Problem 44 - Prealgebra and Introductory Algebra

Question

Write the given numbers in order from smallest to largest. $$11,-8,-1,7,-6$$

Step-by-Step Solution

Answer

The numbers in order from the smallest to the largest are -8, -6, -1, 7, 11.

Step 1: Identify Negative Numbers

First, identify the negative numbers from the list. In this case, the negative numbers are -8, -1, and -6.

Step 2: Order Negative Numbers

Next, order the negative numbers from smallest to largest. Remember, the number with the largest absolute value is the smallest. So, the order is -8, -6, -1.

Step 3: Identify and Order Positive Numbers

In this list, the positive numbers are 7 and 11. They are already in ascending order.

Step 4: Combine Negative and Positive Numbers

To get the final sequence from smallest to largest, combine the ordered negative and positive numbers. So, the final order is -8, -6, -1, 7, 11.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero. They are typically represented with a minus sign (\(-\)) in front of a numeral. Imagine a thermometer: when the temperature is below freezing, it dips into the negative range.
Negative numbers are essential for representing values that lie below a neutral point.Here are a few points to consider:

• Negative numbers can represent things like debt, temperatures below zero, or elevations below sea level.
• On a number line, negative numbers are to the left of zero.
• The further left a negative number is, the smaller its value. For example, \(-8\) is less than \(-1\).

Understanding how negative numbers work is crucial when ordering numbers.Remember: the goal is to find the smallest value, which will be the most negative in the list.

Explaining Positive Numbers

Positive numbers are numbers greater than zero. You don't see a plus sign in front of them because they are the default state for counting and measuring. People use positive numbers to count quantities, describe heights above sea level, or measure temperatures above zero.Here's what to keep in mind:

• All positive numbers are to the right of zero on a number line.
• The larger the positive number, the further to the right it is, meaning it has a larger value.
• In any given context, positive numbers indicate an amount greater than zero.

When ordering a list of numbers, positive numbers usually come after the negative ones because they are always larger. In our example, \(7\) and \(11\) are the positive numbers, and they are automatically ordered smallest to largest when listed.

Demystifying Absolute Value

Absolute value measures how far a number is from zero, regardless of its direction. This means both positive and negative numbers have the same absolute value. For example, both \(7\) and \(-7\) have an absolute value of \(7\).Important features of absolute value:

• It is always a non-negative number.
• The absolute value of zero is, of course, zero.
• On a number line, absolute value represents the distance of a number from zero.

Understanding absolute value helps when ordering negative numbers. Since absolute value ignores the sign, you can compare the sizes of negative numbers by looking at their absolute values. Remember, when ordering negative numbers, the one with the largest absolute value is actually the smallest in the list!