Understanding Integers & Absolute Value - Seventh Grade Mathematics

The number line extends infinitely in both directions from zero, establishing integers as a foundational numerical system for representing magnitude, direction, opposites, and distance.

The Set of Integers

Whole numbers allow us to count discrete objects, but our physical and financial world frequently demands quantities that fall below zero or exist in opposite directions. The set of integers, denoted by the mathematical symbol Z, includes all positive counting numbers, their negative counterparts, and the neutral origin, zero.

                  THE INTEGER NUMBER LINE
<----+----+----+----+----+----+----+----+----+----+----+---->
    -5   -4   -3   -2   -1    0    1    2    3    4    5
   <--- Negative Integers ---|--- Positive Integers --->
                            Zero (Origin)

Positive integers are values strictly greater than zero, positioned to the right of zero on a horizontal number line. Negative integers are values strictly less than zero, positioned to the left of zero, denoted by a negative sign preceding the numeral. Zero is neither positive nor negative; it acts as the neutral balance point or origin of the entire real number system.

Opposite Numbers and Additive Inverses

Every integer has a corresponding opposite located at the exact same distance from zero on the opposite side of the number line. The opposite of positive seven is negative seven, and the opposite of negative twelve is positive twelve. When an integer and its opposite are combined through addition, they cancel each other out completely, producing a sum of zero.

       Opposite (-4)                      Opposite (+4)
  <---------- 4 units ---------- 0 ---------- 4 units ---------->
  +-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
 -5    -4    -3    -2    -1     0     1     2     3     4     5

Because an integer and its opposite always sum to zero, they are known as additive inverses. For any integer a, the additive inverse property states that a plus negative a equals zero. This property forms the bedrock for subtraction and equation solving across all of algebra.

Real-World Representations of Signed Numbers

Signed numbers give precise numerical meaning to physical scenarios that have opposing orientations. In geography, elevation above sea level is expressed with positive integers, while depth below sea level is assigned negative integers. In meteorology, temperatures above freezing are positive, whereas temperatures below freezing are negative. In personal finance, income and deposits are positive, while expenditures, debts, and bank overdrafts are negative.

+-----------------------+-----------------------+-----------------------+
| Domain Context        | Positive Quantity (+) | Negative Quantity (-) |
+-----------------------+-----------------------+-----------------------+
| Altitude & Elevation  | Mountain peak (+2500m)| Ocean trench (-150m)  |
| Temperature           | Warm summer (+28 C)   | Subzero winter (-14 C)|
| Banking & Finance     | Account deposit (+$50)| Overdraft debt (-$35) |
| Coordinate Motion     | Eastward or upward    | Westward or downward  |
+-----------------------+-----------------------+-----------------------+

Absolute Value: Distance and Magnitude

In many scientific and mathematical contexts, we need to know the total magnitude of a quantity without regard to its direction. The absolute value of an integer represents its distance from zero on the number line. Distance is an intrinsically non-negative physical measurement; therefore, the absolute value of any real number is either positive or zero.

           |-5| = 5 units                |3| = 3 units
       <--------------------- 0 ------------->
  +-----+-----+-----+-----+-----+-----+-----+-----+-----+
 -5    -4    -3    -2    -1     0     1     2     3     4

We represent absolute value using vertical bars surrounding the number or expression. For example, the absolute value of negative five is written as vertical bar negative five vertical bar, which evaluates to five. The absolute value of positive three evaluates to three, and the absolute value of zero is zero.

Evaluating Expressions with Absolute Value Bars

Absolute value bars function as grouping symbols in the standard order of operations. You must completely simplify any arithmetic operations inside the bars before determining the non-negative distance. However, when a negative sign appears outside the absolute value bars, that negative sign is applied after the absolute value has been extracted.

Example 1: |-9| = 9
Example 2: -|-9| = -(9) = -9
Example 3: |14 - 20| = |-6| = 6
Example 4: -|7 - 2| = -|5| = -5

Recognizing the distinction between the sign inside the bars and a negative operator outside the bars prevents costly notation errors in multi-step algebraic problems.

Comparing and Ordering Integers

To compare two integers, determine their relative positions on the standard horizontal number line. Any integer located to the right of another integer is greater than that integer. Conversely, any integer located to the left is smaller.

Order from least to greatest: -6, -2, 0, 3
Visual position:   -6  <  -2  <  0  <  3

When comparing two negative integers, intuition can sometimes deceive you if you look only at the numerals. Because negative numbers move further to the left as their absolute values increase, the negative integer with the greater absolute value is actually smaller. For instance, negative eight is less than negative three, because negative eight lies five units further to the left of negative three on the number line.

Exercises

Work through each problem carefully in your notebook before reading the provided solutions.

Exercise 1. Identify the additive inverse of negative nineteen and write a mathematical equation demonstrating the additive inverse property with this value.

Exercise 2. A submarine cruises at a depth of negative two hundred forty meters, while a research drone flies directly above it at an elevation of positive one hundred ten meters. Find the absolute value of each vehicle's elevation and state what each absolute value represents in physical terms.

Exercise 3. Evaluate the following mathematical expression step by step: the absolute value of negative fifteen minus the opposite of the absolute value of negative four.

Exercise 4. Order the following integers from least to greatest: positive seven, negative eleven, negative three, positive zero, negative fourteen, and positive two.

Solutions

Solution to Exercise 1. The additive inverse of negative nineteen is positive nineteen. The mathematical equation demonstrating the additive inverse property is negative nineteen plus nineteen equals zero, or symbolically (-19) + 19 = 0.

Solution to Exercise 2. The absolute value of the submarine's elevation is the absolute value of negative two hundred forty, which equals two hundred forty meters. This represents the submarine's total physical distance below the surface of the ocean. The absolute value of the drone's elevation is the absolute value of positive one hundred ten, which equals one hundred ten meters, representing the drone's physical altitude above the ocean surface.

Solution to Exercise 3. First, evaluate the absolute value of negative fifteen, which simplifies to fifteen. Second, evaluate the absolute value of negative four inside the second term, which simplifies to four. Third, apply the negative sign preceding the second absolute value, producing negative four. Finally, subtract negative four from fifteen, which is equivalent to adding positive four: fifteen minus negative four equals fifteen plus four, which equals nineteen.

Solution to Exercise 4. On a horizontal number line, values increase from left to right. Comparing the given numbers, negative fourteen lies furthest to the left, followed by negative eleven, then negative three, zero, positive two, and positive seven. Therefore, the ordered list from least to greatest is negative fourteen, negative eleven, negative three, zero, two, seven.