Fundamental axiom or theorem for multiplication of an equation by any real number?

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Is there a fundamental axiom or theorem stating that if two quantities are equal, multiplying both quantities by the same scalar real number results in two equal quantities?



I'm imagining something similar to Euclid's "Common Notions," particularly the notion that "If equals are added to equals, then the wholes are equal," as well as "If equals are subtracted from equals, then the remainders are equal." It seems to me that these two axioms in particular could be extended to solve my problem for multiplication by any scalar integer — multiplication by an integer $n$ is just adding $n - 1$ of the same thing — but I don't see how you could use these for multiplication by, say, $1.5$. Is there a more general axiom of some sort covering multiplication of equal quantities by any real number?



Thanks for any insight you might have!




axioms




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    If $f$ is any function, then $a=b$ implies $f(a)=f(b)$ ...
    – Hagen von Eitzen
    Jul 29 at 1:10






  • 1




    See logic : equality.
    – Mauro ALLEGRANZA
    Jul 29 at 15:07














up vote
1
down vote

favorite
1












Is there a fundamental axiom or theorem stating that if two quantities are equal, multiplying both quantities by the same scalar real number results in two equal quantities?



I'm imagining something similar to Euclid's "Common Notions," particularly the notion that "If equals are added to equals, then the wholes are equal," as well as "If equals are subtracted from equals, then the remainders are equal." It seems to me that these two axioms in particular could be extended to solve my problem for multiplication by any scalar integer — multiplication by an integer $n$ is just adding $n - 1$ of the same thing — but I don't see how you could use these for multiplication by, say, $1.5$. Is there a more general axiom of some sort covering multiplication of equal quantities by any real number?



Thanks for any insight you might have!




axioms




share|cite|improve this question















  • 4




    If $f$ is any function, then $a=b$ implies $f(a)=f(b)$ ...
    – Hagen von Eitzen
    Jul 29 at 1:10






  • 1




    See logic : equality.
    – Mauro ALLEGRANZA
    Jul 29 at 15:07












up vote
1
down vote

favorite
1









up vote
1
down vote

favorite
1






1





Is there a fundamental axiom or theorem stating that if two quantities are equal, multiplying both quantities by the same scalar real number results in two equal quantities?



I'm imagining something similar to Euclid's "Common Notions," particularly the notion that "If equals are added to equals, then the wholes are equal," as well as "If equals are subtracted from equals, then the remainders are equal." It seems to me that these two axioms in particular could be extended to solve my problem for multiplication by any scalar integer — multiplication by an integer $n$ is just adding $n - 1$ of the same thing — but I don't see how you could use these for multiplication by, say, $1.5$. Is there a more general axiom of some sort covering multiplication of equal quantities by any real number?



Thanks for any insight you might have!




axioms




share|cite|improve this question











Is there a fundamental axiom or theorem stating that if two quantities are equal, multiplying both quantities by the same scalar real number results in two equal quantities?



I'm imagining something similar to Euclid's "Common Notions," particularly the notion that "If equals are added to equals, then the wholes are equal," as well as "If equals are subtracted from equals, then the remainders are equal." It seems to me that these two axioms in particular could be extended to solve my problem for multiplication by any scalar integer — multiplication by an integer $n$ is just adding $n - 1$ of the same thing — but I don't see how you could use these for multiplication by, say, $1.5$. Is there a more general axiom of some sort covering multiplication of equal quantities by any real number?



Thanks for any insight you might have!





axioms





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 29 at 1:06









Craveable Banana

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  • 4




    If $f$ is any function, then $a=b$ implies $f(a)=f(b)$ ...
    – Hagen von Eitzen
    Jul 29 at 1:10






  • 1




    See logic : equality.
    – Mauro ALLEGRANZA
    Jul 29 at 15:07












  • 4




    If $f$ is any function, then $a=b$ implies $f(a)=f(b)$ ...
    – Hagen von Eitzen
    Jul 29 at 1:10






  • 1




    See logic : equality.
    – Mauro ALLEGRANZA
    Jul 29 at 15:07







4




4




If $f$ is any function, then $a=b$ implies $f(a)=f(b)$ ...
– Hagen von Eitzen
Jul 29 at 1:10




If $f$ is any function, then $a=b$ implies $f(a)=f(b)$ ...
– Hagen von Eitzen
Jul 29 at 1:10




1




1




See logic : equality.
– Mauro ALLEGRANZA
Jul 29 at 15:07




See logic : equality.
– Mauro ALLEGRANZA
Jul 29 at 15:07















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