1.1 Real Numbers and 1.2 Exponents and Radicals (Lecture Notes)

A set of practice flashcards covering key concepts from Section 1.1 Real Numbers and Section 1.2 Exponents and Radicals, including types of real numbers, set notation, intervals, absolute value, distance on the real line, and core exponent/radical rules.

What are natural numbers?

The counting numbers: 1, 2, 3, …

How are integers defined?

The natural numbers together with their negatives and 0; {…, -2, -1, 0, 1, 2, …}.

What is a rational number?

A number that can be expressed as a ratio m/n with integers m and n, n ≠ 0.

What is an irrational number?

A real number that cannot be expressed as a ratio of integers; its decimal expansion is nonrepeating.

What does the symbol R denote in this context?

The set of all real numbers.

What is a repeating decimal?

A decimal where a block of digits repeats forever; occurs for rational numbers.

What is a nonrepeating decimal?

A decimal that does not repeat; these numbers are irrational.

What is the additive identity in real numbers?

0, since a + 0 = a for any real number a.

What is the multiplicative identity in real numbers?

1, since a × 1 = a for any real number a.

What is the reciprocal (inverse) of a nonzero real number a?

1/a, since a × (1/a) = 1.

How do you define the distance between two real numbers a and b on the real line?

The distance is |b − a| (or |a − b|).

What is the meaning of |a| in real numbers?

The absolute value; the distance of a from 0, with properties: |a| ≥ 0, |ab| = |a||b|, |a/b| = |a|/|b|, |−a| = |a|.

What does a ∈ S mean in set notation?

a is an element (member) of the set S.

What do S ∪ T and S ∩ T represent?

S ∪ T is the union (elements in S or T or both); S ∩ T is the intersection (elements in both S and T).

What is an open interval and a closed interval?

Open interval (a, b) excludes endpoints; closed interval [a, b] includes endpoints.

What does the inequality a ≤ b signify on the real line?

a lies to the left of or at b; a is less than or equal to b.

What is the real line representation of a number?

A number can be represented as a point on the real line with origin at 0 and unit distance defining the scale.

What is the principal nth root?

The nonnegative nth root of a number; for even n, the radicand must be ≥ 0; the principal root is denoted by √[n]{a} (or a^(1/n)).

What is the definition of the nth root?

The number b such that b^n = a; for even n, a must be ≥ 0 for real roots.

What is the radical notation for square roots and its sign convention?

√a denotes the principal (nonnegative) square root; the negative root is written as −√a when needed.

What is the Law of Exponents for multiplying powers with the same base?

a^m · a^n = a^(m+n).

What is the Law of Exponents for dividing powers with the same base?

a^m / a^n = a^(m−n).

What is the Law of Exponents for a power raised to another power?

(a^m)^n = a^(mn).

What is the Law of Exponents for (ab)^n?

(ab)^n = a^n b^n.

What is the Law of Exponents for (a/b)^n?

(a/b)^n = a^n / b^n.

What is a zero or negative exponent rule?

a^0 = 1; a^(−n) = 1/a^n; negative exponents move to the denominator.

How is a rational exponent m/n defined (n > 0)?

a^(m/n) is defined as the nth root of a^m, with domain restrictions: if n is even, a ≥ 0; if n is odd, a can be any real number.

What does rationalizing the denominator mean?

Multiplying the numerator and denominator by a suitable expression to eliminate radicals in the denominator.

What is scientific notation?

Expressing a number as x = a × 10^n with 1 ≤ a < 10; used for very large or small numbers.

How do you input scientific notation on many calculators?

Use the E notation, e.g., 3.629 × 10^15 is entered as 3.629E15.

What is the Distributive Property?

a × (b + c) = ab + ac; extends to real numbers and shows interaction of multiplication and addition.