mathmatics concept questions : Real numbers, Estimations and Logic

Define a Rational Number.

.A number that can be written in the form n/m

where m and n are integers and n # 0.

How do you identify a rational number by looking at its decimal?

The decimal either terminates (stops) OR repeats in a regular cycle forever.

Give two examples of irrational numbers.

V2 (or any non-perfect square root) and T.

Their decimals never terminate and never repeat.

What does "Density" mean in the context of Real Numbers?

Between any two distinct real numbers, there is always another real number (and infinitely many more).

True or False: Every real number is a rational number.

False. Real numbers are made of both rational and irrational numbers.

What is the hypothesis and conclusion in: "If it rains, the ground gets wet."

Hypothesis (P): It rains. Conclusion (Q): The ground gets wet.

What is the Converse of P→ Q?

Q → P (If Q, then P). Note: It is not always true just because the original statement is true.

What is the Contrapositive of P → Q?

~ Q →~ P (If not Q, then not P).

True or False: A statement and its contrapositive are logically equivalent.

True. If one is true, the other is absolutely true.

What is the first step of a Proof by Contradiction?

Assume the conclusion (Q) is false, and work to find a mathematical impossibility (a contradiction).

Translate this symbol: Vx

"For all x" or "For every x"

Translate this symbol: E (flipped)y

"There exists a y" (meaning at least one y exists).

What happens to an inequality sign if you multiply both sides by -1?

The sign flips. (e.g., 2 < 4 becomes -2 > - 4).

If x<y, what does that mean regarding their position on the number line?

* is to the left of y.

What is the main purpose of Estimation in calculus?

To act as a "sanity check." It helps you recognize when a calculator answer is wildly incorrect due to a typing error or parenthesis mistake.

Why is division by zero not allowed? (Don't say "because the calculator says so")

Because it is impossible to make sense of the symbols. (You can't divide something into zero groups).

What is the definition of an Integer?

A whole number (positive, negative, or zero). No fractions or decimals.

What is the precise mathematical definition of a Rational Number?

A number that can be written in the form m / n, where m and n are integers and n = with a cross 0.

How can you tell if a decimal represents a Rational Number?

It either terminates (ends) OR repeats a specific cycle of digits forever.

What are the two defining characteristics of an Irrational Number's decimal?

It never terminates AND it never repeats.

Is 0.375 rational or irrational? Why?

Rational, because it terminates. (It can be written as 3(8).

Is 0.375375375.. rational or irrational? Why?

Rational. The '375' repeats forever, so it can be written as a fraction.

Why is (square root) v2 an irrational number?

It cannot be written as the quotient (ratio) of two integers. (Its decimal 1.414213... never terminates or repeats).

What does the "Density Property" of real numbers state?

Between any two distinct real numbers, there is always another real number (and infinitely many more).

How do we approximate an irrational number like square root 2 v?

By using a sequence of rational numbers that get closer and closer to the actual value (e.g., 1. 4,1.41,1.414...).

Why is estimation important before using a calculator?

To act as a "sanity check" to ensure you didn't make a parenthesis or typing error that resulted in a wildly incorrect answer.

What does the symbol ~ mean?

"Approximately equal to." Used in scratch work for estimations.

What is "number sense"?

The ability to work through a problem and recognize if an answer is reasonable or obviously wrong.

When a problem asks for an "exact answer" vs an "approximation," which is generally preferred for subsequent calculations?

The exact answer (e.g., V3 / 2 is preferred over 0.8660254).

What is the formal mathematical definition of x < y?

y - x is a positive number.

What is the formal mathematical definition of x

≤ У?

y - x is positive OR zero.

If t < y, where is x located relative to y on a number line?

X is to the left of y.

State the Transitivity property of inequalities.

If x < y and y < z then x < z

What happens to an inequality sign (<) if you multiply both sides by a negative number?

The inequality sign flips (reverses direction).

E.g.，x < y → -x > -y

In the statement "If P, then Q" (P → Q), what are P and Q called?

P is the hypothesis, Q is the conclusion.

What is the Converse of P→ Q?

Q → P (If Q, then P).

Is the converse of a true statement always true?

No. (e.g., True: "If it is a square, it has 4 sides." False Converse: "If it has 4 sides, it is a square.")

What is the Negation of statement P?

~ P (or - that funny drop end stick P). It is the exact opposite of the statement.

What is the Contrapositive of P→ Q?

~ Q →~ P (If not Q, then not P).

Are a statement and its contrapositive logically equivalent?

Yes. If one is true, the other must be true.

Why is proving the contrapositive sometimes easier than proving the original statement?

Because you get to start with ~ Q as a concrete assumption to work with, rather than starting with P.

What is a Proof by Contradiction (reductio ad absurdum)?

A proof that starts by assuming the conclusion is false, and then showing that this assumption leads to a logical impossibility or contradiction.

What is the Law of the Excluded Middle?

The principle that either a statement R is true, or its negation ~ R is true. There is no middle ground.

True or False: "For all x, x² > 0".

False. If x=0, then x²

= 0, which is not

greater than 0