Problem 11 - Algebra 1

Question

Graph the numbers on a number line. Then write the numbers in increasing order. $$-0.1,-1.1,-1$$

Step-by-Step Solution

Answer

The numbers plotted on a number line in increasing order are -1.1, -1, and -0.1.

Step 1: Understand the Number Line

A number line is a vertical or horizontal line on which every point corresponds to a real number. Usually, it is horizontal, with 0 at the center. Numbers to the right of zero are positive and those to the left are negative. The farther right a number is, the larger it is; the farther left, the smaller.

Step 2: Graph Numbers on the Number Line

Now, let's graph the given numbers on the number line. Start from zero and move to the left, since all given numbers are negative. For -1, move one unit to the left of zero. For -1.1, since it is more negative than -1, move left to a point little further than -1. Finally, for -0.1, since it is less negative compared to both -1 and -1.1, it should be placed slightly to the left of 0, but clearly to the right of both -1 and -1.1.

Step 3: Write Numbers in Increasing Order

After graphing the numbers on the number line, it can be observed that -1.1 is the farthest to the left, thus it is the smallest number. Then comes -1 and finally -0.1. Therefore, the numbers in increasing order are -1.1, -1, -0.1.

Key Concepts

Ordering Numbers

Ordering numbers is a fundamental concept in mathematics, where numbers are arranged from the smallest to the largest. This process is facilitated by understanding and analyzing their positions on a number line. A number to the left is considered smaller than a number to the right. In the context of the exercise, we are dealing with three negative numbers: -1.1, -1, and -0.1.

• -1.1: This number is more negative and thus further to the left than -1 and -0.1.
• -1: Lies between -1.1 and -0.1 on the number line, making it larger than -1.1 but smaller than -0.1.
• -0.1: Being the least negative of the three, it is closest to 0, making it the largest number among them.

By applying this understanding, we arrange the numbers in increasing order as: -1.1, -1, -0.1.

Graphing Numbers

Graphing numbers on a number line provides a visual way to compare and contrast the magnitude of different numbers. The number line is an excellent tool for visualizing the sequence and order of numbers. To graph these numbers, start from zero:

• Move one unit to the left to place -1.
• For -1.1, shift slightly further left from -1, as it is more negative.
• -0.1 is very close to zero; place it just left of 0 but clearly to the right of both -1 and -1.1.

This visual representation helps in easily seeing which numbers are larger or smaller, aiding in better understanding of their order.

Real Numbers

Real numbers include all the numbers you see on the number line. This category encompasses:

• Integers: Positive and negative whole numbers and zero.
• Rational numbers: Numbers that can be expressed as a fraction of two integers, like -0.1 (which is -1/10).
• Irrational numbers: Numbers that cannot be expressed as a simple fraction, like \( \sqrt{2} \).

In this exercise, you are dealing with rational numbers since -1, -1.1, and -0.1 can all be written as fractions of integers. Graphing real numbers on a line helps students understand their properties, relationships, and positions relative to 0 and other numbers. By placing these numbers on the line, students can better memorize and visualize their order and realize their nature as part of the expansive set of real numbers.