Negative numbers and the number line Knowledge-Based Study

Study notes

1. What Are Negative Numbers?

   In everyday life, quantities often have natural reference points. When a value falls below this reference point, we use negative numbers to represent it.

   • Temperature: A temperature of 5 degrees below zero is written as -5°C.
   • Elevation: An elevation of 120 meters below sea level is represented as -120 meters.
   • Financial transactions: A debt or withdrawal of 30 dollars is recorded as -30 dollars.

   A negative number is any number that is strictly less than zero. It is written with a minus sign (-) directly in front of the digit.

   The Structure of the Number Line

   A horizontal number line provides a visual model of real numbers:

   • Zero sits at the center as the origin or reference point. Zero is neither positive nor negative.
   • Positive numbers lie to the right of zero and increase in value as you move further right.
   • Negative numbers lie to the left of zero and decrease in value as you move further left.

   Each unit step to the right adds 1, while each unit step to the left subtracts 1. For example, moving 3 units to the left of 0 brings you to -3.

   Opposite Numbers

   Two numbers are called opposites if they are located at the exact same distance from zero on a number line, but in opposite directions.

   • The opposite of 4 is -4, because both numbers are exactly 4 units away from 0.
   • The opposite of -7 is 7.
   • The opposite of 0 is simply 0.

   In general, for any number $a$, its opposite is $-a$. Notice that the negative sign can be read as "the opposite of". Therefore, $-(-5)$ represents the opposite of -5, which equals 5.

   Using Position to Compare Values

   On a standard horizontal number line, value is determined strictly by relative position:

   Any number located to the right of another number is greater.
   Any number located to the left of another number is smaller.

   Key Observations:

   • Every positive number is greater than every negative number (for example, $2 > -10$).
   • Zero is greater than every negative number (for example, $0 > -4$).
   • When comparing two negative numbers, the number closer to zero is greater because it lies further to the right. For example, -2 lies to the right of -5, which means $-2 > -5$ (or $-5 < -2$).

   Putting It Together

   Consider the set of integers: $\{-6, 3, -1, 0, -4, 5\}$.

   To arrange these numbers from least to greatest, locate each one on the number line from left to right:

   1. The number furthest to the left is -6.
   2. Next to the right is -4.
   3. Then comes -1.
   4. The origin is 0.
   5. Moving into positive territory, the next is 3.
   6. The number furthest to the right is 5.

   Written in ascending order: $-6 < -4 < -1 < 0 < 3 < 5$.

   Rule of thumb: A larger magnitude with a negative sign indicates a position further to the left, which corresponds to a smaller value.