The Other Side of Zero · Lesson 1 of 10
Numbers on Both Sides of Zero
“Use floors above and below ground to understand signed numbers and compare them.”
• Explain why a reference level is useful when describing positions. • Read positive and negative floor numbers. • Explain why zero is neither positive nor negative. • Compare signed numbers using their positions. • Distinguish a floor number from a movement.
When zero is a starting point
Counting tells us how many objects there are: one chair, two chairs, three chairs. Zero tells us there are none. But numbers can also describe a position. When we choose a level as zero, positions on opposite sides of that level need different descriptions.
Imagine entering a building at ground level. A shop might be three floors above the entrance, or three floors below it. The number 3 alone does not tell you which shop is meant. A sign supplies the missing direction: +3 means three floors above ground and −3 means three floors below ground. Both positions are real, even though one has a negative number.
A negative floor number does not mean that the building contains a negative number of floors. It describes where a floor is compared with a chosen reference. This distinction helps us understand why numbers below zero are useful even though we cannot have fewer than no chairs in a collection.
A chosen position called zero, from which other positions are described. In our building, the ground floor is the reference level.
Bela’s Building of Fun
Bela’s building has attractions above ground and below ground. Its Welcome Hall is the entrance, so we call it Floor 0. The lift’s + button moves one floor up each time, while its − button moves one floor down each time.
| Floor | Attraction | Position relative to the Welcome Hall |
|---|---|---|
| +6 | Space Centre | Six floors above |
| +5 | Sports Centre | Five floors above |
| +4 | Ice Cream Shop | Four floors above |
| +3 | Book Store | Three floors above |
| +2 | Art Centre | Two floors above |
| +1 | Food Court | One floor above |
| 0 | Welcome Hall | At ground level |
| −1 | Toy Store | One floor below |
| −2 | Video Games Shop | Two floors below |
| −3 | Cinema Centre | Three floors below |
| −4 | Haunted House | Four floors below |
| −5 | Dinosaur attraction | Five floors below |
From Floor 0, pressing + four times takes you to Floor +4. We can abbreviate those presses as +4. Pressing − three times takes you to Floor −3, so we write −3. Here the floor number and the movement have the same numerical value because the journey starts at zero. From another starting floor, they need not match.
Positive numbers are greater than zero. Negative numbers are less than zero and are written with a minus sign. Zero is neither positive nor negative.
Problem
A cinema is three floors below the Welcome Hall. What is its floor number?
- 1.Take the Welcome Hall as Floor 0.
- 2.Below ground is the negative direction.
- 3.Three floors below is Floor −3, not Floor +3.
Problem
From the Welcome Hall, which buttons reach the Food Court and the Video Games Shop?
- 1.The Food Court is one floor above: press + once, written +1.
- 2.The Video Games Shop is two floors below: press − twice, written −2.
- 3.Each sign tells us the direction from the entrance.
Which number is greater?
In the building, a higher floor has a greater number. This agrees with ordinary positive numbers: +4 is above +3. It also lets us compare numbers that are negative, or numbers on different sides of zero.
Floor −3 is above Floor −4. Therefore −3 > −4 and −4 < −3. The symbols > and < compare numbers: the open side faces the greater number. All negative floors lie below zero, and all positive floors lie above zero, so every positive number is greater than every negative number.
Problem
Jay is at +2, Asin at +5, Binnu at −3, and Aman at −1. Who is lowest?
- 1.Both negative floors are below both positive floors.
- 2.Between −3 and −1, −3 is lower because it is farther below ground.
- 3.Binnu is lowest. In increasing order, the floors are −3, −1, +2, +5.
Problem
Compare −10 and −12 in an imagined larger building.
- 1.Both floors are below zero.
- 2.Floor −12 is two floors below Floor −10.
- 3.Thus −12 < −10, or equivalently −10 > −12.
For negative numbers, −12 is smaller than −10 because it is farther below zero. Comparing 12 and 10 without thinking about their signs reverses the correct comparison.
Quiz
Which floor is two floors below ground?
Which statement about zero is correct?
Which comparison is correct?
Which is the lowest floor?
What does a negative floor number describe?
Which ordering goes from lowest to highest?
Practice Problems
- Describe the button presses from Floor 0 to +4 and to −3.
- Compare −2 and +5; −5 and +4; +6 and −6.
- Compare 0 and −4; 0 and +4; −25 and −7.
- Order −10, +17, 0, −12, and +9 from lowest to highest.
- Draw floors −5 through +6 and label the Welcome Hall, Toy Store, Art Centre, and Sports Centre.
- On a vertical diagram, A is −12, D is −1, and E is +1. Place B at −9, C at −6, F at +2, G at +6, and H at +11. Explain how the spacing relates to the floor numbers.
- Place −7, −4, +3, and −10 on your diagram.
- Explain why −3 can describe a real position without describing a negative count of objects.
+4 means four upward presses; −3 means three downward presses.
Key Takeaways
• A sign distinguishes opposite directions from a chosen zero. • Negative floors are below ground; positive floors are above it. • Zero is neither positive nor negative. • Higher positions represent greater numbers. • A floor number tells where you are; a movement tells how your position changes.
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Next · Lesson 2
Adding Signed Movements