Rational Numbers Quiz

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Questions and Answers

What type of numbers are the square root of 2, $\pi$, $e$, and the golden ratio?

Irrational numbers (correct)
Complex numbers
Real numbers
Rational numbers

In which set are rational numbers usually denoted?

$\mathbb{R}$
$\mathbb{N}$
$\mathbb{Q}$ (correct)
$\mathbb{Z}$

Which of the following is a rational number?

$\pi$
$\sqrt{2}$
$e$
$\frac{3}{7}$ (correct)

What kind of decimal expansion do rational numbers have?

Terminates after a finite number of digits or repeats the same finite sequence of digits over and over (A) Signup and view all the answers

What is the relationship between the countability of rational numbers and the uncountability of real numbers?

The set of rational numbers is countable, while the set of real numbers is uncountable. (B) Signup and view all the answers

Define exponentiation in mathematics and how it is written.

Exponentiation in mathematics is an operation involving two numbers, the base and the exponent or power. It is written as $b^n$, where $b$ is the base and $n$ is the power. Signup and view all the answers

Explain how exponentiation corresponds to repeated multiplication of the base.

When $n$ is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, $b^n$ is the product of multiplying $n$ bases. Signup and view all the answers

What are the different ways to refer to $b^n$?

Different ways to refer to $b^n$ include 'b raised to the nth power', 'b to the power of n', 'the nth power of b', and most briefly as 'b to the n(th)'. Signup and view all the answers

What is the basic rule that exponents add, and what does it imply about $b^0$?

The basic rule is that when multiplying a base raised to one exponent by the same base raised to another exponent, the exponents add. This implies that $b^0$ must be equal to 1 for any $b \neq 0$. Signup and view all the answers

How can it be shown that $b^0 = 1$ for any $b \neq 0$?

For any $n$, $b^0 \times b^n = b^{0+n} = b^n$. Dividing both sides by $b^n$ gives $b^0 = b^n / b^n = 1$. Signup and view all the answers

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Study Notes

Irrational Numbers

The square root of 2, π, e, and the golden ratio are irrational numbers.
Irrational numbers are non-repeating, non-terminating decimals.

Rational Numbers

Rational numbers are usually denoted in the set Q.
A rational number is a number that can be expressed as the ratio of two integers, e.g. 3/4.
Rational numbers have a terminating or repeating decimal expansion.
The set of rational numbers is countable, whereas the set of real numbers is uncountable.

Exponentiation

Exponentiation is a mathematical operation that represents repeated multiplication of a base.
It is written in the form b^n, where b is the base and n is the exponent.
Exponentiation corresponds to repeated multiplication of the base, e.g. b^3 = b × b × b.
The number b^n can be referred to as "b to the power of n" or "b raised to the n th power".
The basic rule of exponents is that exponents add, i.e. b^m × b^n = b^(m+n).
This rule implies that b^0 = 1, for any b ≠ 0.
It can be shown that b^0 = 1 for any b ≠ 0, as any number raised to the power of 0 is 1.