Positive and Negative Numbers: Guide and Examples

What Is Positive and Negative Numbers?

Positive and negative numbers are mathematical values that represent opposite directions or quantities relative to zero.

A positive number is any value greater than zero. It is written with or without a plus sign ($+$). For example, $5$ and $+5$ mean the same thing.

A negative number is any value less than zero. It must always be written with a minus sign ($-$). For example, $-3$ represents a value three units less than zero.

Zero ($0$) is the reference point or anchor. It is the only number that is neither positive nor negative.

Key Ideas and Vocabulary

Understanding how positive and negative numbers work requires knowing a few important natural numbers and broader number sets.

The counting numbers $1, 2, 3$ and so on are positive integers. When we include zero, we get the set of whole numbers. When we extend this set to include the negative counting numbers ($-1, -2, -3$), the complete collection is known as the integers.

• Signed numbers: This is another name for positive and negative numbers because they possess a specific sign ($+$ or $-$) indicating their position relative to zero.
  
• Opposites: Two numbers that are exactly the same distance from zero but on opposite sides are called opposites. For instance, $4$ and $-4$ are opposites. This relationship is also known as an additive inverse.
  
• Absolute value: The absolute value of a number is its total distance from zero, without considering direction. Because distance cannot be negative, absolute value is always positive or zero. We write the absolute value of $-7$ as $|-7| = 7$.
  
• Even and odd: Both positive and negative numbers can be classified as even and odd numbers. For example, $4$ and $-4$ are both even, while $3$ and $-3$ are both odd.

Visual Explanation

The most common way to visualize these values is by placing both positive and negative numbers on a number line. A number line is a straight line that extends endlessly in both directions, divided into equally spaced intervals.

Zero sits at the center. Moving to the right of zero takes you through the positive numbers, which increase in value. Moving to the left of zero takes you through the negative numbers, which decrease in value. The further left a number sits, the smaller its value.

Number lines can also be drawn vertically. On a vertical number line, positive values go up, and negative values go down.

Worked Examples

In the positive and negative numbers examples below, we perform arithmetic by visualizing movement along the number line. Adding moves you right (or up), and subtracting moves you left (or down).

Example 1: Plotting values on a number line

Question: Where do the numbers $-3$, $1$, and $-5$ belong on a number line, and which has the lowest value?

Method:

1. Draw a number line with zero near the center.
   
2. For $-3$, start at zero and move $3$ units to the left. Plot a point.
   
3. For $1$, start at zero and move $1$ unit to the right. Plot a point.
   
4. For $-5$, start at zero and move $5$ units to the left. Plot a point.
   
5. Identify the point furthest to the left.

Answer: The number $-5$ sits furthest to the left, so it has the lowest value.

Check: Since $-5$ is a greater negative distance from zero than $-3$, its value is mathematically smaller.

Here is how adding a positive number looks on the number line:

Example 2: Adding on the number line

Question: Calculate $-2 + 6$.

Method:

1. Locate the starting number, $-2$, on the number line.
   
2. Look at the operation. Adding a positive number means moving to the right.
   
3. Move $6$ units to the right from $-2$.

Answer: Moving $6$ units right from $-2$ lands on $4$. Therefore, $-2 + 6 = 4$.

Check: Since we start at a negative number and add a larger positive amount, the result must be positive.

Example 3: Subtracting a negative number

Question: Evaluate $3 - (-4)$.

Method:

1. Locate the starting number, $3$.
   
2. The operation is subtraction, which normally moves left. However, subtracting a negative number reverses the direction.
   
3. Move $4$ units to the right instead.

Answer: Moving $4$ units right from $3$ lands on $7$. Therefore, $3 - (-4) = 7$.

Check: Subtracting a negative number is mathematically identical to adding its positive opposite: $3 + 4 = 7$.

Common Mistakes and Non-Examples

Working with positive and negative numbers often introduces specific challenges. Watch out for these common errors.

• Confusing larger negative digits with larger values: A common mistake is believing that $-8$ is greater than $-2$ because $8$ is greater than $2$. On the number line, $-8$ is further to the left than $-2$. Therefore, $-8$ is smaller than $-2$.
  
• Treating zero as a positive number: Zero is strictly neutral. A statement like "all numbers are either positive or negative" is a non-example of mathematical truth. The correct statement is "all non-zero numbers are either positive or negative."
  
• Forgetting to reverse direction when subtracting a negative: When calculating $5 - (-2)$, a common error is to subtract $2$ and get $3$. The negative sign on the $-2$ reverses the subtraction direction, turning it into addition: $5 + 2 = 7$.

The further a negative number is from zero, the smaller its value.

Real-World Connections

Positive and negative numbers appear constantly in the real world to measure opposites. These numbers below zero describe physical states like freezing temperatures or deep elevations.

Temperature

Many regions experience temperatures below zero. If the temperature is $-5^\circ\text{C}$, it is five degrees below the freezing point of water. If the temperature drops another $3^\circ\text{C}$, the new temperature is $-8^\circ\text{C}$.

Elevation

Sea level represents an elevation of zero. The height of a mountain peak is a positive number, such as $2{,}500\text{ m}$ above sea level. The depth of an ocean trench is represented by a negative number, such as $-1{,}200\text{ m}$.

Account Balances

If a digital wallet has a positive balance of $150\text{ credits}$, those credits are available to spend. If the account becomes overdrawn by $20\text{ credits}$, the balance is recorded as $-20\text{ credits}$, meaning the account is short by that amount.