arithmetic

Integer Operations

Pre-Algebra20 minDifficulty

Overview

Integer operations extend arithmetic to all whole numbers including negatives. The rules for signs: positive * positive = positive, negative * negative = positive, positive * negative = negative. Subtraction is redefined as adding the opposite: a - b = a + (-b). These rules extend naturally to division.

Intuition

Think of integers as a temperature scale or a bank account. -3 + (-5) = -8: you owe 3 dollars, borrow 5 more, now owe 8. Subtracting a negative: 4 - (-2) = 4 + 2 = 6 (removing a debt is the same as gaining money). Multiplying negatives: (-3)*(-4) = 12 because two 'reversals' return to positive.

Formal Definition

Definition

The integers Z = {..., -2, -1, 0, 1, 2, ...} form a ring under + and *. Rules: a - b = a + (-b); (-a)*b = -(a*b); (-a)*(-b) = a*b; a/(-b) = -(a/b); (-a)/(-b) = a/b. The absolute value |a| is a if a >= 0, -a if a < 0.

a - b = a + (-b)

Subtraction as adding opposite

(-a)(-b) = ab

Negative times negative

|a \cdot b| = |a| \cdot |b|

Absolute value of product

Notation

Notation	Meaning
$\mathbb{Z}$	The set of all integers
$\|a\|$	Absolute value of a

Theorems

Theorem 1: Sign Rules

For integers: (+)(+)=(+), (-)(-)=(+), (+)(-)=(-), (-)(+)=(-). Same rules hold for division. These follow from the distributive law and the requirement that 0*a=0.

Worked Examples

1
Start at -7, move 4 units right.
$-7 + 4 = -(7-4) = -3$

✓ Answer

-3

Practice Problems

Easyfill in blank

Compute -5 - (-3).

Easyfill in blank

Compute (-4) * 7 / (-2).

Common Mistakes

Common Mistake

Thinking (-5) - (-3) = -8 (subtracting magnitudes with same sign).

Subtracting a negative adds: (-5) - (-3) = (-5) + 3 = -2.

Common Mistake

Thinking that three negatives give a negative result overall (e.g., (-2)^3 is positive).

(-2)^3 = (-2)*(-2)*(-2) = 4*(-2) = -8. Odd powers of negatives are negative.

Quiz

What is (-6) * (-3)?

-1818-99

Historical Background

Negative numbers were resisted for centuries. Indian mathematicians (Brahmagupta, 7th century CE) first gave rules for operating with negative quantities ('debts'). European mathematicians accepted them slowly -- Descartes called negative roots 'false' in 1637. By the 19th century, negative numbers were fully accepted as part of the integer system.

628 CE
Brahmagupta gives rules for negative numbers (debts) in Brahmasphutasiddhanta
Brahmagupta
1800s
Negative integers accepted formally in European mathematics

Summary

a - b = a + (-b): subtraction is adding the opposite.
Sign rules for multiplication/division: same signs give positive, different signs give negative.
(-a)(-b) = ab; this follows from distributivity and 0*a = 0.

References

BookLarson, R. Pre-Algebra. Holt McDougal, 2010.