Adding and Subtracting Rational Numbers

Rational numbers include integers, fractions, and decimals that can be positive or negative. When we add or subtract them, we are simply moving along a number line. Adding a positive number moves us to the right, while adding a negative number (which is the same as subtracting) moves us to the left. A key tool here is absolute value, written as $|x|$, which represents the distance a number is from zero. Because distance can't be negative, $|-5|$ and $|5|$ both equal $5$. When adding numbers with different signs, we actually find the difference between their absolute values and keep the sign of the 'larger' number.

Subtraction can often feel confusing when negative numbers are involved. The easiest way to handle it is to remember that subtracting a number is the same as adding its opposite. This is often called the 'Keep-Change-Change' rule. For any rational numbers $a$ and $b$, the formula is $a - b = a + (-b)$. If you encounter a double negative, such as $5 - (-3)$, it transforms into $5 + 3$. This happens because taking away a debt is the same as giving someone credit!

In the real world, we often need to find the distance between two values, like the change in temperature from morning to night. Distance is always a positive value. To find the distance between any two rational numbers $p$ and $q$ on a number line, we use the formula $|p - q|$. It doesn't matter which number you start with; the absolute value will ensure the distance is positive. For example, the distance between $-10$ and $5$ is $|-10 - 5| = |-15| = 15$ units.