High School Math: Proof 1 ≠ 0

First of all, of course, 1 ≠ 0. One is not zero. If they were similar, blimey! Imagine how the bridge construction would be and how many of us would chew beams.

The steps:

chair ≠ banana

Counterexample: Bob ate a chair yesterday.

Hi, Bob. It's your board room meeting suggestion which led to the two-layer right-click menu in Windows 11, innit? 🤔

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Let's dissect the proof from Mudd Math Fun Facts which yields 1 = 0 — the link is gone now. 😂

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Quoted

A bit paraphrased.

Consider two non-zero numbers x and y such that: x = y ------------------------------------------------------------ Then x² = xy (multiplied by x) Subtract both sides with y²: x² - y² = xy - y² ------------------------------------------------------------ Divide by (x-y), we obtain: x + y = y Since x = y, we see that: 2y = y ------------------------------------------------------------ Thus 2 = 1 (divided by y), since we started with nonzero y. ------------------------------------------------------------ Subtracting 1 from both sides: 1 = 0

It looks fine at first — with a baffling final line. BUT! There is a subtle, but actually a giant, flaw in it.

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The Error Occurred at the Division Step

Since x = y, if we use (x - y) as the divisor, that means we're dividing both sides with 0 (zero).

Too, let's take a look at the the quadratic equation (the dividend) which is being divided:

x² - y² = xy - y²

The part which isn't acceptable is when 0 on both sides are divided also by 0.

${(x² - y²)}/{(x - y)} = {(xy - y²)}/{(x - y)}$

Once again, since $x = y$, hence:

$(x - y) = (x² - y²) = (xy - y²) = 0$

If we substitute the manipulation with the actual values:

$0/0 = 0/0$   ⁉️🙋‍♂️🙋‍♀️

$0/0$ is indeterminate. Meaning, it has no actual value.

It's a definition.

indeterminate = indeterminate ⁉️

Not in mathematics, it's indeterminate.

Indeterminate means the expression could represent many possible values depending on context — so we cannot pin down a single answer.

In English (language), in term of sameness, indeed indeterminate is indeterminate, they're the same words — but not in mathematics, as indeterminate is not a value.

It's an uncertainty.

Meaning, it must be stopped there at the subtraction step. Other operation (+, -, exponent, root, logarithmic, integral, etc) besides division can be added, but it will give the same result, that is 0 = 0 (or other same number 1 = 1, 2 = 2, etc.)

We can do another type of different starting manipulation steps for those variables (x and y), and we'll still be using the 0 division to achieve "1 = 0".

In conclusion, the proof manipulation above is a jolly time to check our keenness in algebra and arithmetic. This is categorised under fallacies based on division by zero trope.