Let's look at the values again: * `h` (Planck's constant) ≈ `6.626 x 10⁻³⁴ J·s` * `c` (speed of light) ≈ `3.00 x 10⁸ m/s` * `c²` ≈ `(3.00 x 10⁸ m/s)² ≈ 9.00 x 10¹⁶ m²/s²` **1. Do the *values* suggest a simpler way to express `hf = mc²`?** Not in a way that eliminates the constants or makes the equation look radically different while still using standard units. The equation `hf = mc²` is already considered remarkably simple and elegant for the profound concepts it connects. The specific numerical values of `h` and `c` are what they are in our chosen system of units (SI units: Joules, seconds, meters, kilograms). If we chose different fundamental units, these numbers would change. However, we *can* rearrange the equation to group the constants: `hf = mc²` If we're interested in the relationship between mass (`m`) and frequency (`f`) for this energy equivalence, we can write: `m/f = h/c²` This tells us that the ratio of the equivalent mass (`m`) to the photon's frequency (`f`) is a constant value, `h/c²`. Let's calculate this combined constant: `h/c² ≈ (6.626 x 10⁻³⁴ J·s) / (9.00 x 10¹⁶ m²/s²) ` `h/c² ≈ 0.7362 x 10⁻⁵⁰ J·s³·m⁻²` To make the units more intuitive for mass, remember `1 Joule = 1 kg·m²/s²`. So, `J·s³·m⁻² = (kg·m²/s²) · s³·m⁻² = kg·s`. Thus, `h/c² ≈ 7.362 x 10⁻⁵¹ kg/Hz` (since `1/s = Hz`). So, you could write the relationship as: `m = (h/c²) * f` or `m = K * f`, where `K = h/c² ≈ 7.362 x 10⁻⁵¹ kg/Hz`. This `K` is a new constant derived from `h` and `c`. It tells you how many kilograms of mass-equivalent correspond to one Hertz of photon frequency. While this is a valid rearrangement, it doesn't necessarily make the original `hf = mc²` "simpler" in a fundamental conceptual way. It just bundles two fundamental constants into one derived constant for a specific relationship. **The "Simplest" Way (in Natural Units):** Theoretical physicists often work in "natural units" where fundamental constants like `ħ` (h-bar, `h/2π`), `c`, and sometimes `G` (gravitational constant) and `k_B` (Boltzmann constant) are set to `1`. If `ħ = 1` and `c = 1`: * `E = hf` becomes `E = (2π)f` (or `E = ω` if using angular frequency `ω = 2πf` and `ħ`). * `E = mc²` becomes `E = m`. So, in this system, `(2π)f = m` (or `ω = m`). This is mathematically the simplest form, but it hides the physical scales and units, requiring you to remember to reinsert the constants when converting back to standard units. **2. What is the numerical relationship between `h` and `c` (or `c²`)?** There isn't a simple, clean integer or rational fraction relationship between the *numerical values* of `h` and `c` (or `c²`) in SI units that would lead to a significant simplification of the equation `hf = mc²` just by looking at their values. * `h` is a very small number (`~10⁻³⁴`). * `c` is a very large number (`~10⁸`). * `c²` is an even larger number (`~10¹⁷`). The ratio `h/c²` we calculated above (`~7.362 x 10⁻⁵¹ kg/Hz`) is an extremely tiny number. This reflects the fact that you need an incredibly high frequency (and thus very high energy) photon to have a mass-equivalent that is significant in everyday terms, or conversely, a tiny amount of mass converts into a photon of very high frequency/energy. **Key Takeaway:** The constants `h` and `c` are fundamental because they represent deep aspects of nature: * `h` sets the scale of quantum effects. * `c` sets the cosmic speed limit and the relationship between space and time, and mass and energy. Their specific numerical values in SI units are a consequence of our human-chosen system of measurement. The "simplicity" of `hf = mc²` lies in its powerful unification of concepts from quantum theory and relativity, not in any trivial numerical coincidence between `h` and `c`. The equation shows how these two fundamental constants come together to govern the transformation between light energy and mass-energy.