Problem 14 - Introductory and Intermediate Algebra for College Students 4th

Question

Start by drawing a number line that shows integers from \(-5\) to \(5 .\) Then graph each real number on your number line. $$2 \frac{1}{4}$$

Step-by-Step Solution

Answer

The real number \(2 \frac{1}{4}\) or \(2.25\) is graphed slightly to the right of the point \(2\) on the number line.

Step 1: Draw a Number Line

Draw a number line and label it with the integers from -5 to 5. Be sure to space out the integers evenly.

Step 2: Convert Mixed Number to Decimal

Convert the mixed number \(2 \frac{1}{4}\) to a decimal. To do this, divide the numerator by the denominator and add this to the whole number. So, \(2 \frac{1}{4}\) becomes \(2.25\) when written in decimal form.

Step 3: Graph the Real Number

Now, locate the real number \(2.25\) on the number line. This will be a point slightly to the right of \(2\), and before \(3\). Mark this point on the number line.

Key Concepts

Understanding Integers

Integers are whole numbers that can be either positive, negative, or zero. They do not include fractions or decimals.

• Examples of integers: -3, 0, 4.
• Non-examples: 2.5, 1/2.

Using integers on a number line involves marking these whole number points evenly. In our exercise, we started with integers ranging from -5 to 5. It's essential to understand integers because they are the foundation for many math concepts. They're easy to spot on a number line because they aren't split into smaller parts like fractions or decimals.

Decoding Mixed Numbers

A mixed number is made up of two parts: a whole number and a fraction. It represents a value somewhere between two whole numbers.For instance, let's look at the number 2 \(\frac{1}{4}\). Here, '2' is the whole number, and '1/4' is the fractional part.

• To read a mixed number, first identify the whole number.
• Then, consider the fraction as a portion of the next whole number.

In our task, converting this mixed number to a decimal (like 2.25) helps in graphing since decimals can be placed more precisely on a number line.

Demystifying Decimals

Decimals are numbers with a fractional part, separated by a decimal point. They offer a way to express values more precisely than whole numbers.Upon converting the mixed number 2 \(\frac{1}{4}\) to a decimal, we received 2.25. This conversion allowed us to pin down the number more accurately on our number line.

• Decimals are vital in pinpointing exact positions on a number line.
• They facilitate estimations between integers.

Understanding decimals enriches your ability to handle a broader spectrum of numbers extending beyond simple whole numbers.

Graphing Real Numbers

Real numbers encompass all types of numbers, including integers, whole numbers, decimals, and fractions. When graphing real numbers, you're essentially locating their position on a number line. Begin by drawing a line with evenly spaced integers, such as from -5 to 5. To graph a number like 2.25:

• First, identify the closest whole numbers (between 2 and 3 in this case).
• Next, mark a point slightly past the 2, as 2.25 is closer to 2 than 3.

Graphing on a number line helps visually see where numbers fall in relation to each other, making abstract numbers more concrete.