Pre-Algebra Unit 3 Review: Integers

What Are Integers?

• Integers are whole numbers that can be positive, negative, or zero
  • Positive integers are numbers greater than zero (1, 2, 3, ...)
  • Negative integers are numbers less than zero (-1, -2, -3, ...)
  • Zero (0) is neither positive nor negative
• Integers do not include fractions or decimals
• The set of integers is represented by the symbol $\mathbb{Z}$
• Integers are used to represent quantities that can be counted, such as the number of apples in a basket or the number of people in a room
• Integers are a subset of the real numbers, which include all rational and irrational numbers

Number Line Basics

• A number line is a visual representation of numbers, including integers, on a straight line
• The number line extends infinitely in both the positive and negative directions
• The point where the positive and negative numbers meet is called the origin, represented by zero (0)
• Moving to the right on the number line represents increasing values, while moving to the left represents decreasing values
• The distance between any two consecutive integers on the number line is always equal to 1
• Absolute value represents the distance of a number from zero on the number line, regardless of its sign
  • The absolute value of a positive integer is the integer itself
  • The absolute value of a negative integer is the positive value of that integer

Comparing Integers

• Comparing integers involves determining which integer is greater than, less than, or equal to another integer
• When comparing two positive integers, the integer with the greater absolute value is the greater integer
• When comparing two negative integers, the integer with the lesser absolute value is the greater integer
• When comparing a positive and a negative integer, the positive integer is always greater than the negative integer
• The symbols used for comparing integers are:
  • Greater than (>)
  • Less than (<)
  • Equal to (=)
  • Greater than or equal to (≥)
  • Less than or equal to (≤)
• On a number line, the integer to the right is always greater than the integer to its left

Adding and Subtracting Integers

• Adding integers with the same sign (both positive or both negative) involves adding their absolute values and keeping the common sign
  • Example: $5 + 3 = 8$ and $(-5) + (-3) = -8$
• Adding integers with different signs involves subtracting the absolute values and keeping the sign of the integer with the greater absolute value
  • Example: $5 + (-3) = 2$ and $(-5) + 3 = -2$
• Subtracting integers is equivalent to adding the opposite (additive inverse) of the integer being subtracted
  • To find the opposite of an integer, change its sign
  • Example: $5 - 3 = 5 + (-3) = 2$ and $(-5) - (-3) = (-5) + 3 = -2$
• The commutative property of addition states that the order of the addends does not affect the sum
  • Example: $a + b = b + a$
• The associative property of addition states that the grouping of the addends does not affect the sum
  • Example: $(a + b) + c = a + (b + c)$

Multiplying Integers

• When multiplying two integers with the same sign (both positive or both negative), the product is always positive
  • Example: $5 \times 3 = 15$ and $(-5) \times (-3) = 15$
• When multiplying two integers with different signs (one positive and one negative), the product is always negative
  • Example: $5 \times (-3) = -15$ and $(-5) \times 3 = -15$
• Multiplying an integer by 1 results in the integer itself
• Multiplying an integer by -1 results in the opposite (additive inverse) of the integer
• The commutative property of multiplication states that the order of the factors does not affect the product
  • Example: $a \times b = b \times a$
• The associative property of multiplication states that the grouping of the factors does not affect the product
  • Example: $(a \times b) \times c = a \times (b \times c)$
• The distributive property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products
  • Example: $a \times (b + c) = (a \times b) + (a \times c)$

Dividing Integers

• When dividing two integers with the same sign (both positive or both negative), the quotient is always positive
  • Example: $15 \div 3 = 5$ and $(-15) \div (-3) = 5$
• When dividing two integers with different signs (one positive and one negative), the quotient is always negative
  • Example: $15 \div (-3) = -5$ and $(-15) \div 3 = -5$
• Dividing an integer by 1 results in the integer itself
• Dividing an integer by -1 results in the opposite (additive inverse) of the integer
• Division by zero is undefined for integers
• The quotient of two integers may not always be an integer, but rather a rational number (fraction or mixed number)

Real-World Applications

• Temperature changes can be represented using integers, with positive values indicating temperatures above zero and negative values indicating temperatures below zero
• Elevation can be expressed using integers, with positive values representing heights above sea level and negative values representing depths below sea level
• Profit and loss in financial transactions can be represented using integers, with positive values indicating profits and negative values indicating losses
• Electric charges can be described using integers, with positive values representing positive charges and negative values representing negative charges
• Timelines can be represented using integers, with positive values indicating years after a specific event (AD or CE) and negative values indicating years before the event (BC or BCE)

Common Mistakes and How to Avoid Them

• Forgetting to apply the correct sign rules when adding, subtracting, multiplying, or dividing integers
  • Always remember the rules for operations with same and different signs
• Confusing the absolute value of an integer with the integer itself
  • Remember that the absolute value is always non-negative and represents the distance from zero on the number line
• Incorrectly comparing integers, especially when dealing with negative numbers
  • Keep in mind that when comparing two negative integers, the one with the lesser absolute value is the greater integer
• Misapplying the order of operations (PEMDAS) when working with integers
  • Always follow the correct order: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)
• Attempting to divide by zero, which is undefined for integers
  • Be cautious when dividing integers and ensure that the divisor is not zero
• Oversimplifying real-world applications of integers without considering the context
  • Pay attention to the context and the meaning of positive and negative values in real-world scenarios