How to interpret this mathematical assignment
Hello Philosophers!
Here I am with yet another question about mathematics. I am working through this book: Mathematics for the Nonmathematician, a classic work by professor Kline. I arrived at this question on page 82, if anyone happens to have the book:
9. Is it correct to assert that:
a + (bc) = (a + b)(a + b)
The chapter deals with the axioms of algebra and number in the section before this question is asked. I wasn't sure how to interpret the question. It could be referring to an axiom: would this hold as an axiom?
If a, b and c can be any integer, then the assertion doesn't hold, for three integers can be found where this equality doesn't hold.
If, however, you establish a and b, and solve for c, then you can find three numbers for which the assertion holds, for example:
a = 2
b = 3
then
c = 7 + (2/3)
To arrive at the right c for the equation to hold:
a + (bc) = (a + b)(a + b)
a + (bc) = a^2 + 2ab + b^2
bc = a^2 + 2ab + b^2 - a
c = (a^2 + 2ab + b^2 -a)/b
So, no, as an axiom about three integers it won't hold, but it is solvable, and there are triples of numbers for which it holds. Which is the correct way of looking at this question? I can provide more context from the book if needed.
Here I am with yet another question about mathematics. I am working through this book: Mathematics for the Nonmathematician, a classic work by professor Kline. I arrived at this question on page 82, if anyone happens to have the book:
9. Is it correct to assert that:
a + (bc) = (a + b)(a + b)
The chapter deals with the axioms of algebra and number in the section before this question is asked. I wasn't sure how to interpret the question. It could be referring to an axiom: would this hold as an axiom?
If a, b and c can be any integer, then the assertion doesn't hold, for three integers can be found where this equality doesn't hold.
If, however, you establish a and b, and solve for c, then you can find three numbers for which the assertion holds, for example:
a = 2
b = 3
then
c = 7 + (2/3)
To arrive at the right c for the equation to hold:
a + (bc) = (a + b)(a + b)
a + (bc) = a^2 + 2ab + b^2
bc = a^2 + 2ab + b^2 - a
c = (a^2 + 2ab + b^2 -a)/b
So, no, as an axiom about three integers it won't hold, but it is solvable, and there are triples of numbers for which it holds. Which is the correct way of looking at this question? I can provide more context from the book if needed.
Comments (7)
An equivalent question is "is it correct to assume that x=2x?", and the answer to that equivalent question is no, because there are values of x for which the assumption does not hold
The usual way you deal with bracketed expressions are by axiomatising how products interact with sums and how products interact with products. The sum-product interaction axioms are usually called the distributive laws.
[math]a.(b+c) = a.b + b.c[/math]
[math](b+c).a=b.a+c.a[/math]
There's no need for the second one if the algebra you're dealing with has a commutative product operation, IE where [math]a.b=b.a[/math] for all [math]a[/math] and [math]b[/math]. If only the first one holds. the algebra is called 'left distributive', and the first one is usually called 'the left distributive law', when the second one holds, the algebra is called 'right distributive' and the law correspondingly is called 'the right distributive law'.
The product-product interaction axiom is usually called 'associativity of the product operation' or 'the associative law'.
[math]a.(b.c)=(a.b).c[/math]
When you expand the term [math](a+b)(c+d)[/math], you're covertly using the left distributive law and the right distributive law:
[math](a+b)(c+d)=(a+b).c+(a+b).d=a.c+b.c+a.d+b.d[/math]
the first step uses the left distributive law, the second step uses the right distributive law. Structures which have both addition and multiplication operators which interact using a distributive law are typically called 'rings', if the ring also has [math]a.b=b.a[/math] for all a and b then it's called a 'commutative ring'. You can look up these topics yourself to see how they are axiomatised (there are more required properties than the ones I've presented here).
For a = b = c = 0, the equation is true.
An identity holds true for all possible values of the pronumerals (variable names). An example is [math]\sin^2\theta + \cos^2\theta = 1[/math], which is true for any value of the pronumeral [math]\theta[/math].
An equation, when contrasted with an identity, is something that is to be solved in order to find a value for one of the pronumerals. If there is only one pronumeral, the solution will be a number. Otherwise it will be a formula that uses numbers and the other pronumerals. An example is [math]\sin x = 1[/math], or another one is wrote you wrote above, followed by the instruction 'solve for c'.
For any p and q, pq ? p
Thanks, I think I get it now. I have shown the equation only holds if c has that particular value. If it has any other value, it doesn't hold. So in general, no, it is not right to assert that
a + (bc) = (a + b)(a + b)
I think the author was referring to identity. Cheers.
Thanks.
Not sure what this means. How to interpret ??
It just means equality here. I wrote ? to emphasize that the equality holds unconditionally.
For example, in the standard arithmetic p×1 = p for any p other than 0. This equality is axiomatic. But in an alternative algebra it could be axiomatically true that p×q = p for any p and q. In this algebra
a + (bc) = a + b = (a + b)(a + b)