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Q&A How can I intuit $\dfrac{a - b}{c - d} \equiv \dfrac{{\color{red}{-}}(b - a)}{{\color{red}{-}}(d - c)} \equiv \dfrac{b - a}{d - c}$?

posted 3y ago by Wolgwang‭ · edited 3y ago by Wolgwang‭

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#2: Post edited by $user avatar$ Wolgwang‭ · 2021-08-14T18:38:58Z (over 3 years ago)
Added reference.

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**(1)**
$$\dfrac{a - b}{c - d}$$
![Image 1](https://math.codidact.com/uploads/a8qAWGJyPTajXmh2RNpEnzrn)
As the image depicts, $(a - b)$ means $a$ steps forward and $b$ steps backward which results in net $(a - b)$ steps, and dividing it by $(c - d)$ gives the number of steps (of **length** $(c - d)$) in $(a - b)$.
<hr>
**(2)**
$$\dfrac{{{-}}(b - a)}{{{-}}(d - c)}$$
![Image 2](https://math.codidact.com/uploads/eUMqYu6B5SD2Z2K1N7g7aJv6)
$-(b - a) \implies -[-(a-b)] \implies -(-a)+(-b)$ which means $a$ steps forward by facing backward and similarly $b$ steps backward which again results in net $(a - b)$ steps, dividing it by $-[-(c - d)]$ (here the negative sign indicates the way length $(c - d)$ was measured. See the figure) gives the number of steps (of **length** $(c - d)$) in $(a - b)$.
Just like (1)
<hr>
**(3)**
$$\dfrac{b - a}{d - c}$$
![Image 3](https://math.codidact.com/uploads/JmL7u3u9dJ3R77hMQ5LCfWCU)
Now can you intuite how is $\dfrac{b - a}{d - c}$ related to $\dfrac{a - b}{c - d}$ ?
<hr>
**P.S.:** I have used 'length' in place of 'steps' in some sentences, this was done to not make the explanation complex. They don't really mean the same but are closely related.

**(1)**
$$\dfrac{a - b}{c - d}$$
![Image 1](https://math.codidact.com/uploads/a8qAWGJyPTajXmh2RNpEnzrn)
As the image depicts, $(a - b)$ means $a$ steps forward and $b$ steps backward which results in net $(a - b)$ steps, and dividing it by $(c - d)$ gives the number of steps (of **length** $(c - d)$) in $(a - b)$.
Intuitive Meaning of Division can be found [here](https://betterexplained.com/articles/division-intuition/).
<hr>
**(2)**
$$\dfrac{{{-}}(b - a)}{{{-}}(d - c)}$$
![Image 2](https://math.codidact.com/uploads/eUMqYu6B5SD2Z2K1N7g7aJv6)
$-(b - a) \implies -[-(a-b)] \implies -(-a)+(-b)$ which means $a$ steps forward by facing backward and similarly $b$ steps backward which again results in net $(a - b)$ steps, dividing it by $-[-(c - d)]$ (here the negative sign indicates the way length $(c - d)$ was measured. See the figure) gives the number of steps (of **length** $(c - d)$) in $(a - b)$.
Just like (1)
<hr>
**(3)**
$$\dfrac{b - a}{d - c}$$
![Image 3](https://math.codidact.com/uploads/JmL7u3u9dJ3R77hMQ5LCfWCU)
Now can you intuite how is $\dfrac{b - a}{d - c}$ related to $\dfrac{a - b}{c - d}$ ?
<hr>
**P.S.:** I have used 'length' in place of 'steps' in some sentences, this was done to not make the explanation complex. They don't really mean the same but are closely related.

#1: Initial revision by $user avatar$ Wolgwang‭ · 2021-08-14T18:36:39Z (over 3 years ago)

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**(1)**

$$\dfrac{a - b}{c - d}$$

![Image 1](https://math.codidact.com/uploads/a8qAWGJyPTajXmh2RNpEnzrn)

As the image depicts, $(a - b)$ means $a$ steps forward and $b$ steps backward which results in net $(a - b)$ steps, and dividing it by $(c - d)$ gives the number of steps (of **length** $(c - d)$) in $(a - b)$.

<hr>


**(2)** 
$$\dfrac{{{-}}(b - a)}{{{-}}(d - c)}$$

![Image 2](https://math.codidact.com/uploads/eUMqYu6B5SD2Z2K1N7g7aJv6)

$-(b - a) \implies -[-(a-b)] \implies -(-a)+(-b)$ which means $a$ steps forward by facing backward and similarly $b$ steps backward which again results in net $(a - b)$ steps, dividing it by $-[-(c - d)]$ (here the negative sign indicates the way length $(c - d)$ was measured. See the figure) gives the number of steps (of **length** $(c - d)$) in $(a - b)$.
Just like (1)

<hr>

**(3)**
$$\dfrac{b - a}{d - c}$$


![Image 3](https://math.codidact.com/uploads/JmL7u3u9dJ3R77hMQ5LCfWCU)

Now can you intuite how is  $\dfrac{b - a}{d - c}$ related to $\dfrac{a - b}{c - d}$ ?

<hr>

**P.S.:** I have used 'length' in place of 'steps' in some sentences, this was done to not make the explanation complex. They don't really mean the same but are closely related.

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