Flashcards: Real Numbers and Their Properties — 44 cards

What two sets make up the real numbers?

Rational numbers and irrational numbers.

How can a rational number be expressed?

As a fraction $\frac{a}{b}$ with integers a and b, and $b\ne0$.

Why can't an irrational number be written as $\frac{a}{b}$?

Because a and b are integers with $b\ne0$ and it doesn't fit that form.

What decimal forms can irrational numbers not have?

They cannot be terminating or repeating decimals.

How is subtraction defined using addition?

$a-b = a + (-b)$.

How is division by a nonzero number defined?

$\frac{a}{b} = a \cdot \frac{1}{b}$ with $b\ne0$.

When is the product of two real numbers zero?

If and only if one of the numbers is zero.

When is the product of two real numbers nonzero?

If and only if both numbers are nonzero.

What is an identity in algebra?

An equality true for every allowed value of its variables.

What is the formula for the square of a binomial sum?

$(a+b)^2 = a^2 + 2ab + b^2$

What is the factorization formula for the difference of squares?

$a^2 - b^2 = (a+b)(a-b)$

How is the difference of cubes expressed algebraically?

$a^3 - b^3 = (a-b)(a^2 + ab + b^2)$

To which operations do the laws of integer exponents apply if powers are defined?

Products, quotients, and powers of powers.

How does a direct proof begin and proceed?

It starts from the hypothesis and uses valid transformations to reach the conclusion.

What is sufficient to disprove a universal claim?

Providing one counterexample where the claim is false.

What does a proof by contradiction assume initially?

That the desired conclusion is false.

What is the key step after assuming the conclusion is false in proof by contradiction?

Deriving a contradiction with a known hypothesis or fact.

What conclusion is drawn if a contradiction is found in proof by contradiction?

The initial assumption that the conclusion is false is rejected.

What does $a>b$ mean for real numbers in terms of $a-b$?

$a-b>0$

How is $a>b$ represented geometrically on the real-number axis?

a is to the right of b

What happens to inequality direction when adding the same real number to both sides?

It is preserved

How does multiplying both sides of an inequality by a positive number affect its direction?

The direction is preserved

What effect does multiplying both sides of an inequality by a negative number have on its direction?

It reverses the direction

For positive numbers and positive integer n, when does $a>b$ hold in terms of powers?

If and only if $a^n>b^n$

What is true about $a^2$ for every real number a?

$a^2\ge0$

When does equality hold in $a^2\ge0$?

Only when $a=0$

What defines the closed interval $[a,b]$ in real numbers?

All real numbers x with $a\le x\le b$ including both endpoints.

What defines the open interval $(a,b)$ in real numbers?

All real numbers x with $a < x < b$ excluding both endpoints.

What is the absolute value $|a|$ of a real number a?

Its distance from zero on the real-number axis.

What is the formula for $|a|$ when $a \,\ge\, 0$?

$|a| = a$.

What is the formula for $|a|$ when $a < 0$?

$|a| = -a$.

What is the product rule for absolute values of real numbers a and b?

$|ab| = |a||b|$.

How is the absolute value of a quotient $\frac{a}{b}$ expressed for $b \neq 0$?

$\left|\frac{a}{b}\right| = \frac{|a|}{|b|}$.

What inequality relates $|a+b|$ to $|a|$ and $|b|$?

$|a+b| \leq |a| + |b|$.

How is the distance $d(a,b)$ between real numbers a and b defined?

$d(a,b) = |a - b|$.

What inequality is equivalent to $|x - x_0| < \rho$ for $\rho > 0$?

$x_0 - \rho < x < x_0 + \rho$.

What is the square root of a nonnegative number a?

The nonnegative number whose square is a.

What defines the nth root of a nonnegative number a?

The nonnegative number whose nth power equals a.

How does the nth root of a product relate to the nth roots of factors?

It equals the product of the nth roots of each factor.

How is the nth root of a quotient expressed when denominator is nonzero?

As the quotient of the nth roots of numerator and denominator.

How is a power with rational exponent $m/n$ defined for $a>0$?

As the nth root of $a$ raised to the mth power.

What is the formula for the square of a sum of three terms?

$(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$

How many endpoints do half-open intervals $[a,b)$ and $(a,b]$ include?

Exactly one endpoint.

What do the intervals $[a,+\infty)$ and $(-\infty,a]$ represent?

One-sided unbounded sets.