Types of Numbers and Real Number System Flashcards

Vocabulary flashcards covering categories of numbers, rational and irrational definitions, prime numbers, parity, and mathematical constants based on the lecture transcript.

Natural Numbers

The set of counting numbers starting from $1$, $2$, $3$, and continuing infinitely, spaced equally by $1$. Represented by the symbol $N$.

Whole Numbers

The set of numbers that includes $0$ along with all natural numbers ($0$, $1$, $2$, $3$, …). Represented by the symbol $W$.

Integers

The set of numbers containing all whole numbers and negative natural numbers, extending infinitely in both positive and negative directions spaced by $1$. Symbolized by $I$ or universally $Z$.

Rational Numbers

Any number that can be expressed as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. Universally represented by the symbol $Q$.

Terminating Decimal

A decimal representation of a rational number that ends or stops at a specific decimal place, such as $3.5$ or $2.315$.

Repeating Decimal

A decimal representation of a rational number that does not end but infinitely repeats a digit or a group of digits.

Division by Zero

An operation where a non-zero number is divided by $0$ (e.g., $5 / 0$), which is undefined and results in a non-real number.

Irrational Numbers

Numbers that cannot be expressed as an integer divided by a nonzero integer, having decimal forms that go on infinitely without repeating a pattern. Represented by $Q^c$.

Pi ($\pi$)

An irrational number defined as the ratio of the circumference ($c$) to the diameter ($d$) of any circle ($\frac{c}{d}$).

Prime Number

Any natural number greater than $1$ whose only divisors are $1$ and itself. The smallest prime number is $2$.

Even Number

An integer that is divisible by $2$ without leaving a remainder, such as $-6$, $0$, and $2$.

Odd Number

An integer that is not divisible by $2$ without leaving a remainder, such as $-11$.

Number $e$

An irrational number referred to as 'the natural number' because it occurs naturally in nature, mathematically defined as the limit of $\left(1 + \frac{1}{n}\right)^n$ as $n$ approaches infinity.