# What is the uncertainty principle?

Derived from the work of German theoretical physicist Werner Heisenberg, the Heisenberg Uncertainty Principle postulates that, given a particle such as an electron, one can know its position without knowing its momentum, and one can know its momentum without knowing its position, but not both at the same time.

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The uncertainty principle, formulated by Werner Heisenberg in 1927, states that it is impossible to simultaneously know both the exact position and the exact momentum of a particle with absolute precision. In other words, the more precisely one property (such as position) is measured, the less precisely the other property (such as momentum) can be known, and vice versa. Mathematically, the uncertainty principle is often expressed as:

[ \Delta x \Delta p \geq \frac{\hbar}{2} ]

Where:

- ( \Delta x ) is the uncertainty in position,
- ( \Delta p ) is the uncertainty in momentum,
- ( \hbar ) is the reduced Planck constant (( 1.0545718 \times 10^{-34} , \text{m}^2 \cdot \text{kg/s} )).

This principle has profound implications for the behavior of particles at the quantum level and underlies many aspects of quantum mechanics, including wave-particle duality and the probabilistic nature of quantum systems.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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