用下面句型：Let us first define the function…Let us introduce a new function…Let us consider the function…Let us first investigate the function…Let …Set…Define…Put…Consider…5. 在一个定理中，有几个结论需要证明，其中有些结论比较明显，可不用证明，仅需证明余下结论即可，这时可用下面句型：Since (A) and (B) are obvious, we need only prove (C).Since (A) and (B) are trivial, we need only prove (C).Since (A) and (B) are trivial, it suffices to prove (C)6. 为了证明一个定理，有时我们并不是直接去证明，而是证明一个新的论断，一旦新的论断得到证明，已给定理不难由此而得证，这时可用下面句型：以下各句用于新的论断被证明之前：The theorem will be proved if we can show…The result will be proved if we can show…The theorem will be proved by showing that…If we can prove…then the theorem follows immediately.以下各句用于新的论断被证明之后：The theorem is now a direct consequence of what we have proved.The theorem follows immediately from what we have proved.The theorem is now evident from what we have proved.It is evident to see that the theorem holds.7.在证明过程中，有时要用到一些早已学过的知识或技巧，这时可用下面句子，以提醒读者：Recall that…Notice that…Note that…Observe that…In order to prove the theorem, we need the knowledge of …In order to obtain the following equation, we need…8. 如果需要证明的定理的假设条件是一般条件，但是，只要定理在特殊条件下成立，就不难推出定理在一般条件下也成立，这时仅需要在特殊情况下去证明定理就够了，为此可用下面句型：Without loss of generality, we may consider…Without loss of generality, we may assume…Without loss of generality, we may prove the theorem in the case…It suffices to prove the theorem in the case…We need only consider the case…For simplicity, we may take…9. 如果待证的论断可用以前用过的相似的方法或步骤进行证明，则可用下面句型：This theorem can be proved in the same way as shown before.This statement can be proved in a similar way as shown before.This theorem can be proved by the same method as employed in the last section.This theorem can be completed by the method analogous to that used above.Using the same argument as in the proof of theorem N, we can easily carry out the proof of this theorem.We now proceed as in the proof of theorem N.We shall adopt the same procedure as in the proof of theorem N.10. 如果我们用的是反证法，则其开头及结尾可用下面句型：If the statement(or assertion, conclusion) were false(or not true, not right) then…If the assertion would not hold, then…This is contrary t o…This contradicts the fact that…This leads to a contradiction.11. 表示定理已证毕或者把前面所证的总结为一结论We have thus proved the theorem.This completes the proof.The proof of the theorem is now completed.It is now obvious that the theorem holds.Thus we have derived that …Consequently, we infer that…Thus we conclude that…Thus we are led to the conclusion that …Thus we arrive at the conclusion that …Thus we can summarize what we have proved as the following theorem.12. 其它There exist(s)…such that…We claim…in fact…We are now in a position to…If otherwise…Provided that…。

数学里最经典的一句话1. “两点之间，线段最短。

”就像你要去一个地方，肯定是走直路最快呀，你想想是不是这个道理？比如你着急去商店买东西，肯定会选最近的路走，而不会绕一大圈吧！2. “三角形具有稳定性。

”哎呀，你看那些架子、桥梁，很多不都是做成三角形的嘛，这多牢固啊！就好像你搭积木，三角形的结构是不是很难被弄倒？3. “圆的周长等于直径乘圆周率。

”这就好比你绕着一个圆形的操场跑一圈，操场的直径越大，你跑的路程就越长呀！你说神奇不神奇？4. “勾股定理，直角三角形两直角边的平方和等于斜边的平方。

”哇塞，这在建筑中可太重要啦！工人叔叔们建房子的时候就得靠这个来保证房子稳稳当当的。

就像你搭一个直角的架子，用这个定理就能知道每条边该多长合适啦！5. “同底数幂相乘，底数不变，指数相加。

”嘿，这就像你有一堆同样的糖果，又得到了一些同样的糖果，那总数不就是加起来嘛，只不过这里是指数在相加呢！比如 2 的 3 次方乘以 2 的 2 次方就等于 2 的 5 次方呀！6. “方程是解决问题的好帮手。

”可不是嘛，当你遇到一个难题不知道怎么解决的时候，列个方程试试呀！就好像你想知道自己有多少颗糖，设个未知数，通过一些条件就能算出来啦！7. “概率就是可能性的大小。

”你想想抽奖的时候，你知道自己中奖的概率，不就大概能猜到自己有没有机会嘛！好比扔骰子，扔到 6 的概率就是六分之一呀！8. “函数图像能直观地反映变化。

”哇哦，你看那股市的曲线，不就是函数图像嘛，一下子就能看出是涨是跌呢！就像你记录自己每天的心情分数，画成图像就能看出心情的变化趋势啦！9. “相似三角形对应边成比例。

”这就像你有两个形状差不多的东西，它们的边之间是有一定比例关系的哟！比如两个相似的三角形模型，大的那个的边肯定比小的那个长一些，但比例是一样的呢！10. “数学归纳法，从特殊到一般的证明。

”哎呀呀，这就好像你先发现了一个小规律，然后通过一步步的验证，最后证明了一个大的结论呢！就像你发现每次加 1 都有某种规律，然后用数学归纳法就能证明啦！。

定理证明过程中常见的短语和句子1.下面的句型可用来表达“根据什么即可得到什么”的意思According to definition , it follows …According to hypothesis , it follows …According to asssumptions, it follows …According to theorem(N), it follows …According to lemma (A) , it follows …According to corollary (B) , it follows …According to the remark , it follows …According to the fact that … , it follows …(可以把上面的“according to ”换成“ by” )Since …, it follows …2. 如果一个论断可以通过一些简单运算或简单推理而获得，由于这些运算或推理比较简单，读者可以自行推算，因而只需直接写出论断来，这时可用下面句型：(1) It is easy to see that …It is easy to show that …It is easy to prove that …It is easy to verify that …It is easy to check that …(2) It can easily be seen that …It can easily be shown that …It can easily be proved that …It can easily be v erified that …It can easily be checked that …3.如果所要提及的结论比较显浅，或是众所周知，无需作进一步的证明，这时可用下面句型：(1) It is clear that …It is obvious that …It is evident that …It is well-known that …(2) Clearly, …Obviously, …Evidently,…4.为了证明一个定理有时需要引进辅助函数，这时可用下面句型：Let us first define the function…L et us introduce a new function…Let us consider the function…Let us first investigate the function…Let …Set…Define…Put…Consider…5. 在一个定理中，有几个结论需要证明，其中有些结论比较明显，可不用证明，仅需证明余下结论即可，这时可用下面句型：Since (A) and (B) are obvious, we need only prove (C).Since (A) and (B) are trivial, we need only prove (C).Since (A) and (B) are trivial, it suffices to prove (C)6. 为了证明一个定理，有时我们并不是直接去证明，而是证明一个新的论断，一旦新的论断得到证明，已给定理不难由此而得证，这时可用下面句型：以下各句用于新的论断被证明之前：The theorem will be proved if we can show…The result w ill be proved if we can show…The theorem will be proved by showing that…If we can prove…then the theorem follows immediately.以下各句用于新的论断被证明之后：The theorem is now a direct consequence of what we have proved.The theorem follows immediately from what we have proved.The theorem is now evident from what we have proved.It is evident to see that the theorem holds.7.在证明过程中，有时要用到一些早已学过的知识或技巧，这时可用下面句子，以提醒读者：Recall that…Notice that…Note that…Observe that…In order to prove the theorem, we need the knowl edge of …In order to obtain the following equation, we need…8. 如果需要证明的定理的假设条件是一般条件，但是，只要定理在特殊条件下成立，就不难推出定理在一般条件下也成立，这时仅需要在特殊情况下去证明定理就够了，为此可用下面句型：Without loss of generality, we may consider…Without loss of generality, we may assume…Without l oss of generality, we may prove the theorem in the case…It suffices to prove the theorem in the case…We need only consider the case…For simplicity, we may take…9. 如果待证的论断可用以前用过的相似的方法或步骤进行证明，则可用下面句型：This theorem can be proved in the same way as shown before.This statement can be proved in a similar way as shown before.This theorem can be proved by the same method as employed in the last section.This theorem can be completed by the method analogous to that used above.Using the same argument as in the proof of theorem N, we can easily carry out the proof of this theorem.We now proceed as in the proof of theorem N.We shall adopt the same procedure as in the proof of theorem N.10. 如果我们用的是反证法，则其开头及结尾可用下面句型：If the statement(or assertion, conclusion) were false(or not true, not right) then…If the assertion would not hold, then…This is contrary to…This contradicts the fact that…This leads to a contradiction.11. 表示定理已证毕或者把前面所证的总结为一结论We have thus proved the theorem.This completes the proof.The proof of the theorem is now completed.It is now obvious that the theorem holds.Thus we have derived that …Consequently, we infer that…Thus we conclude that…Thus we are led to the conclusion that …Thus we arrive at the conclusion that …Thus we can summarize what we have proved as the following theorem.12. 其它There exist(s)…such that…We claim…in fact…We are now in a position to…If other wise…Provided that…。

第一章集合与常用逻辑用语（公式、定理、结论图表）1.集合的有关概念（1）集合元素的三大特性：确定性、无序性、互异性．（2）元素与集合的两种关系：属于，记为∈；不属于，记为∉.（3）集合的三种表示方法：列举法、描述法、图示法．（4）五个特定的集合集合自然数集正整数集整数集有理数集实数集符号N N*或N＋Z Q R2.文字语言符号语言集合间的基本关系相等集合A与集合B中的所有元素都相同A＝B子集集合A中任意一个元素均为集合B中的元素A⊆B真子集集合A中任意一个元素均为集合B中的元素，且集合B中至少有一个元素不是集合A中的元素BA⊂≠空集空集是任何集合的子集，是任何非空集合的真子集3.集合的基本运算集合的并集集合的交集集合的补集符号表示 A ∪BA ∩B若全集为U ，则集合A 的补集为∁U A图形表示集合表示 {x |x ∈A ，或x ∈B }{x |x ∈A ，且x ∈B }{x |x ∈U ，且x ∉A }(1)A ∩A ＝A ，A ∩∅＝∅，A ∩B ＝B ∩A . (2)A ∪A ＝A ，A ∪∅＝A ，A ∪B ＝B ∪A . (3)A ∩(∁U A )＝∅，A ∪(∁U A )＝U ，∁U (∁U A )＝A . 5．常用结论（1）空集性质：①空集只有一个子集，即它的本身，∅⊆∅； ②空集是任何集合的子集（即∅⊆A ）； 空集是任何非空集合的真子集（若A ≠∅，则∅A ）．（2）子集个数：若有限集A 中有n 个元素，则A 的子集有2n 个，真子集有2n －1个，非空真子集有22n -个．典例1：已知集合{}2,4,8A =，{}2,3,4,6B =，则A B ⋂的子集的个数为（ ） A ．3 B ．4 C ．7 D ．8【答案】B【详解】因为集合{}2,4,8A =，{}2,3,4,6B =，所以{}2,4A B =， 所以A B ⋂的子集的个数为224=个.故选B.典例2：已知集合{}2N230A x x x =∈--≤∣，则集合A 的真子集的个数为（ ） A ．32 B ．31 C ．16 D ．15【答案】D【详解】由题意得{}{}{}2N230N 130,1,2,3A x x x x x =∈--≤=∈-≤≤=∣∣， 其真子集有42115-=个.故选D.（3）A ∩B ＝A ⇔A ⊆B ；A ∪B ＝A ⇔A ⊇B ．（4）(∁U A )∩(∁U B )＝∁U (A ∪B )，(∁U A )∪(∁U B )＝∁U (A ∩B ) ． 6.充分条件、必要条件与充要条件的概念若p ⇒ q ，则p 是q 的充分条件，q 是p 的必要条件 p 是q 的充分不必要条件 p ⇒ q 且q ⇏ p p 是q 的必要不充分条件 p ⇏ q 且q ⇒ pp 是q 的充要条件p ⇔ qp是q的既不充分也不必要条件p ⇏q且q ⇏p7.充分、必要条件与集合的关系设p，q成立的对象构成的集合分别为A，B．（1）p是q的充分条件⇔A⊆B，p是q的充分不必要条件⇔A B；（2）p是q的必要条件⇔B⊆A，p是q的必要不充分条件⇔B A；（3）p是q的充要条件⇔A＝B．8.全称量词和存在量词量词名称常见量词符号表示全称量词所有、一切、任意、全部、每一个等∀存在量词存在一个、至少有一个、有些、某些等∃9.全称命题和特称命题名称全称命题特称命题形式语言表示对M中任意一个x，有p(x)成立M中存在元素x0，使p(x0)成立符号表示∀x∈M，p(x)∃x0∈M，p(x0)10.全称命题与特称命题的否定<知识记忆小口诀>集合平时很常用，数学概念有不同，理解集合并不难，三个要素是关键，元素确定和互译，还有无序要牢记，空集不论空不空，总有子集在其中，集合用图很方便，子交并补很明显．<解题方法与技巧>集合基本运算的方法技巧：（1）当集合是用列举法表示的数集时，可以通过列举集合的元素进行运算，也可借助Venn 图运算；（2）当集合是用不等式表示时，可运用数轴求解．对于端点处的取舍，可以单独检验.集合常与不等式，基本函数结合，常见逻辑用语常与立体几何，三角函数，数列，线性规划等结合.充要条件的两种判断方法（1）定义法：根据p⇒q，q⇒p进行判断.（2）集合法：根据使p，q成立的对象的集合之间的包含关系进行判断.充分条件、必要条件的应用，一般表现在参数问题的求解上.解题时需注意：（1）把充分条件、必要条件或充要条件转化为集合之间的关系，然后根据集合之间的关系列出关于参数的不等式(或不等式组)求解.（2）要注意区间端点值的检验.尤其是利用两个集合之间的关系求解参数的取值范围时，不等式是否能够取等号决定端点值的取舍，处理不当容易出现漏解或增解的现象.（3）数学定义都是充要条件.。

数学证明题专业术语
在数学证明题中，通常会使用一些专业术语来表达证明的过程和结论。

以下是一些常用的数学证明题专业术语：
1. 定义（Definition）：对某一数学概念或符号进行明确的解释和规定。

2. 定理（Theorem）：经过严格证明的数学命题，通常以简短的语句形式
给出。

3. 证明（Proof）：对某一数学命题进行逻辑推导和验证的过程，以证明该
命题的真假。

4. 推导（Derivation）：从已知的数学事实或定理出发，通过逻辑推理得到新的结论或定理的过程。

5. 引理（Lemma）：在证明过程中使用的辅助性命题，通常用于支持主要
定理的证明。

6. 条件（Condition）：数学命题中的附加要求或限制，通常用于限制结论的范围或适用性。

7. 反证法（Proof by Contradiction）：通过假设与已知事实相矛盾的结论，来证明某一命题为真的方法。

8. 归纳法（Inductive Proof）：通过证明基础步骤和归纳步骤，来证明无
限序列或集合中的所有元素都满足某一性质的方法。

9. 数学归纳法（Mathematical Induction）：基于归纳法的原理，通过证明基础步骤和归纳步骤，来证明与自然数有关的数学命题的方法。

10. 构造法（Proof by Construction）：通过具体构造满足某一性质的对象或实例，来证明某一命题的方法。

这些术语可以在数学证明题中用来表达证明过程和结论，使证明更加准确、严谨和易于理解。

数学常用短语与从句(Phrases or clauses frequently used in mathematics)常见的涉及运算的短语或句子1. Differentiate both sides of the equation and we get...Integrate both sides of the equation and we get...2. Differentiating both sides of the equation , we get...Integrating both sides of the equation , we get...3.Add (A) to (B) and we have…(其中(A) 、(B)为某些表达式，如不等式、等式、方程等，下同) 4．Subtract(B) from (A) and we have …5. Multiplying each term of the equation by …, we obtain…6. Dividing the equation through by ..., we have ...7. (A) and (B) together give …(A) and (B) together yield …(A) and (B) together imply …8. Comparing (A) with (B), it is easy to see that …9. Substituting (A) into (B), we obtain …10. Eliminating (the parameter) t from (A) and (B),we have …11. By introducing a new variable …, we can then rewrite (A) as followsBy introducing a new variable …, we can then rewrite (A) in the following form12. By a simple calculation, we obtain from (A) …定理证明过程中常见的短语和句子1.下面的句型可用来表达“根据什么即可得到什么”的意思According to definition , it follows …According to hypothesis , it follows …According to assumptions, it follows …According to theorem(N), it follows …According to lemma (A) , it follows …According to corollary (B) , it follows …According to the remark , it follows …According to the fact that … , it follows …(可以把上面的“according to ”换成“by” )Since …, it follows …2. 如果一个论断可以通过一些简单运算或简单推理而获得，由于这些运算或推理比较简单，读者可以自行推算，因而只需直接写出论断来，这时可用下面句型：(1) It is easy to see that …It is easy to show that …It is easy to prove that …It is easy to verify that …It is easy to check that …(2) It can easily be seen that …It can easily be shown that …It can easily be proved that …It can easily be verified that …It can easily be checked that …3.如果所要提及的结论比较显浅，或是众所周知，无需作进一步的证明，这时可用下面句型：(1) It is clear that …It is obvious that …It is evident that …It is well-known that …(2) Clearly, …Obviously, …Evidently,…4.为了证明一个定理有时需要引进辅助函数，这时可用下面句型：Let us first define the function…Let us introduce a new function…Let us consider the function…Let us first investigate the function…Let …Set…Define…Put…Consider…5. 在一个定理中，有几个结论需要证明，其中有些结论比较明显，可不用证明，仅需证明余下结论即可，这时可用下面句型：Since (A) and (B) are obvious, we need only prove (C).Since (A) and (B) are trivial, we need only prove (C).Since (A) and (B) are trivial, it suffices to prove (C)6. 为了证明一个定理，有时我们并不是直接去证明，而是证明一个新的论断，一旦新的论断得到证明，已给定理不难由此而得证，这时可用下面句型：以下各句用于新的论断被证明之前：The theorem will be proved if we can show…The result will be proved if we can show…The theorem will be proved by showing that…If we can prove…then the theorem follows immediately.以下各句用于新的论断被证明之后：The theorem is now a direct consequence of what we have proved.The theorem follows immediately from what we have proved.The theorem is now evident from what we have proved.It is evident to see that the theorem holds.7.在证明过程中，有时要用到一些早已学过的知识或技巧，这时可用下面句子，以提醒读者：Recall that…Notice that…Note that…Observe that…In order to prove the theorem, we need the knowledge of …In order to obtain the following equation, we need…8. 如果需要证明的定理的假设条件是一般条件，但是，只要定理在特殊条件下成立，就不难推出定理在一般条件下也成立，这时仅需要在特殊情况下去证明定理就够了，为此可用下面句型：Without loss of generality, we may consider…Without loss of generality, we may assume…Without loss of generality, we may prove the theorem in the case…It suffices to prove the theorem in the case…We need only consider the case…For simplicity, we may take…9. 如果待证的论断可用以前用过的相似的方法或步骤进行证明，则可用下面句型：This theorem can be proved in the same way as shown before.This statement can be proved in a similar way as shown before.This theorem can be proved by the same method as employed in the last section.This theorem can be completed by the method analogous to that used above.Using the same argument as in the proof of theorem N, we can easily carry out the proof of this theorem.We now proceed as in the proof of theorem N.We shall adopt the same procedure as in the proof of theorem N.10. 如果我们用的是反证法，则其开头及结尾可用下面句型：If the statement(or assertion, conclusion) were false(or not true, not right) then…If the assertion would not hold, then…This is contrary to…This contradicts the fact that…This leads to a contradiction.11. 表示定理已证毕或者把前面所证的总结为一结论We have thus proved the theorem.This completes the proof.The proof of the theorem is now completed.It is now obvious that the theorem holds.Thus we have derived that …Consequently, we infer that…Thus we conclude that…Thus we are led to the conclusion that …Thus we arrive at the conclusion that …Thus we can summarize what we have proved as the following theorem.12. 其它There exist(s)…such that…We claim…in fact…We are now in a position to…If otherwise…Provided that…第九章微分方程Chapter12 Differential Equation解微分方程solve a dirrerential equation常微分方程ordinary differential equation偏微分方程partial differential equation,PDE微分方程的阶order of a differential equation微分方程的解solution of a differential equation微分方程的通解general solution of a differential equation初始条件initial condition微分方程的特解particular solution of a differential equation初值问题initial value problem微分方程的积分曲线integral curve of a differential equation可分离变量的微分方程variable separable differential equation隐式解implicit solution隐式通解inplicit general solution衰变系数decay coefficient衰变decay齐次方程homogeneous equation一阶线性方程linear differential equation of first order非齐次non-homogeneous齐次线性方程homogeneous linear equation非齐次线性方程non-homogeneous linear equation常数变易法method of variation of constant暂态电流transient stata current稳态电流steady state current伯努利方程Bernoulli equation全微分方程total differential equation积分因子integrating factor高阶微分方程differential equation of higher order悬链线catenary高阶线性微分方程linera differential equation of higher order自由振动的微分方程differential equation of free vibration强迫振动的微分方程differential equation of forced oscillation串联电路的振荡方程oscillation equation of series circuit二阶线性微分方程second order linera differential equation线性相关linearly dependence线性无关linearly independce二阶常系数齐次线性微分方程second order homogeneour linear differential equation with constant coefficient二阶变系数齐次线性微分方程second order homogeneous linear differential equation with variable coefficient特征方程characteristic equation无阻尼自由振动的微分方程differential equation of free vibration with zero damping 固有频率natural frequency简谐振动simple harmonic oscillation,simple harmonic vibration微分算子differential operator待定系数法method of undetermined coefficient共振现象resonance phenomenon欧拉方程Euler equation幂级数解法power series solution数值解法numerial solution勒让德方程Legendre equation微分方程组system of differential equations常系数线性微分方程组system of linera differential equations with constant coefficient。

prove的用法和例句一级段落标题：prove的用法和例句二级段落标题1：prove的基本定义与含义prove，是一个常见的动词，其基本含义是证明（某事物）是真实、有效或正确的。

在英语中，当我们使用prove时，通常是希望通过提供证据、理由或例子来验证某个观点或明确执行某种动作的结果。

二级段落标题2：prove作为“证明”使用时的句子与短语在很多情况下，我们会使用prove这个词来描述我们想要通过一些方式来证明某个事实。

以下是一些例句，展示了如何使用prove来表示证明：1. The scientific research proves that regular exercise is beneficial to our health.这项科学研究证明定期锻炼对我们的健康有益。

2. The witness provided strong evidence to prove the defendant's guilt.该证人提供了确凿的证据，以证明被告人有罪。

3. Historical records can prove that this historic event took place in the 19th century.历史记录可以证明这一历史事件发生在19世纪。

4. She was able to prove her innocence by producing a reliable alibi.她通过提供可靠的不在场证明她的清白无辜。

5. His hard work and dedication proved him worthy of the promotion.他的辛勤工作和奉献精神证明他值得晋升。

以上例句展示了prove在表示“证明”的意义下的应用。

它可以用于科学研究、法律案件、历史事件、个人辩解以及工作表现等各种情境中，都能准确传达出验证真实性或合理性的含义。

二级段落标题3：prove作为“竟然是”、“结果是”使用时的句子与短语除了表示证明的意义外，prove还可以用来强调某个结果与我们预期或想像的不同。

初中数学定义定理公理公式证明汇编
一、定义
1、定义（Definition）：是严格地把其中一概念确定下来，使之具有明确的含义，以便在日后的讨论中能够不产生歧义的基础。

2、直线（Line）：它是一种无穷长的无限短的线段的集合，可以把它看作一维空间中一条无穷无尽的曲线。

3、圆（Circle）：它是一种由其中一点作原点，同一点作圆心，其中一条直线作半径所围成的图形。

4、三角形（Triangle）：是由三条线段所构成的多边形，并且三条线段相互间没有共有点和共线部分。

5、平行四边形（Parallelogram）：是由四条线段构成的四边形，其中两条对角线相互平行且等长，两条边相互平行。

6、正方形（Square）：是一种特殊的四边形，四条边长度相等，且角度相等，其中两条对角线相互平行且相等。

二、定理
1、相似三角形定理（Theorem of Similar Triangles）：如果两个三角形的相对应的角都相等，则这两个三角形也相等。

2、勾股定理（Pythagorean Theorem）：对于任意直角三角形，其斜边的平方等于它的两条直边的平方之和。

3、余弦定理（Cosine Theorem）：在任意三角形内，两边的平方和减去第三边的平方，等于这两边乘积的两倍乘以余弦值（cosθ）。

4、三角形内角和定理（Theorem of Triangle Angle Sum）：在任意三角形中，三角形的三个内角和等于180度。

三、公理。

初中数学词汇数学mathematics, mathsBrE, mathAmE公理 axiom定理 theorem计算calculation运算operation证明prove假设hypothesis, hypothesespl.命题proposition算术arithmetic加 plusprep., addv., additionn.被加数 augend, summand加数 addend和 sum减minusprep., subtractv., subtractionn.被减数 minuend减数 subtrahend差 remainder乘timesprep., multiplyv., multiplicationn.被乘数multiplicand, faciend乘数 multiplicator积 product 除 divided byprep., dividev., divisionn.被除数 dividend 除数 divisor商 quotient等于 equals, is equal to, is equivalent to大于 is greater than小于 is lesser than大于等于is equal or greater than小于等于is equal or lesser than运算符 operator数字 digit数 number自然数natural number整数integer小数decimal小数点decimal point分数 fraction分子 numerator 分母 denominator比 ratio正 positive负 negative零 null, zero, nought, nil十进制 decimal system二进制 binary system十六进制hexadecimal system权weight, significance进位carry截尾truncation四舍五入round下舍入 round down上舍入 round up有效数字 significant digit无效数字 insignificant digit 代数algebra公式formula, formulaepl.单项式monomial多项式polynomial, multinomial系数coefficient未知数unknown, x-factor, y-factor, z-factor等式,方程式 equation一次方程simple equation二次方程 quadratic equation三次方程 cubic equation四次方程 quartic equation不等式 inequation阶乘 factorial对数 logarithm 指数,幂 exponent乘方 power二次方,平方 square三次方,立方 cube四次方 the power of four, the fourth powern次方 the power of n, the nth power开方 evolution, extraction二次方根,平方根 square root三次方根,立方根 cube root四次方根the root of four, the fourth rootn次方根 the root of n, the nth root集合 aggregate元素 element空集 void子集 subset交集 intersection 并集 union补集 complement映射 mapping函数 function定义域domain, field of definition值域 range常量 constant变量 variable 单调性monotonicity奇偶性parity周期性 periodicity图象 image 数列,级数series微积分calculus微分differential导数derivative极限 limit无穷大 infinitea. infinityn.无穷小infinitesimal积分integral定积分 definite integral不定积分 indefinite integral有理数 rational number无理数irrational number实数real number虚数imaginary number复数complex number矩阵matrix行列式determinant几何geometry点 point线 line面 plane体 solid线段 segment射线 radial平行 parallel相交 intersect 角 angle角度 degree弧度 radian锐角 acute angle 直角 right angle钝角 obtuse angle平角straight angle周角perigon底 base边 side高 height三角形 triangle 锐角三角形 acute triangle直角三角形 right triangle直角边 leg斜边 hypotenuse勾股定理 Pythagorean theorem 钝角三角形 obtuse triangle不等边三角形 scalene triangle 等腰三角形 isosceles triangle 等边三角形equilateral triangle四边形 quadrilateral 平行四边形 parallelogram矩形rectangle长 length宽 width菱形rhomb, rhombus, rhombipl., diamond正方形 square梯形 trapezoid直角梯形 right trapezoid等腰梯形 isosceles trapezoid 五边形 pentagon六边形hexagon七边形heptagon八边形octagon九边形enneagon十边形decagon十一边形hendecagon十二边形dodecagon多边形polygon正多边形 equilateral polygon 圆 circle圆心 centreBrE, centerAmE半径radius直径 diameter圆周率 pi弧 arc半圆 semicircle扇形 sector环 ring椭圆ellipse圆周circumference周长 perimeter面积 area轨迹locus, locapl.相似similar全等congruent四面体tetrahedron五面体pentahedron六面体hexahedron平行六面体 parallelepiped立方体 cube七面体heptahedron八面体octahedron九面体enneahedron十面体decahedron十一面体 hendecahedron十二面体 dodecahedron 二十面体icosahedron多面体polyhedron棱锥 pyramid棱柱 prism棱台 frustum of a prism旋转rotation轴 axis圆锥 cone圆柱 cylinder圆台 frustum of a cone球 sphere半球 hemisphere底面undersurface表面积surface area体积 volume空间 space坐标系coordinates坐标轴x-axis, y-axis, z-axis横坐标x-coordinate纵坐标y-coordinate原点 origin双曲线 hyperbola 抛物线parabola三角trigonometry正弦 sine余弦 cosine正切 tangent余切 cotangent正割 secant余割 cosecant反正弦arc sine反余弦arc cosine反正切 arc tangent反余切 arccotangent反正割 arc secant 反余割 arc cosecant相位 phase 周期 period 振幅amplitude内心incentreBrE, incenterAmE 外心 excentreBrE, excenterAmE 旁心 escentreBrE, escenterAmE 垂心orthocentreBrE,orthocenterAmE重心barycentreBrE, barycenterAmE 内切圆 inscribed circle 外切圆 circumcircle 统计 statistics 平均数average加权平均数 weighted average 方差 variance 标准差root-mean-square deviation, standard deviation 比例 propotion 百分比 percent 百分点percentage百分位数 percentile 排列 permutation组合 combination 概率,或然率 probability分布 distribution 正态分布 normal distribution非正态分布 abnormal distribution 图表 graph 条形统计图 bar graph 柱形统计图 histogram折线统计图 broken line graph 曲线统计图 curve diagram 扇形统计图 pie diagram 数学公式的英文读法&数学英文词汇there existfor allpq p implies q / if p, then q pq p if and only if q /p isequivalent to q / p and q areequivalentx ∈ A x belongs to A / x is an element or a member of A xA x does notbelong to A / x is not an element or a member of AAB A is contained in B / A is a subset of BAB A contains B / B is a subset of AA∩B A cap B / A meet B / A intersection BA∪B A cup B / A join B / A union B A\B A minus B / the diference between A and B A×B A cross B / the cartesian product of A and B3. Real numbersx+1 x plus onex-1 x minus onex±1x plus or minus onexy xy /3. Real numbers x+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by yx - yx + y x minus y, x plus yx y x over y= the equals signx = 5 x equals 5 / x is equal to 5x≠5 x is not equal to 5x≡y x is equivalent to or identical with yx ≡ y x is not equivalent to or identical with yx > y x is greater than y x≥y x is greater than or equal to yx < y x is less than yx≤y x is less than or equal to y0 < x < 1 zero is less than x is less than 10≤x≤1 zero is less than or equal to x is less than or equal to 1| x | mod x / modulus xx 2 x squared / x raised to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to the power nx n x to the power minus n x square root x / the square root of xx 3 cube root of xx 4 fourth root of xx n nth root of xx+y 2 x plus y all squared x y 2 x over y all squared n n factorialx ^ x hatx ˉ x barx x tildex i xi / x subscript i / x suffix i / x sub i ∑ i=1 n a i the sum from i equals one to n a i / the sum as i runs from 1 to n of the a i multiplied by yx - yx + y x minus y,3. Real numbersx+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by yx - yx + y x minus y, x plus yx y x over y= the equals signx = 5 x equals 5 / x is equal to 5x≠5 x is not equal to 5x≡y x is equivalent to or identical with yx ≡ y x is not equivalent to or identical with yx > y x is greater than yx≥y x is greater than or equal to yx < y x is less than yx≤y x is less than or equal to y0 < x < 1 zero is less than x is less than 10≤x≤1 zero is less than or equal to x is less than or equal to 1| x | mod x / modulus xx 2 x squared / x raised to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to the power nx n x to the power minus n x square root x / the square root of xx 3 cube root of xx 4 fourth root of xx n nth root of xx+y 2 x plus y all squared x y 2 x over y all squared n n factorialx ^ x hatx ˉ x barx x tildex i xi / x subscript i / x suffix i / x sub i∑ i=1 n a i the sum from i equals one to n a i / the sum as i runs from 1 to n of the a i plus yx y x over y = the equals signx = 5 x equals 5 /3. Real numbersx+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by yx - yx + y x minus y, x plus yx y x over y= the equals signx = 5 x equals 5 / x is equal to 5x≠5 x is not equal to 5x≡y x is equivalent to or identical with yx ≡ y x is not equivalent to or identical with yx > y x is greater than y x≥y x is greater than or equal to yx < y x is less than yx≤y x is less than or equal to y0 < x < 1 zero is less than x is less than 10≤x≤1 zero is less than or equal to x is less than or equal to 1| x | mod x / modulus xx 2 x squared / x raised to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to the power nx n x to the power minus n x square root x / the square root of xx 3 cube root of xx 4 fourth root of xx n nth root of xx+y 2 x plus y all squared x y 2 x over y all squared n n factorialx ^ x hatx ˉ x barx x tildex i xi / x subscript i / x suffix i / x sub i∑ i=1 n a i the sum from i equals one to n a i / the sum as i runs from 1 to n of the a i is equal to 5x≠5x is not equal to 5x≡y x is equivalent to or identical withyx ≡ y x is not equivalent to or identical with yx > y x is greater than yx≥y x is greater than or equal to yx < y x is less than yx≤y x is less than or equal to y0 <3. Real numbersx+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by yx - yx + y x minus y, x plus yx y x over y= the equals signx = 5 x equals 5 / x is equal to 5 x≠5 x is not equal to 5x≡y x is equivalent to or identical with yx ≡ y x is not equivalent to or identical with yx > y x is greater than y x≥y x is greater than or equal to yx < y x is less than yx≤y x is less than or equal to y0 < x < 1 zero is less than x is less than 10≤x≤1 zero is less than or equal to x is less than or equal to 1| x | mod x / modulus xx 2 x squared / x raised to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to the power nx n x to the power minus nx square root x / the squareroot of xx 3 cube root of xx 4 fourth root of xx n nth root of xx+y 2 x plus y all squared x y 2 x over y all squared n n factorialx ^ x hatx ˉ x barx x tildex i xi / x subscript i / x suffix i / x sub i∑ i=1 n a i the sum from i equals one to n a i / the sum as i runs from 1 to n of the a i < 1 zero is less than3. Real numbersx+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by yx - yx + y x minus y, x plus y x y x over y= the equals signx = 5 x equals 5 / x is equal to 5x≠5 x is not equal to 5x≡y x is equivalent to or identical with yx ≡ y x is not equivalent to or identical with yx > y x is greater than y x≥y x is greater than or equal to yx < y x is less than yx≤y x is less than or equal to y0 < x < 1 zero is less than x is less than 10≤x≤1 zero is less t han or equal to x is less than or equal to 1| x | mod x / modulus xx 2 x squared / x raised to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to the power nx n x to the power minus n x square root x / the square root of xx 3 cube root of xx 4 fourth root of xx n nth root of xx+y 2 x plus y all squared x y 2 x over y all squared n n factorialx ^ x hatx ˉ x barx x tildex i xi / x subscript i / x suffix i / x sub i∑ i=1 n a i the sum from i equals one to n a i / the sum as i runs from 1 to n of the a i is less than 10≤x≤1zero is less than or equal to3. Real numbers x+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by yx - yx + y x minus y, x plus yx y x over y= the equals signx = 5 x equals 5 / x is equal to 5x≠5 x is not equal to 5x≡y x is equivalent to or identical with yx ≡ y x is not equivalent to or identical with yx > y x is greater than y x≥y x is greater than or equal to yx < y x is less than yx≤y x is less than or equal to y0 < x < 1 zero is less than x is less than 10≤x≤1 zero is less than or equal to x is less than or equal to 1| x | mod x / modulus xx 2 x squared / x raised to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to the power nx n x to the power minus n x square root x / the square root of xx 3 cube root of xx 4 fourth root of xx n nth root of xx+y 2 x plus y all squared x y 2 x over y all squared n n factorialx ^ x hatx ˉ x barx x tildex i xi / x subscript i / x suffix i / x sub i ∑ i=1 n a i the sum from i equals one to n a i / the sum as i runs from 1 to n of the a i is less than or equal to 1|3. Real numbersx+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by yx - yx + y x minus y, x plus yx y x over y= the equals signx = 5 x equals 5 / x is equal to 5x≠5 x is not equal to 5x≡y x is equivalent to or identical with yx ≡ y x is not equivalent to or identical with yx > y x is greater than y x≥y x is greater than or equal to yx < y x is less than yx≤y x is less than or equal to y0 < x < 1 zero is less than x is less than 10≤x≤1 zero is less than or equal to x is less than or equal to 1| x | mod x / modulus xx 2 x squared / x raised to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to the power nx n x to the power minus n x square root x / the square root of xx 3 cube root of xx 4 fourth root of xx n nth root of xx+y 2 x plus y all squared x y 2 x over y all squared n n factorial x ^ x hatx ˉ x barx x tildex i xi / x subscript i / x suffix i / x sub i∑ i=1 n a i the sum from i equals one to n a i / the sum as i runs from 1 to n of the a i | mod3. Real numbersx+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by yx - yx + y x minus y, x plus yx y x over y= the equals signx = 5 x equals 5 / x is equal to 5x≠5 x is not equal to 5x≡y x is equivalent to or identical with yx ≡ y x is not equivalent to or identical with yx > y x is greater than y x≥y x is greater than or equal to yx < y x is less than yx≤y x is less than or equal to y0 < x < 1 zero is less than x is less than 10≤x≤1 zero is less than or equal to x is less than or equal to 1| x | mod x / modulus xx 2 x squared / x raised to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to the power nx n x to the power minus n x square root x / the square root of xx 3 cube root of x x 4 fourth root of xx n nth root of xx+y 2 x plus y all squared x y 2 x over y all squared n n factorialx ^ x hatx ˉ x barx x tildex i xi / x subscript i / x suffix i / x sub i∑ i=1 n a i the sum from i equals one to n a i / the sum as i runs from 1 to n of the a i / modulus3. Real numbersx+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by yx - yx + y x minus y, x plus yx y x over y= the equals signx = 5 x equals 5 / x is equalto 5x≠5 x is not equal to 5x≡y x is equivalent to or identical with yx ≡ y x is not equivalent to or identical with yx > y x is greater than y x≥y x is greater than or equal to yx < y x is less than yx≤y x is less than or equal to y0 < x < 1 zero is less than x is less than 10≤x≤1 zero is less than or equal to x is less than or equal to 1| x | mod x / modulus xx 2 x squared / x raised to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to the power n x n x to the power minus n x square root x / the square root of xx 3 cube root of xx 4 fourth root of xx n nth root of xx+y 2 x plus y all squared x y 2 x over y all squared n n factorialx ^ x hatx ˉ x barx x tildex i xi / x subscript i / x suffix i / x sub i∑ i=1 n a i the sum from i equals one to n a i / the sum as i runs from 1 to n of the a ix 2 x squared /3. Real numbersx+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by yx - yx + y x minus y, x plus yx y x over y= the equals signx = 5 x equals 5 / x is equal to 5x≠5 x is not equal to 5x≡y x is equivalent to or identical with yx ≡ y x is not equivalent to or identical with yx > y x is greater than y x≥y x is greater than or equal to yx < y x is less than yx≤y x is less than or equal to y0 < x < 1 zero is less than x is less than 10≤x≤1 zero is less than or equal to x is less than or equal to 1| x | mod x / modulus xx 2 x squared / x raised to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to the power nx n x to the power minus n x square root x / the square root of xx 3 cube root of xx 4 fourth root of xx n nth root of xx+y 2 x plus y all squared x y 2 x over y all squared n n factorialx ^ x hatx ˉ x barx x tildex i xi / x subscript i / x suffix i / x sub i∑ i=1 n a i the sum from i equals one to n a i / the sum as i runs from 1 to n of the a i raised to the power 2x 3 x cubedx 4 x to the fourth /3. Real numbersx+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by yx - yx + y x minus y, x plus yx y x over y= the equals signx = 5 x equals 5 / x is equal to 5x≠5 x is not equal to 5x≡y x is equivalent to or identical with yx ≡ y x is not equivalent to or identical with yx > y x is greater than y x≥y x is greater than or equal to yx < y x is less than yx≤y x is less tha n or equal to y 0 < x < 1 zero is less than x is less than 10≤x≤1 zero is less than or equal to x is less than or equal to 1| x | mod x / modulus xx 2 x squared / x raised to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to the power nx n x to the power minus n x square root x / the square root of xx 3 cube root of xx 4 fourth root of xx n nth root of xx+y 2 x plus y all squared x y 2 x over y all squared n n factorialx ^ x hatx ˉ x barx x tildex i xi / x subscript i / x suffix i / x sub i∑ i=1 n a i the sum from i equals one to n a i / the sum as i runs from 1 to n of the a i to the power fourx n x to the nth /3. Real numbersx+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by yx - yx + y x minus y, x plus yx y x over y= the equals signx = 5 x equals 5 / x is equal to 5x≠5 x is not equal to 5x≡y x is equi valent to or identical with yx ≡ y x is not equivalent to or identical with y x > y x is greater than y x≥y x is greater than or equal to yx < y x is less than yx≤y x is less than or equal to y0 < x < 1 zero is less than x is less than 10≤x≤1 zero is less than or equal to x is less than or equal to 1| x | mod x / modulus xx 2 x squared / x raised to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to the power nx n x to the power minus n x square root x / the square root of xx 3 cube root of xx 4 fourth root of xx n nth root of xx+y 2 x plus y all squared x y 2 x over y all squared n n factorialx ^ x hatx ˉ x barx x tildex i xi / x subscript i / x suffix i / x sub i∑ i=1 n a i the sum from i equals one to n a i / the sum as i runs from 1 to n of the a i to the power nx −n x to the power minus nx square root3. Real numbersx+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by yx - yx + y x minus y, x plus yx y x over y = the equals signx = 5 x equals 5 / x is equal to 5x≠5 x is not equal to 5x≡y x is equivalent to or identical with yx ≡ y x is not equivalent to or identical with yx > y x is greater than y x≥y x is greater than or equal to yx < y x is less than yx≤y x is less than or equal to y0 < x < 1 zero is less than x is less than 10≤x≤1 zero is less tha n or equal to x is less than or equal to 1| x | mod x / modulus xx 2 x squared / x raised to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to thepower nx n x to the power minus n x square root x / the square root of xx 3 cube root of xx 4 fourth root of xx n nth root of xx+y 2 x plus y all squared x y 2 x over y all squared n n factorialx ^ x hatx ˉ x barx x tildex i xi / x subscript i / x suffix i / x sub i∑ i=1 n a i the sum from i equals one to n a i / the sum as i runs from 1 to n of the a i / the square root of3. Real numbersx+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by y x - yx + y x minus y, x plus yx y x over y= the equals signx = 5 x equals 5 / x is equal to 5x≠5 x is not equal to 5x≡y x is equivalent to or identical with yx ≡ y x is not equivalent to or identical with yx > y x is greater than y x≥y x is greater than or equal to yx < y x is less than yx≤y x is less than or equal to y0 < x < 1 zero is less than x is less than 10≤x≤1 zero is less than or equal to x is less than or equal to 1| x | mod x / modulus xx 2 x squared / x raised to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to the power nx n x to the power minus n x square root x / the square root of xx 3 cube root of xx 4 fourth root of xx n nth root of xx+y 2 x plus y all squared x y 2 x over y all squared n n factorialx ^ x hatx ˉ x barx x tildex i xi / x subscript i / x suffix i / x sub i∑ i=1 n a i the sum from i equals one to n a i / the sum as i runs from 1 to n of the a i x 3 cube root of3. Real numbersx+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by yx - yx + y x minus y, x plus yx y x over y= the equals signx = 5 x equals 5 / x is equal to 5x≠5 x is not equal to 5x≡y x is equivalent to or identical with yx ≡ y x is not equivalent to or identical with yx > y x is greater than y x≥y x is greater than or equal to yx < y x is less than yx≤y x is less tha n or equal to y0 < x < 1 zero is less than x is less than 10≤x≤1 zero is less than or equal to x is less than or equal to 1| x | mod x / modulus xx 2 x squared / x raised to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to the power nx n x to the power minus n x square root x / the square root of xx 3 cube root of xx 4 fourth root of xx n nth root of xx+y 2 x plus y all squared x y 2 x over y all squared n n factorialx ^ x hatx ˉ x barx x tilde x i xi / x subscript i / x suffix i / x sub i∑ i=1 n a i the sum from i equals one to n a i / the sum as i runs from 1 to n of the a ix 4 fourth root of3. Real numbersx+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by yx - yx + y x minus y, x plus yx y x over y= the equals signx = 5 x equals 5 / x is equal to 5x≠5 x is not equal to 5x≡y x is equivalent to or identical with yx ≡ y x is not equivalent to or identical with yx > y x is greater than y x≥y x is greater than or equal to yx < y x is less than yx≤y x is less than or equal to y0 < x < 1 zero is less than x is less than 10≤x≤1 zero is less than or equal to x is less than or equal to 1| x | mod x / modulus xx 2 x squared / x raised to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to the power nx n x to the power minus n x square root x / the square root of xx 3 cube root of xx 4 fourth root of xx n nth root of x x+y 2 x plus y all squared x y 2 x over y all squared n n factorialx ^ x hatx ˉ x barx x tildex i xi / x subscript i / x suffix i / x sub i∑ i=1 n a i the sum from i equals one to n a i / the sum as i runs from 1 to n of the a ix n nth root of3. Real numbersx+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by yx - yx + y x minus y, x plus yx y x over y= the equals signx = 5 x equals 5 / x is equalto 5x≠5 x is not equal to 5x≡y x is equivalent to or identical with yx ≡ y x is not equivalent to or identical with yx > y x is greater than y x≥y x is greater than or equal to yx < y x is less than yx≤y x is less than or equal to y0 < x < 1 zero is less than x is less than 10≤x≤1 zero is less than or equal to x is less than or equal to 1| x | mod x / modulus xx 2 x squared / x raised to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to the power n x n x to the power minus n x square root x / the square root of xx 3 cube root of xx 4 fourth root of xx n nth root of xx+y 2 x plus y all squared x y 2 x over y all squared n n factorialx ^ x hatx ˉ x barx x tildex i xi / x subscript i / x suffix i / x sub i∑ i=1 n a i the sum from i equals one to n a i / the sum as i runs from 1 to n of the a ix+y 2 x plus y all squared3. Real numbersx+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by yx - yx + y x minus y, x plus yx y x over y= the equals signx = 5 x equals 5 / x is equal to 5x≠5 x is not equal to 5x≡y x is equivalent to or identical with yx ≡ y x is not equivalent to or identical with yx > y x is greater than y x≥y x is greater than or equal to yx < y x is less than yx≤y x is less than or equal to y0 < x < 1 zero is less than x is less than 10≤x≤1 zero is less than or equal to x is less than or equal to 1| x | mod x / modulus x x 2 x squared / x raised to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to the power nx n x to the power minus n x square root x / the square root of xx 3 cube root of xx 4 fourth root of xx n nth root of xx+y 2 x plus y all squared x y 2 x over y all squared n n factorialx ^ x hatx ˉ x barx x tildex i xi / x subscript i / x suffix i / x sub i∑ i=1 n a i the sum from i equals one to n a i / the sum as i runs from 1 to n of the a i y 2 x overy all squaredn n factorialx ^ x hatx ¯x barx ˜x tilde x i xi /3. Real numbersx+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by yx - yx + y x minus y, x plus yx y x over y= the equals signx = 5 x equals 5 / x is equal to 5x≠5 x is not equal to 5x≡y x is equivalent to or identical with yx ≡ y x is not equivalent to or identical with y x > y x is greater than y x≥y x is grea ter than or equal to yx < y x is less than yx≤y x is less than or equal to y0 < x < 1 zero is less than x is less than 10≤x≤1 zero is less than or equal to x is less than or equal to 1| x | mod x / modulus xx 2 x squared / x raised to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to the power nx n x to the power minus n x square root x / the square root of xx 3 cube root of xx 4 fourth root of xx n nth root of xx+y 2 x plus y all squared x y 2 x over y all squared n n factorialx ^ x hatx ˉ x barx x tildex i xi / x subscript i / x suffix i / x sub i∑ i=1 n a i the sum from i equals one to n a i / the sum as i runs from 1 to n of the a i subscript i /3. Real numbers x+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by yx - yx + y x minus y, x plus yx y x over y= the equals signx = 5 x equals 5 / x is equal to 5x≠5 x is not equal to 5 x≡y x is equivalent to or identical with yx ≡ y x is not equivalent to or identical with yx > y x is greater than y x≥y x is greater than or equal to yx < y x is less than yx≤y x is less than or equal to y0 < x < 1 zero is less than x is less than 10≤x≤1 zero is less than or equal to x is less than or equal to 1| x | mod x / modulus xx 2 x squared / x raised to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to the power nx n x to the power minus n x square root x / the square。

关于定理的名言1.定理是数学中的基本概念，是经过证明并得到广泛接受的数学断言。

2.“数学是一门科学，定理是它的灵魂。

”- 黎曼3.“定理是数学中的瑰宝，揭示了自然界的奥秘。

”- 康托尔4.“定理是从真理而生的旗帜，在人类思维的灿烂星空中闪耀。

”- 勒贝格5.“每一个定理都是一个宝贵的收获，是人类智慧的结晶。

”- 狄利克雷6.“定理是人类智慧的结晶，是对数学真理的坚定追求。

”- 高斯7.“定理是人类思维的辉煌成果，是数学世界中的明亮星辰。

”- 康威8.“定理是数学中永恒的丰碑，是神秘之门的钥匙。

”- 弗雷德霍尔姆9.“每一个定理都是人类智慧的巅峰，是人类对宇宙规律的理解。

”- 希尔伯特10.“定理是数学的光辉，是抵达真理之岛的航标。

”- 康威11.“定理是数学舞台上的明星，点亮了整个思维的大舞台。

”- 华罗庚12.“每一个定理的证明都是一次攀登巅峰的征程。

”- 波利亚13.“定理是黑暗中的闪电，将真理照亮人类前行的道路。

”- 约翰逊14.“定理是数学的玫瑰，用自身的尖刺穿透了谜题的迷雾。

”- 康威15.“定理是思想的奇迹，展示了数学思维的无穷魅力。

”- 科尔曼16.“每一个定理都是一次胜利的战役，是智慧与勇气的结晶。

”- 康威17.“定理是贯穿时空的纽带，连接了过去、现在和未来。

”- 希尔伯特18.“定理是数学之旅的里程碑，引导我们穿越宇宙的星空。

”- 康托尔19.“每一个定理都是一个谜题的解码器，带领我们揭示世界的奥秘。

”- 康威20.“定理是思维的灯塔，照亮了我们在知识海洋中航行的方向。

”- 黎曼21.如果每个人都听从命运的指引，那么那个实际上并不存在的命运就会消失。

22.定理是思维的宝藏，而命题则是探宝者的地图。

23.定理就像是数学世界的石碑，用永恒的语言记录着人类智慧的足迹。

24.人们常常用定理来证明自己的观点，而用名言来彰显自己的智慧。

25.定理有如火炬，照亮了思维的前行道路。

26.定理是思维的几何图景，让理性之花在其中绽放。