一、英文表示：
1.13。25 thirteen-point—two-five 2. the logarithm of n to the ba b equals a
oblivious>拿破仑英语名言3.5/8 five eighths 4。 the fifth root of a squared plus x to the minus one third
5。 x to the minus two thirds plus the fifth root of a squared equals zero
副词的位置6。 a cubed plus b cubed equals a plus b into a squared minus ab plus b squared
二、英语名词定义
1.Equation
An equation is a statement of the equality between two equal number symbols.
2.Function
The modern definition of a function y of x is simply a mapping from a space X to another space Y。A mapping defined when every point x of X has a definite image y， a point of Y。
3.The limit of a quence (for example ) ：
A quence is said to have a limit L if ，for every positive number ，there is another positive number N (which may depend on ）such that for all n〉=N In this ca, we say the quence converges to L and we write ,or as .A quence which does not converge is called divergent。
4.The derivative of function f(x)
The derivative is defined by the equation provided the limit exists.英文起名 The number is also called the rate of change of f at x.
5。Statistical population
A statistical population is the t of measurements (or record of some qualitative trait)corresponding to the entire collection of units about which information is sought。
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三 几何图形的名称
1.圆 Circle 2.椭圆 ellip 3.长方形 rectangle 4。正方形 square 5。平行四边形 parallelogram 6.三角形 triangle 7。立方体 cube 8.圆锥 cone 9.曲线 curve 10.双曲线 hyperbola
四．英译汉
1。This device for reprenting real numbers geometrically is a very worthwhile aid that helps us to discover and understand better certain properties of real numbers. However， the readers should realize that all properties of real numbers that are to be accepted as theorems must be deducible from the axioms without any reference to geometry。 This does not mean that one should not make u of geometry in studying properties of real numbers. On the contrary ，the geometry often suggests the method of proof of particular theorem, and sometimes a geometric argument is more illuminating than a purely analytic proof（one depending entirely on the axioms for the real numbers)。In this book， geometric arguments are ud to large to help motivate or clarify a particular discuss。
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译文:几何化地表示实数的方法是一种非常有益的辅助手段，它可以帮助我们发现和更好的了解实数的某些性质。然而,读者应该意识到，那些将要被采用作为定理的所有有关实数的性质时不应该使用几何学。相反地,几何学经常启发特殊定理的证明方法,而且,有时候,几何学方面的论点比纯分析(它完全依赖与实数的公理）的证明更直观。在本书中，几何学的论点会大范围地使用,以帮助人们推动或阐明一些特殊的讨论。
2。Equations are of very great u. We can u equations in many mathematical problems. We may notice that almost every problem gives us one or more statement that something is equal to something； this gives us equations， with which we may work if we need to。 To solve an equation means to find the value of the unknown term。 To do this， we must change the terms about until the unknown term stand alone on one side of the unknown and the answer to the question. To solve the equation, therefore, means to move and change the terms about without making the equation untrue, until only the unknown quantity is left on one side， on matter which side。
译文：方程的用处很大.我们能将方程用于许多数学问题。我们或许注意到几乎每一个问题
都给了我们以一种或多种表示某物和某物相等的说明；这就是给出了方程，如果我们需要的话,我们就可以进行运算、解方程就是找出未知数的值。要做到这点我们必须一项直到使未知项单独处于方程的一边为止，这样一来，就是使得它等于方程另一边的那些项。然后，我们就得到未知数的值也就是问题的答案。因此解方程意味着进行移项，而不是方程失真，直到方程的一边(无论那一边）只留下一个未知数时为止。
3.The study of differential equation is one part of mathematics that, perhaps more than any other，the way we were has been directly inspired by mechanics, astronomy，loops and mathematical physics。 Its history began in the 17th century when Newton, Leibniz， and the Bernoullis solved some simple differential equations arising from problems in geometry and mechanics。 The early discoveries, daringbeginning about 1690,gradually led to the development of a lot of “special tricks” for solving cewideopenrtain special kings of differential equations， Although the special tricks are applicable in relatively few cas, they do enable us to solve many differential equations that ari in mechanics and geometry.
译文：微分方程的研究是数学的一个部分，它可能比其他部分，更多地直接受到了理学，
天文学和数学物理的推动。它的历史开始于17世纪，当时,牛顿,莱布尼茨和伯努利家族解决了一些来源于几何学和力学的简单的微分方程。这些早期发现，大约开始于1690年，逐渐导致了解决一些特定类型的微分方程的大量的“特殊窍门”的发展。尽管这些特殊窍门只适用于相当少的情形，它们确实能使我们解决许多起源与力学和几何学的微分方程.
4. A large variety of scientific problems ari in which one tries to determine something from its rate of change。 For example， we could try to compute the position of a moving particle from a knowledge of its velocity or acceleration。 Or a radioactive substance may be disintegrating at a known rate and we may be required to determine the amount of material pret after a give time。 In example like the, we are trying to determine an unknown function from prescribed information expresd in the form of an equation involving are least one of the derivatives of the unknown function. The equations are called differential equations, and their study forms one of the most challenging branches of mathematics。
