Showing that $-2(-b)=2b$

Multiplication of:

1. $$3\times 4=12$$, obvious

2. $$-3 \times 4=-12$$, obvious

3. $$-3 \times -4=12$$, not so obvious. We are just told to memorises it.

I would like to demonstrate that the multiplication of two negatives numbers is positive.

Here is my steps:

$$5-3=2$$

let $$a-b=1$$

$$5-2(a-b)=3$$

$$5-2a-2(-b)=3$$

$$-2(-b)=3-5+2a$$

$$-2(-b)=2(a-1)$$ where $$a-1=b$$

$$-2(-b)=2b$$

Are my steps correct? If not valid, I would like to see your method of showing the multiplication of two negative numbers is positive.

• Usually, I think multiplication with negative number e.g; $-2$ as multiplying the number by $2$ and rotating $180$ deg. in the complex plane, keeping the length same. As, $i^2=-1$ and multiplying by $i$ can be interpreted as rotating by $90$ deg. keeping the magnitude unchanged. Apr 16, 2019 at 10:55
• That it is a very interesting way of looking at it Apr 16, 2019 at 10:57
• By associativity, $-a\cdot -b=(a)(-1)(b)(-1)=ab\cdot(-1)(-1)$. So the only real question is, why is it that $(-1)(-1)=1$? Apr 16, 2019 at 13:51
• is repeated addition giving the same result as multiplication obvious ? is subtraction of a negative being addition of a positive obvious ?
– user645636
Apr 16, 2019 at 14:15
• Why is 2 in your opinion more obvious than 3?
– user
Apr 16, 2019 at 21:55

$$\overbrace{\bf\ Law\ of\ Signs}^{\rm\Large {(-x)(-y)}\ =\ xy}$$ proof: $$\rm\,\ (-x)(-y) = (-x)(-y) + \color{#c00}x(\overbrace{\color{#c00}{-y} + y}^{\Large =\,0}) = (\overbrace{-x+\color{#c00}x}^{\Large =\,0})(\color{#c00}{-y}) + xy = xy.$$

• He is aware of that law, just want a convincing explanation ig. Apr 16, 2019 at 10:45
• His approach lacks rigour and proper notation. It is handwavey at best. If his argument used my approach he would not need convincing. Apr 16, 2019 at 10:50

I would like to see your method

Multiplication of $$p$$(suppose in $$\mathbb{Z}$$) by $$i=\sqrt{-1}$$ can be interpreted geometrically as rotating the line joining $$(p,0)$$ and $$(0,0)$$ by $$90$$ deg. without scaling the line. Suppose we are multiplying $$3$$ and $$-1$$. As, multiplying by $$-1=i^2$$ is same as multiplying by $$i$$ twice, hence we can interpret multiplication by $$-1$$ as rotating the line made by $$(0,0)$$ and $$(3,0)$$ by $$180$$ deg. without scaling it and looking at the point where the line cuts the imaginary axis.

Now, if you multiply $$3$$ with $$-2$$, think it like $$3\times 2\times -1$$, i.e; multiply the multiplicand by the magnitude of the multiplier and then rotate the line $$(6,0)$$ by $$180$$ deg.

Same for $$-3\times -2$$. We can rewrite like $$3\times 2\times -1\times -1$$. Multiplying $$3$$ and $$2$$ gives $$6$$, and multiplying by $$-1$$ twice means rotating the line joining $$(0,0)$$ and $$(6,0)$$ by $$180$$ deg twice, or a total $$360$$ rotation and you will land on $$(6,0)$$ again!

I think this way will convince you.