实数的几何解释

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Proof the Propertis of Real Number
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worthwhile property theorem deducible reference illuminating analytic extent
a.值得做的 n.性质 n.定理 a.可推论的 n.涉及；参考 a.有启发性的；v.阐释 a.分析的；解析的 n.范围，长度
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Proof the Propertis of Real Number
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The Introducion of Real Number
重点句型
be often called ... 经常被称为
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The Ordering Relation of Real Number
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interpretation n.翻译；解释 positive numbers n.正数 negative numbers n.负数 inequalities n.不等式 if and only if 当且仅当
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Proof the Propertis of Real Number
In this book,geometric arguments are used to a large extent to help motivate or clarify a particular discuss.Nevertheless,the proofs of all the important theorems are presented in analytic form. 在这本书中，几何学被广泛的被用于激发和阐明一个 特别的问题。然而，所有重要定理的证明还是用分析 形式展现出来的。
Positive numbers lies to the right of 0 and negative numbers to the left of 0.If a<b,a point x satisfies the inequalities a<x<b if and only if x is between a and b. 正数在0的右边，负数在0的左边。如果a<b，点x满 足不等式a<x<b当且仅当x在a和b的中间。
实数的几何解释——直线上的 点
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The Introducion of Real Number
vocabularys and expressions



geometric a.几何学的 illustrate vt.举例说明 scale n.数值范围；比例 axiom n.公理 Euclidean geometric 欧几里得几 何 correspond vi.对应 conversely a.相反的 real axis n.实数轴 customary a.通常的，习惯的 interchangeably ad.可交换地
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The Introducion of Real Number
For this reason the line is often called the real line or the real axis,and it is customary to use the words real number and point interchangeably.Thus we often speak of the point x rather than the point corresponding to the real numbers. 因为这个原因，直线经常被称为实数轴，并且实数和 点是可以互相表示的。因此我们经常说点x而不是点对 应的实数。
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The Ordering Relation of Real Number
The ordering relation among the real numbers has a simple geometric interpretation.If x<y,the point x lies to the left of the point y as shown in Figure 2-4-1.
实数之间的顺序关系有一个简单的几何解释。如果 x<y,点x就在点y的左侧，正如图2-4-1所展示的。 0 1 x y
Fig.2-4-1 Real numbers represented geometrically on a line
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The Ordering Relation of Real Number
motivate vt.刺激；激发... clarify vt.阐明 nevertheless ad.然而；虽然如此 make use of 充分利用 on the contrary 正相反
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Proof the Propertis of Real Number
The device for representing real numbers geometrically is a very worthwhile aid that helps us to discover and understand better certain properties of real numbers.However,the reader should realize that all properties of real number that are to be accepted as theorems must be deducible from the axioms without any reference to geometry.
这种实数几何的表示方式对于更好的发现和理解实数 的某一性质是非常有意义的助手。然而，读者们应该 意识到实数所有的性质像定理一样一定能从不涉及任 何几何学公理中推论得出。
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Proof the Propertis of Real Number
This does not mean that one should not make use of geometry in studying properties of real numbers.On the contrary,the geometry often suggests the method of proof of a particular theorem,and sometimes a geometric argument is more illminating than a purely analytic proof(one depending entirely on the axioms for the real numbers). 这并不是意味着学习性质时不应该充分利用几何学。 相反地，几何学经常提供证明特别的定理的方法，并 且几何学论证有时比证明分析（完全依赖于实数的公 理）更有启发性。
Fig.2-4-1 Real numbers represented geometrically on a line
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The Introducion of Real Number
This choice determines the scale.If one adopts an appropriate set of axioms for Euclidean geometry,then each real number corresponds to exactly one and only one real number. 这个方式决定了数值范围。如果一个采用了欧几里得 几何中合适的集合，那么每一个实数对应且只对应一 个实数。
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The Introducion of Real Number
The reader is undoubtedly familiar with the representation of real number by means of points on a straight line.A point is selected to represent 0 and another, to the right of 0,to represent 1,as illustrated in Figure 2-41. 毫无疑问地，读者们对于实数就是直线上的点这样的几何 表示十分熟悉。一个点被选择表示为0和其他的数字，在 0的右边就表示为1，正如图2-4-1举例说明的。 0 1 x y