Number System

Which rational number is located exactly midway between $\frac{2}{7}$ and $\frac{3}{7}$ on the number line?

What is the integer located immediately to the left of the rational number $-\frac{13}{3}$ on the number line?

What rational number is represented by the point that lies exactly halfway between $0$ and $-1$ on the number line?

Between which two consecutive integers does the rational number $\frac{13}{5}$ lie on the number line?

If the segment between integers $2$ and $3$ on the number line is divided into $5$ equal parts, what is the value of the rational number at the $4^{th}$ division mark starting from $2$ and moving towards $3$?

Simplify the expression: $7^{1/2} \cdot 8^{1/2}$

Calculate $\sqrt{13^2-12^2}$.

Point $X$ represents $\frac{1}{3}$ and point $Y$ represents $\frac{1}{2}$ on the number line. What is the value of the rational number represented by point $Z$ that lies exactly one-third of the way from $X$ to $Y$?

If $a = \frac{\sqrt{2}+1}{\sqrt{2}-1}$ and $b = \frac{\sqrt{2}-1}{\sqrt{2}+1}$, find $a^2-b^2$.

If $x = 2^{1/3} + 2^{2/3}$, find the value of $x^3 - 6x$.

Which fraction is equivalent to a rational number between $2$ and $3$?

Between the rational numbers $x$ and $y$, is there always another rational number?

What is the result of $\frac{1}{2}(\frac{1}{2} + \frac{1}{4})$?

Which of these is a rational number between $-5$ and $-6$?

What is the rational number that lies exactly one-third of the distance from $\frac{1}{4}$ toward $\frac{1}{2}$ on the number line?

Is the number $\frac{a+b}{2}$ always a rational number if $a$ and $b$ are rational?

If you are asked to find 99 rational numbers between 2 and 3, what is a convenient denominator to use for both numbers written as fractions?

Is the set of all rational numbers between $\frac{1}{2}$ and $\frac{1}{2}$ a dense set?

To find $n$ rational numbers between $x$ and $y$, we can use the formula $d = \frac{y-x}{n+1}$. What is $d$ for finding 9 rational numbers between $0.1$ and $0.2$?

Which of the following is a rational number between $\frac{1}{13}$ and $\frac{1}{12}$?

What is the decimal value of $\frac{7}{8}$?

A terminating decimal can always be written in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q$ is:

Which type of decimal expansion does $\frac{1}{20}$ have?

The sum of a rational number and an irrational number is always:

If the mixed recurring decimal $0.1\overline{36}$ is written as a fraction $\frac{p}{q}$ in its simplest form, where $p$ and $q$ are integers, then the value of $q - p$ is:

The value of $0.1\overline{4} + 0.2\overline{7}$ expressed as a rational number $\frac{p}{q}$ in its simplest form is:

The $p/q$ form of $0.\overline{36}$ is:

Express $0.\overline{123}$ in the form $\frac{p}{q}$ in its simplest version.

Find the value of the difference $0.\overline{45} - 0.\overline{44}$ as a recurring decimal.

Represent $0.\overline{001}$ as a fraction.

The number $0.232323...$ (where the block of digits $23$ repeats indefinitely) is classified as a/an:

Which of the following numbers is irrational?

The value of $2\sqrt{3} + 4\sqrt{3}$ is:

What is the type of the number $\sqrt{1000}$?

Classify the decimal numbers $a = 0.565656\dots$ and $b = 0.565665666\dots$ as rational or irrational based on their digit patterns.

Classify the numbers $x = \sqrt{0.4}$ and $y = \sqrt{0.04}$ as rational or irrational.

Determine the nature of the number obtained by simplifying the expression $(\sqrt{5} + 2)(\sqrt{5} - 2) + \sqrt{3}$.

If $a = \sqrt{2} + 1$ and $b = \sqrt{2} - 1$, find the value of $a^2 + b^2 - 4ab$.

Solve for $x$: $\sqrt{x+3} - \sqrt{x-2} = 1$.

If $a = 3 + 2\sqrt{2}$, find the value of $\frac{a^6 + a^4 + a^2 + 1}{a^3}$.

Which of the following is a common factor of $a$ and $b$ found during the contradiction proof of $\sqrt{3}$?

Which of the following is a common factor of $p$ and $q$ found during the contradiction proof of $\sqrt{2}$?

In the proof for $\sqrt{2}$, if $p^2 = 2q^2$, we conclude $p$ is even. If $p=2k$, we find $q^2 = 2k^2$, concluding $q$ is:

If $\sqrt{3}$ was rational, then $\sqrt{3} = p/q$ would imply $p^2 = 3q^2$. This means $p^2$ is divisible by:

What does the symbol $\sqrt{2} \notin \mathbb{Q}$ represent in the context of Number Systems?

What contradiction is specifically encountered in the proof for $\sqrt{3}$ being irrational?

If $p$ is a prime number that divides $k^2$, what is the relationship between $p$ and $k$?

If the proof for $\sqrt{2}$ is applied to $\sqrt{p}$ where $p$ is any prime, what is the conclusion about the set of all square roots of primes?

When proving $\sqrt{2}$ is irrational, why can we not assume $p$ and $q$ are both odd at the beginning?

In the proof of $\sqrt{3}$ being irrational, the contradiction is that $a$ and $b$ have a common factor of $3$. If we assumed $a/b$ was in simplest form, this contradicts what property of the fraction?

If a point $X$ represents $5.75$, then $X$ is the midpoint of the interval:

Which integer is immediately to the right of $-1$?

What is the distance between $-5$ and $-2$ on the number line?

The number $0.0001$ is located to the _____ of $0.001$.

To locate $\sqrt{14}$ using the spiral method, which value must be constructed immediately before it?

In the geometric construction of $\sqrt{x}$, if $OA$ is the radius, then $OA^2 - OB^2$ equals:

When representing $6.123$ on the number line, the interval $[6.12, 6.13]$ is divided into how many equal parts?

In the construction of $\sqrt{x}$, if $BD = \sqrt{x}$, what is the value of $BD^2 + OB^2$ where $O$ is the center of the semicircle and $B$ is the point such that $AB=x, BC=1$?

When magnifying $2.34567$ to $5$ decimal places, the $4^{th}$ step involves dividing which interval?

If we represent $\sqrt{2}, \sqrt{3}, \sqrt{4}, ...$ as points $P_2, P_3, P_4, ...$ on the number line, which point is at a distance of exactly $1$ unit from $P_9$?