Problem 16 - Introductory Algebra for College Students

Question

Start by drawing a number line that shows integers from \(-5\) to \(5 .\) Then graph each real number on your number line. $$\frac{7}{3}$$

Step-by-Step Solution

Answer

The real number \(\frac{7}{3}\) is graphed on the number line between the integers 2 and 3, slightly towards '3'.

Step 1: Drawing a Number Line

Draw a straight horizontal line. On this line, mark a spot in the middle as '0', on the left of '0', mark five more spots representing '-1', '-2', '-3', '-4' and '-5', each with equal distance. Then, on the right side of '0', mark another five spots representing '1', '2', '3', '4' and '5'.

Step 2: Identify the Real Number

The real number we need to graph is \(\frac{7}{3}\) which approximately equals to \(2.33\) when rounded to two decimal places. This number lies between 2 and 3 on our number line.

Step 3: Graph the Real Number on Number Line

Draw a small vertical line or a dot on the number line slightly to the right of '2', but before reaching '3'. This represents the real number \(\frac{7}{3}\).

Key Concepts

Understanding Real Numbers

Real numbers are all the numbers you can find on the number line, including rational numbers (like fractions and integers) and irrational numbers (like the square root of 2). They form a continuous line with no gaps.
Real numbers can be infinite, including positive, negative, and zero.

• Rational numbers can be expressed as a fraction (like \( \frac{7}{3} \)).
• Irrational numbers cannot be expressed as a simple fraction.

When graphing a real number like \( \frac{7}{3} \) on a number line, it’s essential to estimate its position accurately. This helps visualize and understand where fractions lie in relation to whole numbers.
In this case, \( \frac{7}{3} \) is approximately 2.33, placing it between 2 and 3, slightly closer to 3.

Exploring Integers

Integers are whole numbers that can be positive, negative, or zero. They do not include fractions or decimals.
Here’s a quick breakdown of integers:

• Positive integers: 1, 2, 3, ...
• Negative integers: -1, -2, -3, ...
• Zero, which is neither positive nor negative

On a number line, integers are represented as points at equal intervals. In our example, we marked points between -5 and 5.
Understanding integers is crucial for building math skills, as they form the backbone for more complex concepts like real numbers and algebra.

Graphing on a Number Line

Graphing on a number line helps visualize mathematical concepts and understand relationships between numbers. When we graph numbers:

• We draw a horizontal line and mark equal intervals.
• Each point on the line represents an integer, with zero usually at the center.

To graph a real number, find its approximate location among the integers. For \( \frac{7}{3} \), place it between 2 and 3 because it’s roughly 2.33.
This method aids in visual comprehension and demonstrates how different numbers relate spatially. It's a powerful tool for enhancing numerical understanding.