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3

Do not confuse rigorous with foundations-first. Highschool math is neither, but it is perfectly possible to do rigorous math without starting with the foundations. A key aspect here is being utterly clear on what is taken for granted as a prerequisite. Developing mathematics foundations-first is challenging enough when addressing an audience of ...

1

Your $2 c + 5 f = -90$ can be solved using Bézout's identity: As $\gcd(2, 5) = 1$, you know that there are $2 u + 5 v = 1$ (for example, $u = 3, v = -1$), and manufacturing your $c, f$ if trivial, $c = u \cdot (-90) = -180, f = v \cdot (-90) = 90$. Replacing in the first equation this gives you a similar equation for $a, d$.

0

Your approach to automating the product of such problems seems workable, but it does require you take care to remember all the details of your set-up. Let me suggest a far less clever, but very portable method. Stand in front of board. Write the solution you want, let's say $x= 13$. Write an arithmetic sentence which is true off the top of your head like: ...

7

As a consequence of this, we get a lot of definitions that are not really definitions but are more like "defining things into existence", for example this year we learned about the square root function and this was the definition: "Let x≥0. The square root of x is the unique number r≥0 such that r2=x" You're just mistaken about what ...

7

For the same reason that elementary counting numbers of more than a single digit are explained as ones, tens, hundreds, etc. . The concept of powers and exponents has not yet been developed. Later it is re-cast in terms of powers of ten with a decimal point delineating the positive/negative powers. Even then, it's often never further defined as powers of a ...

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