positive-negative-numbers

Positive and Negative Numbers

Summary: Numbers preceded by + (positive) or − (negative) represent quantities greater or less than zero respectively; sign rules govern how they combine under multiplication and division.

Sources: chapter-1.1.2, chapter-1.1.3, chapter-1.1.5

Last updated: 2026-04-24

---

Definition

• Positive quantities: preceded by +; represent real possessions, increases, values greater than zero.
• Negative quantities: preceded by −; represent debts, decreases; less than nothing.

Euler’s financial illustration: a man with 100 crowns and a 50-crown debt has $100-50=50$ (source: chapter-1.1.2).

The Integer Number Line

$\dots ,-3,-2,-1,0,+1,+2,+3,\dots$

Positive integers arise by adding 1 successively from 0; negative integers by subtracting 1 successively. Between any two consecutive integers lie infinitely many values — this leads to fractions (source: chapter-1.1.2).

Sign Rules

For multiplication and division, the rules are identical (source: chapter-1.1.3, chapter-1.1.5):

Signs	Result
$(+)\times (+)$	$+$
$(+)\times (-)$	$-$
$(-)\times (+)$	$-$
$(-)\times (-)$	$+$

Mnemonic: Like signs give +; unlike signs give −.

The rule $(-)(-)=+$ is proved by contradiction: since $(-a)(+b)=-ab$, $(-a)(-b)$ must give the opposite sign, $+ab$ (source: chapter-1.1.3).

Key Identities

$+a-a=0,+2-5=-3,7-12=-5$

For $+a-b$: result is $+(a-b)$ if $a>b$, and $-(b-a)$ if $a<b$.

Related pages

• ch1.1.2-signs-plus-minus
• ch1.1.3-multiplication-simple-quantities
• ch1.1.5-division-simple-quantities
• magnitude
• fractions