Comparing Integers: Guide and Examples

What Is Comparing Integers?

Integers are all the whole numbers and their negative opposites, including zero. When comparing integers, we establish the mathematical relationship between two values. We express this relationship using greater than less than equal to symbols: $>$ (greater than), $<$ (less than), and $=$ (equal to).

Before learning to compare negative numbers, you practiced comparing whole numbers. That logic is the same for positive integers: $5$ is greater than $2$. However, negative integers introduce a new direction. The value of a negative number decreases as it moves farther away from zero on the left side of the number line.

When to Use It

Integer comparison is essential for solving problems involving directions or opposites. You use it when analyzing temperature drops, tracking financial credits and debits, or reading elevations above and below sea level.

Once you can accurately establish which of two integers is greater, you can confidently apply the same principles to ordering numbers in sequences, sorting data from lowest to highest, or identifying extreme values in a set.

Step-by-Step Method

To compare any two integers, use the rightward-is-greater rule on a number line.

1. Identify the signs of the two integers.
   
2. If one integer is positive and the other is negative, the positive integer is always greater.
   
3. If one integer is zero and the other is negative, zero is always greater.
   
4. If both integers are negative, picture them on a number line. The number located farther to the right has the greater value.

When comparing two negative integers, the number closer to zero is the greater number.

Visual Worked Examples

Applying the number line rules systematically makes it easy to compare any pair of integers.

Example 1: Positive and negative integers

Question: Compare $4$ and $-6$ using an inequality symbol.

Method:

1. Identify the signs: $4$ is positive and $-6$ is negative.
   
2. Recall the rule: Any positive integer is greater than any negative integer.
   
3. Write the inequality symbol pointing the open side toward the greater value.

Answer: $4>-6$.

Check: On a number line, $4$ is to the right of zero, and $-6$ is to the left of zero. Therefore, $4$ is farther right and greater.

Example 2: Two negative integers

Question: Compare $-7$ and $-2$ using an inequality symbol.

Method:

1. Identify the signs: Both integers are negative.
   
2. Locate both numbers on a mental or physical number line.
   
3. Observe that $-2$ is closer to zero and lies to the right of $-7$.
   
4. Determine that $-2$ is the greater value.

Answer: $-7<-2$.

Check: $-2$ represents a smaller debt or a warmer temperature than $-7$, confirming it is the greater integer.

Example 3: Comparing with zero

Question: Compare $0$ and $-5$ using an inequality symbol.

Method:

1. Identify the values: One is zero and the other is negative.
   
2. Recall that zero separates the positive and negative numbers.
   
3. Every negative number lies to the left of zero.

Answer: $0>-5$.

Check: The number $0$ has a greater value because it sits further to the right on the number line than $-5$.

How to Check the Answer

While skills like rounding whole numbers and estimation in math focus on approximate amounts, checking integer comparisons requires strict attention to the negative sign.

You can check negative integer comparisons by looking at their absolute values (their distance from zero without the negative sign). When both numbers are negative, the number with the larger absolute value is actually the smaller integer because it lies farther away from zero in the negative direction.

For example, the absolute value of $-8$ is $8$, and the absolute value of $-3$ is $3$. Since $8$ is larger than $3$, $-8$ must be farther into the negatives, making it the smaller number.

Common Mistakes

A very common mistake is ignoring the negative signs and assuming that a visually "larger" number always represents a greater value. For instance, a learner might look at $-10$ and $-2$, see that $10$ is bigger than $2$, and incorrectly write $-10>-2$.

Always remember the number line rule. Because $-10$ is farther to the left than $-2$, the true relationship is $-10<-2$. Treating negative numbers as if they were positive will lead to backwards comparisons.