Chapter II – Explanation of the Signs + Plus and − Minus
Summary: Introduces the + and − signs for addition and subtraction, defines positive and negative quantities, and establishes integers (whole numbers) as the series extending in both directions from zero.
Sources: chapter-1.1.2
Last updated: 2026-04-24
§8–10: The + Sign and Addition
The sign + (read plus) indicates addition. Thus . In algebra, numbers are generalised by letters: denotes the sum of two quantities and , and expressions like denote the sum of all four (source: chapter-1.1.2).
§11–15: The − Sign and Subtraction
The sign − (read minus) indicates subtraction. Thus . When multiple numbers are subtracted, one may subtract their sum at once: . With mixed signs, collect the positives, collect the negatives, and subtract:
The order of terms is arbitrary provided signs are preserved (source: chapter-1.1.2).
§16–18: Positive and Negative Quantities
- Positive quantities carry the sign +.
- Negative quantities carry the sign −.
Euler’s financial analogy: assets are positive, debts are negative. A man with 100 crowns and a 50-crown debt has real possession . Negative quantities are therefore less than nothing (source: chapter-1.1.2).
§19–20: The Integer Number Line
Positive integers arise by successively adding 1 to 0:
Negative integers arise by successively subtracting 1 from 0:
Together, all whole numbers (positive, zero, and negative) are called integers. Between any two consecutive integers there are infinitely many intermediate values — the first hint that fractions will be needed (source: chapter-1.1.2).
§21–22: Key Identities
- for any .
- : if , the result is positive ; if , the result is negative .
Example: (source: chapter-1.1.2).