Comment on I just cited myself.

• ytg@sopuli.xyz ⁨5⁩ ⁨months⁩ ago

  The substitution property of equality is a part of its definition; you can substitute anywhere.

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  • SmartmanApps@programming.dev ⁨5⁩ ⁨months⁩ ago

    you can substitute anywhere

    And if you are rearranging algebra you have to do the exact same thing on both sides, always

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    • SmartmanApps@programming.dev ⁨5⁩ ⁨months⁩ ago

      And if you don’t then you can no longer claim they are still equal.

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      • ytg@sopuli.xyz ⁨5⁩ ⁨months⁩ ago

        For any a, b, c, if a = b and b = c, then a = c, right? The transitive property of equality.
        For any a, b, x, if a = b, then x + a = x + b. The substitution property.
        By combining both of these properties, for any a, b, x, y, if a = b and y = b + x, it follows that b + x = a + x and y = a + x.

        In our example, a is x’ (notice the ') and b is 0.999… (by definition). y is 10x’ and x is 9. Let’s fill in the values.

        If x’ = 0.9999… (true by definition) and 10x = 0.999… + 9 (true by algebraic manipulation), then 0.999… + 9 = x’ + 9 and 10x’ = x’ + 9.

        if you are rearranging algebra you have to do the exact same thing on both sides

        If you actually change any of the sides. Since, after substitution, the numeric value doesn’t change (literally the definition of equality), I don’t have to do anything – as I’m not rearranging. I’m merely presenting the same value in an equivalent manner. By contrast, when multiplying both sides by 10, since multiplication by 10 changes the concrete numeric value, I have to do it on both sides to maintain the equality relation (ditto for subtracting x’). But substitution never changes a numeric value – only rearranges what we already know.

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• zarkanian@sh.itjust.works ⁨5⁩ ⁨months⁩ ago

  They multiplied both sides by 10.

  0.9999… times 10 is 9.9999…

  X times 10 is 10x.

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  • SmartmanApps@programming.dev ⁨5⁩ ⁨months⁩ ago

    X times 10 is 10x

    10x is 9.9999999…

    As I said, they didn’t substitute on both sides, only one, thus breaking the rules around rearranging algebra. Anything you do to one side you have to do to the other.

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