Intereting Posts

Is there a prime between every pair of consecutive triangular numbers?
Compute cos(5°) to 5 decimal places with Maclaurin's Series
Transforming the cubic Pell-type equation for the tribonacci numbers
What is $\int_{0}^{\pi}\frac{x}{x^2+\ln^2(2\sin x)}\:\mathrm{d}x$?
Integral of $\log(\sin(x))$ using contour integrals
Statement that is provable in $ZFC+CH$ yet unprovable in $ZFC+\lnot CH$
Average waiting time in a Poisson process
sum of series involving coth using complex analysis
On a certain basis of an order of a quadratic number field
Solve for ? – undetermined inequality symbol
Control / Feedback Theory
Confusion about Homotopy Type Theory terminology
Closed points are dense in $\operatorname{Spec} A$
Proving $\,f$ is constant.
composition of $L^{p}$ functions

Possible Duplicate:

Why negative times negative = positive?

An Abstract Algebra text book has a sentence on its 1st chapter about natural numbers that i cannot get around easily. The sentence reads “Why minus multiplied by minus needs to be plus is something you might reflect on now.” I cannot answer the question, why minus multiplied by minus needs to be plus?

thanks

- $p = x^2 + xy + y^2$ if and only if $p \equiv 1 \text{ mod }3$?
- An irreducible polynomial in a subfield of $\mathbb{C}$ has no multiple roots in $\mathbb{C}$
- Seeking an example in module theory — (In)decomposable modules
- Prime which is not irreducible in non-commutative ring with unity without zero divisors
- Prove that an infinite ring with finite quotient rings is an integral domain
- There is no “operad of fields”

- If $|G|=p^nq$, then $G$ contains a unique normal subgroup of index $q$
- Showing that $X^2$ and $X^3$ are irreducible but not prime in $K$
- Show that $ a，b, \sqrt{a}+ \sqrt{b} \in\mathbb Q \implies \sqrt{a},\sqrt{b} \in\mathbb Q $
- What specific algebraic properties are broken at each Cayley-Dickson stage beyond octonions?
- Is there a systematic way of finding the conjugacy class and/or centralizer of an element?
- In an Integral Domain, if $a^2=1$, then $a=1$ or $a=-1$
- Must an ideal generated by an irreducible element be a maximal ideal?
- Show that $m\mathbb{Z}/md\mathbb{Z} \cong \mathbb{Z}/d\mathbb{Z}$ if and only if $(d,m)=1$
- Proving that $\cos(2\pi/n)$ is algebraic
- Is $\mathbb{Q}(\sqrt{3}, \sqrt{3})$ a Galois extension of $\mathbb{Q}$

In a ring, $-a$ is the unique element that, when added to $a$, is equal to $0$. And $a$ is the unique element that, when added to $-a$, is equal to $0$, because additive inverses are unique.

In order to show that $-(-a)$ *must* equal $a$, we need to show that when we add it to $-a$, we get $0$; because then *both* $a$ and $-(-a)$ have the property “when you add me to $-a$ you get zero”, and there’s supposed to be only one element with that property. Indeed, by the very definition of the symbol $-$, we have that $-(-a) + (-a) = 0$; so $-(-a)=a$. This is an instance of “minus multiplied by minus is plus”.

In order to show that $(-a)(-b)$ *must* equal $ab$, we need to show that $(-a)(-b)$ has the property that, when added to $-(ab)$, we get $0$; for the same reason as above. First, note that $(-a)x = -(ax)$ for any $x$:

$$(-a)x + ax = \Bigl((-a)+a\Bigr)x = 0x = 0,$$

so $(-a)x$ and $-(ax)$ both have the property “when you add me to $ax$ you get $0$”, so they must be equal. Similarly, $x(-b) = -(xb)$ for any $x$.

And now we have:

$$\begin{align*}

(-a)(-b) &= -\Bigl(a(-b)\Bigr) &\text{(since }(-a)x=-(ax)\text{ for any }x)\\

&= -\Bigl( -(ab)\Bigr) &\text{(since }x(-b)=-(xb)\text{ for any }x)\\

&= -(-(ab))\\

&= ab &\text{(since }-(-y) = y\text{ for any }y).

\end{align*}$$

So if we want the basic properties of rings to hold, we need $(-a)(-b)$ to be the same thing as $ab$.

Hint $\rm\ xy,\ (-x)(-y)\ $ are both inverses of $\rm\ x(-y)\ $ so they’re equal by **uniqueness of inverses**.

As I often emphasize, uniqueness theorems provide powerful tools for proving equalities.

- Fourier Analysis textbook recommendation
- If $p$ divides $a^n$, how to prove/disprove that $p^n$ divides $a^n$?
- Slight generalisation of the distribution of Brownian integral
- Introductory textbooks on Morse-Kelley set theory
- Understanding the definition of a compact set
- Enumerations of the rationals with summable gaps $(q_i-q_{i-1})^2$
- Products of Infinitesimals
- Limit in Definition of Riemann Integral is one-sided?
- Distribution of sum of iid cos random variables
- Alternative proof that $(a^2+b^2)/(ab+1)$ is a square when it's an integer
- Prove Minkowski's inequality directly in finite dimensions
- Lie derivative along time-dependent vector fields
- Semisimple Lie algebras are perfect.
- product of two uniformly continuous functions is uniformly continuous
- Fixed field of automorphism $t\mapsto t+1$ of $k(t)$