Review the key concepts, formulae, and examples before starting your quiz.
🔑Concepts
Integers are a collection of whole numbers and their negatives, represented by the symbol . This set includes . On a horizontal number line, zero is the central point; positive integers are located to the right of zero, and negative integers are to the left.
The number line is a visual representation where integers are marked at equal intervals. The distance between any two consecutive integers is constant. Visually, as you move to the right on the number line, the values of the integers increase, and as you move to the left, the values decrease.
Ordering of integers is determined by their position on the number line. Any integer that lies to the right of another integer is greater. For example, because is to the right of . Similarly, because is closer to zero and further to the right than .
The absolute value of an integer , denoted by , is its numerical value regardless of its sign. Visually, it represents the distance of the number from zero. Since distance cannot be negative, absolute value is always non-negative. For instance, both and equal because both points are units away from zero.
The Additive Inverse of an integer is . When an integer is added to its additive inverse, the result is the additive identity, zero (). On a number line, an integer and its additive inverse are located at the same distance from zero but on opposite sides.
Addition of integers follows two main rules: 1) If the signs are the same, add their absolute values and keep the common sign (e.g., ). 2) If the signs are different, subtract the smaller absolute value from the larger one and use the sign of the integer with the larger absolute value (e.g., ).
Subtraction of integers is performed by adding the additive inverse of the number being subtracted (). Visually, subtracting a positive integer means moving to the left on the number line, while subtracting a negative integer (which is the same as adding a positive) means moving to the right.
📐Formulae
Definition of Absolute Value: if and if
Additive Inverse Property:
Subtraction Rule:
Double Negative Rule:
Commutative Law for Addition:
💡Examples
Problem 1:
Evaluate the following expression:
Solution:
Step 1: Simplify the expression by handling the subtraction of a negative number. Using the rule , the expression becomes . \nStep 2: Add the positive integers together: . The expression is now . \nStep 3: Since the signs are different, subtract the smaller absolute value from the larger: . \nStep 4: Use the sign of the number with the larger absolute value ( is positive), so the result is .
Explanation:
This problem demonstrates the transformation of subtraction into addition and the rule for adding integers with different signs.
Problem 2:
Arrange the following integers in ascending order:
Solution:
Step 1: Identify and compare the negative integers: . On the number line, is furthest to the left, followed by , then . So, . \nStep 2: Place zero: is greater than all negative integers but less than all positive integers. \nStep 3: Identify and compare the positive integers: . Clearly, . \nStep 4: Combine all parts to form the ascending sequence: .
Explanation:
Ascending order means moving from the smallest (most negative) to the largest (most positive) value, corresponding to a left-to-right movement on the number line.