Understanding Integers

Learn about positive and negative numbers and how to compare them

Understanding Integers

What are Integers?

Integers are whole numbers and their opposites, including zero: ...,−3,−2,−1,0,1,2,3,......, -3, -2, -1, 0, 1, 2, 3, ...

  • Positive integers: 1, 2, 3, 4, ...
  • Negative integers: -1, -2, -3, -4, ...
  • Zero: Neither positive nor negative

The Number Line

Integers can be shown on a number line:

⟵-5 -4 -3 -2 -1 0 1 2 3 4 5⟶\longleftarrow \text{-5 -4 -3 -2 -1 0 1 2 3 4 5} \longrightarrow

  • Numbers increase as you move right
  • Numbers decrease as you move left

Comparing Integers

On a number line:

  • Numbers to the right are greater
  • Numbers to the left are less

Examples:

  • 5>−35 > -3 (5 is to the right of -3)
  • −1>−4-1 > -4 (-1 is to the right of -4)
  • −2<0-2 < 0 (-2 is to the left of 0)

Absolute Value

The absolute value of a number is its distance from zero (always positive):

∣−5∣=5and∣5∣=5|-5| = 5 \quad \text{and} \quad |5| = 5

Both -5 and 5 are 5 units from zero.

Real-World Examples

  • Temperature: −10°-10° (below zero)
  • Elevation: −50-50 feet (below sea level)
  • Money: -\20$ (debt/owe money)

📚 Practice Problems

1Problem 1easy

❓ Question:

Which is greater: −7-7 or −3-3?

💡 Show Solution

Solution:

On a number line, -3 is to the right of -7:

-7 -6 -5 -4−3 -2 -1 0\text{-7 -6 -5 -4} \boxed{-3} \text{ -2 -1 0}

Numbers to the right are greater.

Answer: −3-3 is greater than −7-7 (written: −3>−7-3 > -7)

2Problem 2medium

❓ Question:

Order from least to greatest: −2,5,−8,0,3,−1-2, 5, -8, 0, 3, -1

💡 Show Solution

Solution:

Think of the number line or compare:

  • Most negative (furthest left): −8-8
  • Then: −2-2
  • Then: −1-1
  • Then: 00
  • Then: 33
  • Most positive (furthest right): 55

Answer: −8,−2,−1,0,3,5-8, -2, -1, 0, 3, 5

3Problem 3hard

❓ Question:

The temperature at midnight was −5°-5°C. By noon, it had risen 12 degrees. What was the temperature at noon?

💡 Show Solution

Solution:

Start at -5 on the number line. Move right (increase) by 12:

−5+12=7-5 + 12 = 7

You can also think: 5 degrees to get to 0, then 7 more degrees.

Answer: The temperature at noon was 7°7°C

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