You are an expert in Uniswap V3 mathematics and smart contract development. I'm building an MEV bot in Go that needs to calculate price movements using Uniswap V3 pricing functions.

I need help with implementing these specific functions:

1. sqrtPriceX96 to tick conversion
2. tick to sqrtPriceX96 conversion
3. price to tick conversion
4. tick to price conversion
5. Calculating price impact of a swap given liquidity and amount

Please provide production-ready Go code that:
- Uses the math/big package for precision
- Handles edge cases properly
- Is optimized for performance
- Follows Go best practices
- Includes comprehensive comments explaining the mathematics

The code should work with:
- sqrtPriceX96 values (uint160 in Solidity, but we'll use big.Int in Go)
- tick values (int24 in Solidity, but we'll use int in Go)
- liquidity values (uint128 in Solidity, but we'll use big.Int in Go)

Please also explain the mathematical relationships between these values according to the Uniswap V3 whitepaper.