Chapter Summary

Integers and ordering

• Integers: $\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$. Zero is neither positive nor negative.
• On a number line, numbers increase from left to right.
• "Negative" describes numbers $< 0$; "minus" describes the operation of subtraction, these are different.

Operations with negatives

• Adding a negative is subtracting: $a + (-b) = a - b$.
• Subtracting a negative is adding: $a - (-b) = a + b$.
• Same signs multiply/divide to give positive; different signs give negative.
• Even number of negatives → positive result. Odd number → negative result.
• Sign rules for multiplication do NOT apply to addition.
• BODMAS determines the order of operations, always apply it before handling signs.

Number classification

• Rational numbers: can be written as $\frac{p}{q}$ ($p, q$ integers, $q \neq 0$). Includes all integers, terminating decimals, and recurring decimals.
• Irrational numbers: non-terminating, non-repeating decimals (e.g. $\sqrt{2}$, $\pi$).
• Real numbers = rational + irrational.