Real Numbers: Vocabulary Flashcards

These vocabulary flashcards cover key concepts about real numbers, their properties, and absolute value as discussed in the lecture notes.

Real numbers (R)

The set of all rational and irrational numbers; all numbers on the number line that can be expressed as fractions, integers, or irrational values.

Commutative property

Order of numbers does not affect the result for addition and multiplication: a + b = b + a; a · b = b · a.

Associative property

Grouping of numbers does not affect the result for addition and multiplication: (a + b) + c = a + (b + c); (a · b) · c = a · (b · c).

Identity property of addition

Adding zero leaves a number unchanged: a + 0 = a.

Identity property of multiplication

Multiplying by one leaves a number unchanged: a · 1 = a.

Additive inverse (inverse property of addition)

A number plus its opposite equals zero: a + (−a) = 0.

Multiplicative inverse (reciprocal)

For a ≠ 0, a · (1/a) = 1; the reciprocal of a number.

Distributive property

Multiplication distributes over addition: a · (b + c) = a·b + a·c.

Absolute value

The distance from zero on the number line; |a| ≥ 0; |a| = a if a ≥ 0; |a| = −a if a < 0.

Additive inverse / Opposite

The opposite of a number; the number that sums with it to zero; example: opposite of 5 is −5.

Distance on the number line

Absolute value represents distance from zero; distance is nonnegative.

Negative outside the absolute value

The expression −|a| equals the negative of the distance; it is always ≤ 0.

Absolute value of a negative number

|−a| = |a|; the sign disappears inside the absolute value.

Zero/division rules

0/x = 0 for x ≠ 0; x/0 is undefined.

Inequality symbols with absolute value