Units and orders of magnitude

Climb the prefix ladder, let units cancel, and size any number by its power of ten

Math toolbox K1 of Energy Systems, Step by Step: convert between kW, MW and GW, add a real day’s hourly MW into MWh, let units cancel like numbers, write energy from a phone charge to a nation’s year as powers of ten, and catch a slipped prefix with two bounds.
  • Math toolbox
  • 10 min guided · about 10 min practice
  • No prerequisites

Units and powers of ten, in six moves

  1. Each prefix is 1,000 times the one below: Prairie Town's busiest hour of 2025, 450 MW, is 450,000 kW or 0.45 GW.
  2. Energy is power × time: on 3 September 2025 the town's 24 hourly values add up to 6,957 MWh, or 6.957 GWh.
  3. Units cancel like numbers: at a flat $40/MWh, 1,965 MWh × $40/MWh = $78,600; and 6,957 MWh ÷ 24 h ≈ 289.9 MW, the day's average power.
  4. Scientific notation writes a number as a × 10ⁿ, with a at least 1 and less than 10: 1,000 = 10³, and the day's 6,957,000,000 Wh ≈ 6.96 × 10⁹ Wh.
  5. Written in Wh as powers of ten, a phone charge (1.9 × 10¹) and the U.S. year (4.2 × 10¹⁵) are 14 powers of ten apart.
  6. The day lies between 235 MW × 24 h = 5,640 MWh and 353 MW × 24 h = 8,472 MWh, so a report of 6,957 GWh has a slipped prefix.
Amount of energy As published In Wh Scientific notation (3 figures)
Charging a smartphone once 0.019 kWh 19 1.9 × 10¹ Wh
An average U.S. home's year 10,791 kWh 10,791,000 1.08 × 10⁷ Wh
Prairie Town's day, 3 September 2025 6,957 MWh 6,957,000,000 6.96 × 10⁹ Wh
The Southwest Power Pool's year, 2025 298.6 TWh 298,600,000,000,000 2.99 × 10¹⁴ Wh
The United States' year, 2025 about 4,200 TWh 4,200,000,000,000,000 4.2 × 10¹⁵ Wh

With JavaScript on, you work through this step by step with a slider, a log scale and your own answers.

  • Source. Prairie Town's day is the real hourly demand shape of 3 September 2025 on the Southwest Power Pool grid (EIA-930).
  • Illustrative. The shape is scaled as in lesson 0 so that the town's busiest hour of 2025 is 450 MW. The town, its scale and the flat $40/MWh gas running cost are illustrative.
  • Not a schedule. $78,600 prices all 1,965 MWh of gas at $40/MWh. Lesson 0's schedule also runs the $100/MWh peaker for 20 MWh, so its gas costs $80,400.
  • Anchors. A phone charge: EPA (checked 2 November 2026). A home's year: EIA (2022 data). The Southwest Power Pool's year: EIA-930 (2025 data). The U.S. year: EIA (2025 data).

What you just did, as diagrams

Prefix
A letter that multiplies a unit: k by 1,000, M by a million, G by a billion, T by a trillion.
Power and energy
Power (MW) is a rate; energy (MWh) is power held over time: MW × h = MWh.
Units cancel
A unit on top cancels the same unit underneath: $/MWh × MWh leaves $.
Scientific notation
A number at least 1 and less than 10, times a power of ten: the town's day is about 6.96 × 10⁹ Wh.
Order-of-magnitude check
Bound a number from both sides; missing by a power of ten points to a slipped prefix or unit.
Go deeper · AnalystExponents, logarithms and the joule
  • Multiply: add the exponents. \(10^3 \times 10^6 = 10^9\): a thousand MW is a GW.
  • Divide: subtract them. \(\dfrac{4.2 \times 10^{15}}{1.9 \times 10^{1}} \approx 2.2 \times 10^{14}\): the U.S. year in phone charges.
  • The logarithm is the exponent. \(\log_{10}(6.96 \times 10^9) \approx 9.84\). A log scale spaces 1, 10 and 100 Wh evenly, so one chart holds a phone and a nation.
  • The joule. A watt is one joule per second, so 1 Wh = 3,600 J and 1 kWh = 3.6 MJ, exactly. Heat units such as the MMBtu come in toolbox K2.
Go deeper · ModelerUnits inside a model (a preview)
  • Choose units that keep numbers modest. Models write MW and $/MWh, not W and $/Wh, so their coefficients (the fixed numbers in front of each unknown) stay within a few powers of ten of each other.
  • Why it matters. A solver treats values within a small tolerance, an allowed rounding error, as equal or zero; coefficients spread over many powers of ten can hide an error or slow a solve.
  • Put the unit in the name. demand_mw * hours is MWh and gas_mwh * price_usd_per_mwh is dollars. Terms added together must share a unit: MW plus MW, never MW plus MWh.
  • Where next. Bridge B3 checks that every term of a model’s balance has one unit; M1 builds a dispatch model in MW and $/MWh.

Check yourself

Myth or fact? “A kWh is a kilowatt per hour.”

  • Myth
  • Fact
Answer

Myth. A kWh is kilowatts × hours: energy. “Per hour” is how fast something changes, like lesson 0's ramp limit of 40 MW per hour.

Myth or fact? “450 MW is more power than 0.45 GW.”

  • Myth
  • Fact
Answer

Myth. They are the same power: a bigger unit takes a smaller number.

The Southwest Power Pool's demand in 2025 was 298.6 TWh. How many GWh is that?

  • 0.2986 GWh
  • 298,600 GWh
  • 298,600,000 GWh
  • 2,986 GWh
Answer

298,600 GWh: a TWh is 1,000 GWh, and a smaller unit takes a bigger number. 298,600,000 is the same energy in MWh.

About how many powers of ten from an average U.S. home's year (1.08 × 10⁷ Wh) to the Southwest Power Pool's year (2.99 × 10¹⁴ Wh)?

  • 5
  • 7
  • 14
  • 21
Answer

7: the exponents differ by 14 − 7 = 7, and 2.99 ÷ 1.08 is about 2.8, so the ratio is about 2.8 × 10⁷. 2.8 is below 3.16, half a power of ten, so the nearest power stays 10⁷.

Numbers, sources and what’s next

Numbers to remember

Number What it means Source
× 1,000 Each step of the prefix ladder: W, kW, MW, GW, TW. SI Brochure (BIPM), 9th edition, 2019
1 MW × 1 h = 1 MWh: power held for an hour is energy. Definition
10,791 kWh Electricity an average U.S. home buys in a year: about 10⁷ Wh. EIA FAQ · 2022 data
298.6 TWh The Southwest Power Pool's demand in 2025: about 3 × 10¹⁴ Wh. EIA-930 · 2025 data
about 4,200 TWh U.S. electricity use in 2025: about 4 × 10¹⁵ Wh. EIA · 2025 data

Next

K2 · Rates and efficiencies

Sources and further reading

Spotted an error, or stuck? Write to hlli@ksu.edu. Corrections are made in place and dated here.

Text licensed CC BY 4.0, code MIT: teachers and students may reuse and adapt it, with credit to Honglin Li.