Introduction to Integers and Number Line

TL;DR

Integers are all the whole numbers, including positive numbers, negative numbers, and zero. You can visualise integers and their order using a number line, with zero in the middle. The further right an integer is on the number line, the larger its value.

1. The Mental Model

Think of integers as steps on an infinitely long straight road. You can step forwards (positive numbers), backwards (negative numbers), or stand still (zero).

2. The Core Material

Integers are whole numbers (no fractions or decimals) that can be positive, negative, or zero.

• Positive integers are numbers greater than zero, like 1, 2, 3, and so on. We usually don't write a plus sign (+) in front of them unless we're comparing them to negative numbers.
• Negative integers are numbers less than zero, like -1, -2, -3, and so on. We always write a minus sign (-) in front of them.
• Zero (0) is neither positive nor negative. It's the middle point.

The Number Line

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A number line is a straight line that shows numbers in order. It's super helpful for understanding integers.

     <----------------------|-----------------------|----------------------->
... -4   -3   -2   -1   0   1   2   3   4 ...

Here's how it works:

• Zero (0) is usually in the middle.
• Positive numbers are to the right of zero. As you move right, the numbers get bigger.
• Negative numbers are to the left of zero. As you move left, the numbers get smaller (even though the number itself might look bigger, like -5 is smaller than -2).

Comparing Integers

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You can easily compare integers using a number line. The integer further to the right is always greater. The integer further to the left is always smaller.

Let's look at some comparisons:
- Is 3 greater or smaller than -2? On the number line, 3 is to the right of -2, so 3 > -2.
- Is -4 greater or smaller than -1? On the number line, -4 is to the left of -1, so -4 < -1.
- How about -5 and 0? 0 is to the right of -5, so 0 > -5.

It's all about position on that line!

graph TD
    start["You want to compare two integers?"] --> Visualize["Visualise them on a number line"]
    Visualize --> RuleCheck{"Which integer is further to the right?"}
    RuleCheck -- "The one on the right" --> Greater["That integer is GREATER"]
    RuleCheck -- "The one on the left" --> Smaller["That integer is SMALLER"]
    Greater --> Done
    Smaller --> Done
    Done["Comparison complete!"]

3. Worked Example

Let's order the following integers from smallest to greatest: 5, -3, 0, 2, -1

1. Draw a number line (you can just sketch a simple one).
2. Mark zero (0).

3. Place the other integers on the line relative to zero and each other.

   
   

        <------|------|----------------|------|------|------>
   ... -3   -2   -1   0   1   2   3   4   5 ...
   

   
   

   • 5 is to the right of 0.
   • -3 is to the left of 0.
   • 2 is to the right of 0, but to the left of 5.
   • -1 is to the left of 0, but to the right of -3.

4. Read the numbers from left to right:

   The order from smallest to greatest is: -3, -1, 0, 2, 5.

4. Key Takeaways

• Integers include all whole numbers: positive, negative, and zero.
• A number line helps you visualise integers and their relative values.
• Numbers increase in value as you move to the right on a number line.
• Numbers decrease in value as you move to the left on a number line.
• Negative numbers are always smaller than positive numbers and zero.
• Zero is neither positive nor negative.

Common mistakes to avoid:
- Thinking that larger-looking negative numbers are greater (e.g., -5 is less than -2).
- Forgetting that zero is an integer.
- Mixing up integers with fractions or decimals (integers are always whole numbers).
- Not using the minus sign (-) for negative numbers.

5. Now Try It

Draw a number line from -5 to 5. Then, plot the following numbers and list them in order from smallest to largest: 4, -2, 1, -5, 0, 3.

Success looks like a correctly drawn number line with all points marked, and a list showing the correct order: -5, -2, 0, 1, 3, 4.