<Question>
What is the smallest number which is one less than twice its reciprocal?
</Question>

<think>
<Plan_1>
Okay, so I need to find the smallest number that's one less than twice its reciprocal. Hmm, let's break this down step by step. First, let me parse the problem correctly. The number we're looking for is such that when you take twice its reciprocal and subtract one, you get the number itself. 

Let me write that out as an equation to make it clearer. Let's let the number be x. The reciprocal of x is 1/x. Twice the reciprocal would be 2*(1/x), which is 2/x. Then, one less than that would be 2/x - 1. According to the problem, this expression equals the number itself. So the equation is:

x = (2/x) - 1

Now, I need to solve this equation for x. Since it's a bit of a non-linear equation because of the x in the denominator, I might need to rearrange terms and maybe multiply both sides by x to eliminate the fraction. Let me try that.
Multiplying both sides by x gives:

x*x = 2 - x

Which simplifies to:

x² = 2 - x

Now, let's bring all terms to one side to form a quadratic equation. I'll add x to both sides:

x² + x - 2 = 0

Okay, so now we have a quadratic equation: x² + x - 2 = 0. To solve this, I can use the quadratic formula. The quadratic formula is x = [-b ± sqrt(b² - 4ac)]/(2a), where a, b, and c are coefficients from the equation ax² + bx + c = 0. In this case, a = 1, b = 1, and c = -2.

Plugging those values into the quadratic formula:

<prompt_1.1> Solve the quadratic equation using the quadratic formula. </prompt_1.1>
-----

But wait, let me think again. The problem says "the smallest number". So if there are two solutions, we need the smaller one. The solutions are 1 and -2.
So 1 is smaller than -2? Wait, no. Wait, 1 is greater than -2. So the smallest number would be -2.
-----

But let's go back to the original equation and check if x = -2 satisfies it.

Original equation: x = (2/x) - 1

<prompt_1.2> Re-evaluate the problem statement to ensure the correct interpretation of "smallest number". </prompt_1.2>
-----

But let me just confirm once more. The problem says "the smallest number". If we consider real numbers, negative numbers are smaller than positive numbers.

<prompt_1.3> Perform a final verification of the solution and conclude with the final answer. </prompt_1.3>
</Plan_1>

<Execution_1>
<execution_1.1>
x = [-1 ± sqrt(1² - 4*1*(-2))]/(2*1)
x = [-1 ± sqrt(1 + 8)]/2
x = [-1 ± sqrt(9)]/2
sqrt(9) is 3, so:

x = (-1 ± 3)/2

This gives two solutions:

1. x = (-1 + 3)/2 = 2/2 = 1
2. x = (-1 - 3)/2 = (-4)/2 = -2
</execution_1.1>

<execution_1.2>
Left side: x = -2

Right side: (2/-2) - 1 = (-1) - 1 = -2

So yes, it does satisfy the equation. But wait, earlier I thought maybe 1 is the answer, but since -2 is smaller, that's the one. However, the problem says "the smallest number", so if negative numbers are allowed, then -2 is the answer. But sometimes "number" can be considered as positive unless specified. Let me check the problem again. It just says "the smallest number", so if there are two real solutions, 1 and -2, then the smallest is -2.
</execution_1.2>

<execution_1.3>
So -2 is indeed smaller than 1. Therefore, -2 is the correct answer. 

But wait, let me just check if I interpreted the problem correctly. The problem says "the smallest number which is one less than twice its reciprocal". So the number is equal to twice the reciprocal minus one. 

Yes, that's how I set it up. So x = (2/x) - 1. Solving that gives x = 1 and x = -2. Since we're looking for the smallest number, -2 is the answer.
</execution_1.3>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Wait, but the problem says "the smallest number". So if there are two solutions, 1 and -2, then the smallest one is -2. But wait, is -2 actually a valid solution here? Let me check.
If x = -2, then the reciprocal is -1/2. Twice the reciprocal is -1. Then, one less than that is -1 - 1 = -2, which equals x. So yes, that works. 

But what about x = 1? Let's check that too. If x = 1, the reciprocal is 1. Twice the reciprocal is 2. One less than that is 2 - 1 = 1, which equals x. So that works as well. 

So both numbers satisfy the equation. But since the problem asks for the smallest number, between 1 and -2, the smallest is -2.
-----

But maybe there's a misunderstanding here. Let me re-examine the problem statement again: "What is the smallest number which is one less than twice its reciprocal"?

<prompt_2.1> Re-examine the problem statement and verify the solutions again. </prompt_2.1>
-----

Based on execution_1.2:
But maybe the problem expects a positive number. Let me check.

<prompt_2.2> Confirm the final answer based on the problem's context and expectations. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
So the number (x) is equal to (twice its reciprocal) minus one. So x = 2*(1/x) - 1. That's exactly what we set up. So both solutions are valid. But since the question asks for the smallest number, -2 is smaller than 1, so -2 is the answer. However, sometimes in math problems, unless specified otherwise, numbers can be negative. So unless there's a context implying positivity, both solutions are valid. 

Therefore, the answer should be -2. But wait, maybe the problem is expecting a positive number. Let me check again. If we consider only positive numbers, then x = 1 is the only solution. But the problem doesn't specify positivity. Hmm. Let me confirm with another approach. 

Alternatively, maybe I made a mistake in setting up the equation. Let me re-read the problem statement: "What is the smallest number which is one less than twice its reciprocal"? 

So the number is equal to twice its reciprocal minus one. So x = 2*(1/x) - 1. That's correct. So solving gives x = 1 and x = -2. So both are valid solutions. Therefore, the smallest number is -2. 

But wait, maybe I need to check if there are other solutions or if I made an error in solving. Let's see. The quadratic equation x² + x - 2 = 0 factors into (x + 2)(x - 1) = 0, which gives x = -2 or x = 1. So that's correct. 

Therefore, the answer is -2. However, depending on the context, sometimes problems like this might look for positive numbers. But since the problem doesn't specify, and mathematically both solutions are valid, the smallest number is -2. 

Wait a second, but maybe I should check if there's a mistake in the quadratic equation. Let me go back. Starting from the original equation:

x = (2/x) - 1

Multiply both sides by x:

x² = 2 - x

Bring all terms to one side:

x² + x - 2 = 0

Yes, that's correct. So the quadratic is correct. Factoring gives (x + 2)(x - 1) = 0, so roots at x = -2 and x = 1. 

Therefore, the answer is -2. But let me just check once more with substitution. 

For x = -2:

Twice the reciprocal is 2*(1/-2) = 2*(-1/2) = -1. Then, one less than that is -1 - 1 = -2, which is equal to x. 

For x = 1:

Twice the reciprocal is 2*(1/1) = 2. One less than that is 2 - 1 = 1, which is equal to x. 

Both check out. So since the problem asks for the smallest number, and -2 is smaller than 1, the answer is -2.
</execution_2.1>

<execution_2.2>
If the problem is from a context where numbers are positive, like in some word problems, then maybe 1 is the answer. But the problem as stated doesn't specify that the number has to be positive. So strictly speaking, both 1 and -2 are solutions, and the smallest is -2. 

However, sometimes in such problems, if not specified, they might expect the positive solution. But since the problem says "smallest number", which in mathematics can refer to the least value regardless of sign, then -2 is smaller than 1. Therefore, the answer should be -2. 

Wait, but let me check if there's any other constraints. For example, if x is a number, can it be negative? The reciprocal of a negative number is negative, so twice the reciprocal is negative, and subtracting one would make it more negative. But in the case of x = -2, that works. So there's no restriction in the problem that x has to be positive. Therefore, -2 is a valid solution and is indeed smaller than 1.
</execution_2.2>
</Execution_2>

<Final_answer>
Therefore, the smallest number which is one less than twice its reciprocal is $\boxed{-2}$.


Note: If the problem context requires a positive number, then $\boxed{1}$ would be the answer. However, based on the problem as stated, $-2$ is the correct answer.
</Final_answer>
</think>