uniswap-math

Uniswap Concentrated Liquidity Math

Q64.96 Fixed-Point Arithmetic

Uniswap V3/V4 stores prices as Q64.96 fixed-point numbers representing the square root of the price ratio. This encoding fits in uint160 and enables efficient swap math without division.

sqrtPriceX96 = √(price) × 2⁹⁶

• 64 bits for the integer part, 96 bits for the fractional part
• Stored as uint160 — fits alongside int24 tick and uint128 liquidity in the pool's Slot0
• Price of token1 in terms of token0: price = (sqrtPriceX96 / 2⁹⁶)²
• Inverse conversion: sqrtPriceX96 = √(price) × 2⁹⁶

Why Store √P Instead of P

1. The core swap formulas only need √P, never P directly
2. amount1 = L × Δ√P is a simple multiplication — no square root at runtime
3. amount0 = L × Δ(1/√P) avoids computing reciprocals of prices
4. Avoids the precision loss of squaring and rooting during swaps

Decimal Normalization

Prices are always in raw token units — you must account for decimals manually.

token0 = WETH (18 decimals)
token1 = USDC (6 decimals)

Human-readable price: 1 ETH = 3000 USDC
Raw price (token1/token0) = 3000 × 10⁶ / 10¹⁸ = 3000 × 10⁻¹²

sqrtPrice = √(3000 × 10⁻¹²) = √(3 × 10⁻⁹) ≈ 5.47722558 × 10⁻⁵
sqrtPriceX96 = 5.47722558 × 10⁻⁵ × 2⁹⁶ ≈ 4_339_505_179_874_779_489_431_521

For a pair where both tokens have 18 decimals (e.g., WETH/DAI at price 3000):

Raw price = 3000 (decimals cancel)
sqrtPrice = √3000 ≈ 54.7722558
sqrtPriceX96 = 54.7722558 × 2⁹⁶ ≈ 4_339_505_179_874_779_489_431_521_786_241

Converting sqrtPriceX96 to Human Price

// In Solidity — use FullMath to avoid overflow
uint256 priceX192 = FullMath.mulDiv(sqrtPriceX96, sqrtPriceX96, 1);
// priceX192 is price * 2^192, divide by 2^192 to get raw price
// For display: rawPrice * 10^(decimals0 - decimals1) = human price

# In Python (offchain)
def sqrtPriceX96_to_price(sqrtPriceX96, decimals0, decimals1):
    price = (sqrtPriceX96 / 2**96) ** 2
    adjusted = price * 10 ** (decimals0 - decimals1)
    return adjusted

# ETH/USDC: sqrtPriceX96 = 4_339_505_179_874_779_489_431_521
sqrtPriceX96_to_price(4_339_505_179_874_779_489_431_521, 18, 6)
# ≈ 3000.0