Today's soundtrack is Starflyer 59: Leave Here a Stranger. I still haven't found what I'm looking for.
This evening, I'm working on the first part of the second chapter of my math course, "Absolute Value of a Real Number."
A number's absolute value is its distance from 0, regardless of whether it is positive or negative. So -6 has an absolute value of 6, as does +6. A number's absolute value is also its magnitude, which is also "its distance from 0" (Pearson, Pre-Calculus 11, p. 85). With both absolute values and orders of magnitude, we deal with real numbers, which are whole positive numbers not including zero.
When we look at square roots, we always deal with the principal square root, which is an understanding that both 5x5 and -5 x -5 give us +25. So if we are told to find the square root of (7-9)^2, we know that even though 7-9 = -2, we treat our answer as +2, since that is the principal square root.
The absolute value of a negative number is the positive version of that number.
The absolute value of any positive number (or 0) is itself.
If we are told to determine the absolute value of an equation, we only find the absolute value at the end. While we are solving the equation itself, we still deal with regular numbers (positive and negative) as normal.
If we are asked to use absolute value to determine the distance between a pair of numbers on a number line when one of the numbers is negative, we can determine the difference between the first number and 0, then find the difference between the second number and 0. Then we add those numbers together to find the distance. For example, if given -3 and 4, we say that -3 is 3 away from 0, and 4 is 4 away from 0. 3+4=7; therefore, -3 is 7 digits away from 4.
There are two ways that we can write expressions showing the distance between two numbers on a number line: we can either subtract the second number from the first, or the first number from the second.
If we are told to order numbers from least to greatest by their absolute values, we order them by their absolute values, but they retain their negative symbols if they had them to begin with.