MATH 19 - Review on Integers

Vocabulary flashcards covering core definitions, mathematical properties, operation rules, divisibility rules, and order of operations from MATH 19: Number Theory Unit 1 lecture notes.

Integers

Numbers that include negative whole numbers, zero, and positive whole numbers, excluding fractions or decimals.

Set of Integers ($\mathbb{Z}$)

The collection of all positive whole numbers, negative whole numbers, and zero, represented by the symbol $\mathbb{Z}$ as $\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.

Positive Integers

Whole numbers greater than $0$, represented by the set $\{1, 2, 3, 4, \dots\}$.

Negative Integers

Whole numbers less than $0$, represented by the set $\{\dots, -4, -3, -2, -1\}$.

Zero

An integer that is neither positive nor negative, represented as $\{0\}$.

Whole Numbers

Non-negative integers, including zero, represented by the set $\{0, 1, 2, 3, \dots\}$.

Natural Numbers

Counting numbers, commonly represented by the set $\{1, 2, 3, \dots\}$.

Visualizing Integers

Representing integers on a number line to show their positions and values, with $0$ placed in the center, positive integers to the right, and negative integers to the left.

Commutative Property

A property stating that when adding or multiplying integers, changing the order of the numbers does not change the answer ($a + b = b + a$ and $a \times b = b \times a$).

Associative Property

A property stating that when adding or multiplying three or more integers, changing the grouping of the numbers does not change the result ($a + (b + c) = (a + b) + c$ and $a \times (b \times c) = (a \times b) \times c$).

Distributive Property

A property stating that multiplication can be distributed over addition or subtraction ($a \times (b + c) = (a \times b) + (a \times c)$ and $a \times (b - c) = (a \times b) - (a \times c)$).

Identity Property

A property stating that adding zero to an integer keeps the number unchanged ($a + 0 = a$), while multiplying an integer by one also keeps it unchanged ($a \times 1 = 1 \times a = a$).

Inverse Property

A property stating that when an integer is added to its opposite (additive inverse), the result is always zero ($a + (-a) = 0$).

Addition of Integers (Same Signs)

A rule stating that when adding integers with the same sign, you add the numbers and keep the same sign.

Addition of Integers (Different Signs)

A rule stating that when adding integers with different signs, you subtract the smaller absolute value from the larger one, then use the sign of the number with the greater absolute value.

Keep, Change, Change

The rule for subtracting integers: Keep the first number, Change the subtraction sign to addition, and Change the sign of the second number.

Multiplication and Division Sign Rules

Rules stating that multiplying or dividing integers with the same signs yields a positive answer, while different signs yield a negative answer.

Divisibility Rules

Shortcuts used to determine whether a whole number can be divided by another number without leaving a remainder using patterns in its digits.

Divisibility Rule for 2

A number is divisible by $2$ if its last digit is $0$, $2$, $4$, $6$, or $8$.

Divisibility Rule for 3

A number is divisible by $3$ if the sum of its digits is divisible by $3$.

Divisibility Rule for 4

A number is divisible by $4$ if its last two digits form a number divisible by $4$.

Divisibility Rule for 5

A number is divisible by $5$ if its last digit is $0$ or $5$.

Divisibility Rule for 6

A number is divisible by $6$ if it is divisible by both $2$ and $3$.

Divisibility Rule for 7

Double the last digit, subtract it from the remaining number, and check whether the result is divisible by $7$.

Divisibility Rule for 8

A number is divisible by $8$ if its last three digits form a number divisible by $8$.

Divisibility Rule for 9

A number is divisible by $9$ if the sum of its digits is divisible by $9$.

Divisibility Rule for 10

A number is divisible by $10$ if its last digit is $0$.

Divisibility Rule for 11

Add and subtract the digits alternately; if the result is divisible by $11$, the original number is divisible by $11$.

Order of Operations (PEMDAS)

The set of rules used to determine which operation to perform first in a mathematical expression: Parentheses/Grouping Symbols, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).