Foundation lesson
An equals sign states a balance
In physics an equals sign usually states that two quantities are the same, like the two sides of a balanced account. It does not say which side to work out first.
Written for this edition, not translated from Einstein.
Is an equals sign an instruction to calculate, or a statement?
In school, 7 + 5 = asks for an answer. In physics the equals sign more often states a fact: two ways of counting the same thing agree.
Here is an account in made-up units. A body starts with 100 units of energy, sends 30 away as light and keeps 70. Energy before equals energy after plus energy sent out: 100 = 70 + 30. The line is a balance to check, not a sum to do, and it is just as true written 100 − 70 = 30.
The mass-energy paper writes the same kind of balance: E₀ − E₁ = L. The body's energy before, minus its energy after, equals the energy L that the light carries away. It is a statement about the two accounts. Nothing in it says to work out E₀ first.
Because an equation states a balance, it can be read from either side and rearranged without changing what it says. E₁ = E₀ − L tells the same story.
Worked example: Rearranging a balance without changing it
- Start from 100 = 70 + 30, in made-up units.
- Take 70 from both sides: 100 − 70 = 30. Still true, and still the same account.
- Put the paper's letters in place of the numbers: E₀ = E₁ + L, or E₀ − E₁ = L.
- Whichever form is written, it states one fact: what the body lost is what the light carried away.
The two sides of the account, drawn at one scale
The upper bar is the energy the body started with. The lower bar is the same length, built from two pieces: what it kept, and what the light carried away.
What it shows, in words
Both bars run from the same starting edge to the same finishing edge, so both stand for 100 units. The lower one is cut at 70, leaving a stretch of 30 beside it. Read on the lower bar that stretch is the energy sent out; read on the upper bar it is what is left of the 100 after 70 is set aside. One length, two readings, which is why 100 = 70 + 30 and 100 − 70 = 30 say the same thing.
Nothing in the drawing says which piece to work out first, and there is no step in it that has to be done in an order. That is the difference between a statement and an instruction.
Where this lesson stops
An equals sign states that two quantities are the same. Which quantities, measured how, is what the paper around it has to say.
If you came here from a passage, Back returns you to the exact place you left.