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edited Apr 13, 2017 at 12:21

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You can define limits of functions from $\mathbb{Q}$ to $\mathbb{Q}$ and also define limits of sequences with rational values and thus in general any concept based on limit of functions and sequences can be developed easily (almost in line with the way usual calculus is developed with real numbers). In particular it is possible to define derivatives and Riemann integrals.

However such a system of calculus over rationals will be almost totally uninteresting because it will lack all the significant theorems of calculus all the significant theorems of calculus which are based on completeness property of real numbers. In essence the "calculus over rationals" will be nothing more than just fancy algebra. In general most of the interesting limits will not exist in this system.

It is a rather deep misconception that the power of calculus is because of ideas like derivatives and integrals (differential and integral calculus). The power of calculus comes entirely out of the structure of real numbers and by a sheer act of smartness/shrewdness the importance of real numbers is never emphasized in introductory calculus textbooks and students are left with the feeling that the strange new techniques of derivatives / integrals and their applications are the backbone of calculus.

You can define limits of functions from $\mathbb{Q}$ to $\mathbb{Q}$ and also define limits of sequences with rational values and thus in general any concept based on limit of functions and sequences can be developed easily (almost in line with the way usual calculus is developed with real numbers). In particular it is possible to define derivatives and Riemann integrals.

However such a system of calculus over rationals will be almost totally uninteresting because it will lack all the significant theorems of calculus which are based on completeness property of real numbers. In essence the "calculus over rationals" will be nothing more than just fancy algebra. In general most of the interesting limits will not exist in this system.

It is a rather deep misconception that the power of calculus is because of ideas like derivatives and integrals (differential and integral calculus). The power of calculus comes entirely out of the structure of real numbers and by a sheer act of smartness/shrewdness the importance of real numbers is never emphasized in introductory calculus textbooks and students are left with the feeling that the strange new techniques of derivatives / integrals and their applications are the backbone of calculus.

You can define limits of functions from $\mathbb{Q}$ to $\mathbb{Q}$ and also define limits of sequences with rational values and thus in general any concept based on limit of functions and sequences can be developed easily (almost in line with the way usual calculus is developed with real numbers). In particular it is possible to define derivatives and Riemann integrals.

However such a system of calculus over rationals will be almost totally uninteresting because it will lack all the significant theorems of calculus which are based on completeness property of real numbers. In essence the "calculus over rationals" will be nothing more than just fancy algebra. In general most of the interesting limits will not exist in this system.

It is a rather deep misconception that the power of calculus is because of ideas like derivatives and integrals (differential and integral calculus). The power of calculus comes entirely out of the structure of real numbers and by a sheer act of smartness/shrewdness the importance of real numbers is never emphasized in introductory calculus textbooks and students are left with the feeling that the strange new techniques of derivatives / integrals and their applications are the backbone of calculus.

answered Aug 4, 2016 at 7:35

Paramanand Singh ♦

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You can define limits of functions from $\mathbb{Q}$ to $\mathbb{Q}$ and also define limits of sequences with rational values and thus in general any concept based on limit of functions and sequences can be developed easily (almost in line with the way usual calculus is developed with real numbers). In particular it is possible to define derivatives and Riemann integrals.

However such a system of calculus over rationals will be almost totally uninteresting because it will lack all the significant theorems of calculus which are based on completeness property of real numbers. In essence the "calculus over rationals" will be nothing more than just fancy algebra. In general most of the interesting limits will not exist in this system.

It is a rather deep misconception that the power of calculus is because of ideas like derivatives and integrals (differential and integral calculus). The power of calculus comes entirely out of the structure of real numbers and by a sheer act of smartness/shrewdness the importance of real numbers is never emphasized in introductory calculus textbooks and students are left with the feeling that the strange new techniques of derivatives / integrals and their applications are the backbone of calculus.