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Denomination Scale Visualization

Understanding the relationships between Wei, Gwei, and Ether with visual guides and scale comparisons.

The Denomination Scale

1 Ether (ETH)
     |
     | = 10^9 Gwei
     |
     v
1,000,000,000 Gwei
     |
     | = 10^9 Wei per Gwei
     |
     v
1,000,000,000,000,000,000 Wei

Linear Scale (Powers of 10)

Wei to Ether
Code Scale

Wei:                           Gwei:                    Ether:
1                              0.000000001              0.000000000000000001
10                             0.00000001               0.00000000000000001
100                            0.0000001                0.0000000000000001
1,000                          0.001                    0.000000000001
10,000                         0.00001                  0.00000000001
100,000                        0.0001                   0.0000000001
1,000,000                      0.01                     0.000000001
10,000,000                     0.1                      0.00000001
100,000,000                    1                        0.000000001
1,000,000,000                  10                       0.0000001
10,000,000,000                 100                      0.000001
100,000,000,000                1,000                    0.00001
1,000,000,000,000              10,000                   0.0001
10,000,000,000,000             100,000                  0.001
100,000,000,000,000            1,000,000                0.01
1,000,000,000,000,000          10,000,000               0.1
10,000,000,000,000,000         100,000,000              1
100,000,000,000,000,000        1,000,000,000            10
1,000,000,000,000,000,000      10,000,000,000           100

// Wei to Ether scale (log10)
const scales = [
  { wei: 1n, gwei: 0.000000001, ether: 0.000000000000000001, name: 'Single Wei' },
  { wei: 1_000_000_000n, gwei: 1, ether: 0.000000001, name: '1 Gwei' },
  { wei: 1_000_000_000_000_000_000n, gwei: 1_000_000_000, ether: 1, name: '1 Ether' },
  { wei: 21_000_000_000_000_000n, gwei: 21_000_000, ether: 0.021, name: 'Transfer' },
  { wei: 1_000_000_000_000_000_000_000n, gwei: 1_000_000_000_000, ether: 1_000, name: '1000 Ether' }
]

scales.forEach(scale => {
  console.log(`${scale.name}: ${scale.wei} Wei = ${scale.gwei} Gwei = ${scale.ether} ETH`)
})

Real-World Comparison

Daily Transactions

Type	Wei	Gwei	Ether	Real Value
Dust (spam blocker)	1,000	0.000001	0.000000001	~$0.000002
Minimum gas (1 gwei)	1,000,000,000	1	0.000000001	~$0.000002
Min transaction (21k gas @ 1 gwei)	21,000,000,000	0.021	0.000000021	~$0.00004
Cheap transfer (20 gwei)	420,000,000,000,000	420,000	0.00042	~$0.84
Normal transfer (50 gwei)	1,050,000,000,000,000	1,050,000	0.00105	~$2.10
Fast transfer (200 gwei)	4,200,000,000,000,000	4,200,000	0.0042	~$8.40
Expensive swap (100k gas @ 200 gwei)	20,000,000,000,000,000	20,000,000	0.02	~$40

Assuming ETH = $2,000

Conversion Flow Diagrams

Direct Conversions (Reversible)

┌─────────┐         ×10^9         ┌──────────┐         ×10^9         ┌────────┐
│  1 Wei  │ ─────────────────────> │  1 Gwei  │ ─────────────────────> │ 1 Ether│
└─────────┘                        └──────────┘                        └────────┘
     ^                                   ^                                  ^
     │                                   │                                  │
     │         ÷10^9                     │         ÷10^9                    │
     └───────────────────────────────────┴──────────────────────────────────┘

All direct conversions are reversible without data loss:

1 Wei → toGwei → Wei = 1 Wei ✓
1 Gwei → toWei → Gwei = 1 Gwei ✓
1 Ether → toWei → Ether = 1 Ether ✓

Fractional Conversions (Data Loss)

┌──────────────────┐
│ 1.5 Gwei in Wei  │  (1,500,000,000 Wei)
│ = 1.5 × 10^9 Wei │
└──────────────────┘
         │
         │ toGwei (÷10^9)
         v
┌──────────────────┐
│ = 1 Gwei         │  (truncates .5)
│ (data loss!)     │
└──────────────────┘
         │
         │ toWei (×10^9)
         v
┌──────────────────┐
│ = 1,000,000,000  │  (lost 500,000,000 Wei)
│ (NOT 1.5 Gwei)   │
└──────────────────┘

Safe pattern: Always work in smallest unit (Wei) when possible

Magnitude Comparison

How Small is Wei?

1 Ether = 1,000,000,000,000,000,000 Wei

Like currency: 1 Dollar = 100 Cents = 10,000 Mills

Wei is 18 decimal places smaller than Ether:
Ether.0000000000000000 = Wei

Range: 0 to 115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,935
(2^256 - 1 Wei = ~1.16e77)

Scientific Notation

// Understanding big numbers

// 1 Ether
const oneEther = 1_000_000_000_000_000_000n  // 1e18 in decimal
// = 10^18 Wei

// 1000 Ether
const thousandEther = 1_000_000_000_000_000_000_000n  // 1e21
// = 10^21 Wei

// 1 Million Ether
const millionEther = 1_000_000_000_000_000_000_000_000n  // 1e24
// = 10^24 Wei

// Useful conversions
const wei = 1_500_000_000_000_000_000n       // 1.5 Ether in Wei
const exponent = 18  // Decimal places for Ether
const ether = Number(wei) / Math.pow(10, exponent)  // 1.5

console.log(ether)  // 1.5

Gas Price Scale

Understanding how gas prices affect transaction costs:

Gas Price Levels (typical mainnet):

Urgent:     200 Gwei ──────────────────┐
                                       │
Fast:       100 Gwei ─────────────┐   │
                                   │   │
Standard:    50 Gwei ──────────┐  │   │
                                │  │   │
Safe:        20 Gwei ────────┐ │  │   │
                              │ │  │   │
Slow:        10 Gwei ──────┐ │ │  │   │
                            │ │ │  │   │
                            v v v  v   v
Transfer (21k gas):
Slow:    210,000 Gwei     = 0.00021 ETH
Safe:    420,000 Gwei     = 0.00042 ETH
Standard: 1,050,000 Gwei  = 0.00105 ETH
Fast:    2,100,000 Gwei   = 0.0021 ETH
Urgent:  4,200,000 Gwei   = 0.0042 ETH

Cost Impact
Visual Graph

import * as Gwei from 'tevm/Gwei'
import * as Wei from 'tevm/Wei'
import * as Uint from 'tevm/Uint'

function showCostScale(gasUsed: bigint) {
  const prices = [10, 20, 50, 100, 200]

  console.log(`Gas amounts: ${gasUsed}`)
  console.log('─'.repeat(50))

  prices.forEach(gwei => {
    const gasPriceWei = Gwei.toWei(Gwei(BigInt(gwei)))
    const costWei = Uint.times(gasPriceWei, Uint(gasUsed))
    const costGwei = Uint.dividedBy(costWei, Uint(1_000_000_000n))
    const costEther = Number(costWei) / 1e18

    console.log(`${gwei.toString().padStart(3)} Gwei: ${costGwei} Gwei = ${costEther.toFixed(6)} ETH`)
  })
}

// Show scale for transfer (21,000 gas)
showCostScale(21_000n)

console.log()

// Show scale for complex interaction (200,000 gas)
showCostScale(200_000n)

Transfer (21,000 gas) - Cost in Ether:

0.0042 ETH │     ■
           │     ■
0.0035 ETH │     ■
           │     ■
0.0028 ETH │     ■
           │     ■
0.0021 ETH │     ■
           │     ■
0.0014 ETH │   ■ ■
           │   ■ ■
0.0007 ETH │   ■ ■
           │   ■ ■
0      ETH │ ■ ■ ■
           └─────────────────
             10 20 50 100 200 Gwei
                 Gas Price

Practical Size Comparisons

What fits in different units?

import * as Wei from 'tevm/Wei'
import * as Gwei from 'tevm/Gwei'
import * as Ether from 'tevm/Ether'
import * as Uint from 'tevm/Uint'

// How much in Wei?
const weiCapacity = Uint(2n ** 256n - 1n)  // Max Uint256
// = 115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,935 Wei
// = 115,792,089,237 Billion Billion Ether
// Enough for everyone on Earth to have 14+ Ether

// Practical limits
const dappBudget = Ether(1_000_000n)          // 1M Ether (huge budget)
const dappBudgetWei = Ether.toWei(dappBudget)      // 1e24 Wei

const liquidEthereum = 120_000_000n                // All ETH ever (approx)
const liquidWei = Ether.toWei(Ether(liquidEthereum))  // All Wei in existence

Conversion Reference Table

From     To       Formula              Example
─────────────────────────────────────────────────────────────
Wei   → Gwei     ÷ 10^9               1e9 Wei → 1 Gwei
Wei   → Ether    ÷ 10^18              1e18 Wei → 1 Ether
Gwei  → Wei      × 10^9               1 Gwei → 1e9 Wei
Gwei  → Ether    ÷ 10^9               1e9 Gwei → 1 Ether
Ether → Wei      × 10^18              1 Ether → 1e18 Wei
Ether → Gwei     × 10^9               1 Ether → 1e9 Gwei

Calculator: Power of 10 Growth

How values grow with each power of 10:

const base = 1_000_000_000_000_000_000n  // 1 Ether in Wei

// Each step multiplies by 10^9
const wei = base                    // 10^18 (1 Ether)
const megaWei = base * 1_000n       // 10^21
const gigaWei = base * 1_000_000n   // 10^24
const teraWei = base * 1_000_000_000n  // 10^27

console.log('Growth scale (base = 1 Ether):')
console.log('1x:', wei)           // 1 Ether
console.log('1K×:', megaWei)      // 1,000 Ether
console.log('1M×:', gigaWei)      // 1,000,000 Ether
console.log('1B×:', teraWei)      // 1,000,000,000 Ether

Denomination - Type definitions
Conversions - Conversion functions
Gas Calculator - Real-world costs
Usage Patterns - Common patterns

Try it Live

​Denomination Scale Visualization

​The Denomination Scale

​Linear Scale (Powers of 10)

​Real-World Comparison

​Daily Transactions

​Conversion Flow Diagrams

​Direct Conversions (Reversible)

​Fractional Conversions (Data Loss)

​Magnitude Comparison

​How Small is Wei?

​Scientific Notation

​Gas Price Scale

​Practical Size Comparisons

​What fits in different units?

​Conversion Reference Table

​Calculator: Power of 10 Growth

​Related