Problem 32 - Elementary Algebra

Question

For the following problems, locate the numbers on a number line by placing a point at their (approximate) position. $$ 3.6 $$

Step-by-Step Solution

Answer

Answer: To locate 3.6 on a number line, divide the space between the consecutive integers into ten equal parts, representing tenths. Count 6 equal parts from the tick mark at 3. The point marked represents the location of 3.6 on the number line.

Step 1: Draw a number line

Draw a horizontal line and mark evenly spaced points or ticks on it. Each point represents an integer value. Place the values in ascending order starting from a value less than the given number (3.6) to a value greater than the given number. In this case, we can start from 0 and continue up to 5.

Step 2: Mark the integer values on the number line

Mark the integer values of the numbers on the number line below the tick marks. Write '0' below the first tick, '1' below the second tick, '2' below the third tick, and so on.

Step 3: Divide the space between the integers into ten equal parts

Since 3.6 has a single digit after the decimal point, divide the space between each pair of consecutive integer values on the number line (e.g., space between 3 and 4) into ten equal parts. These will be your tenths (0.1, 0.2, 0.3, etc.).

Step 4: Identify the position of 3.6

Now, you need to locate the position of 3.6 on the number line. Since it is between 3 and 4, focus on that segment. We know that 3.6 is 6 tenths greater than 3, hence count 6 equal parts from the tick at 3 and mark that point.

Step 5: Finalize the location of 3.6 on the number line

The point you've marked in the previous step represents the location of 3.6 on the number line. Clearly label the point as "3.6" to complete the task.

Key Concepts

Decimal Numbers

Decimal numbers are a way of expressing fractions or parts of whole numbers in our number system, which is the base-10 system. They are written with a decimal point dividing the whole number part from the fractional part. For example, in the decimal number 3.6, the '3' represents the whole part, while the '.6' represents the fractional part. This is simpler than using a fraction, like 6/10.

• Decimal point: A dot that separates the whole number part from the fractional part
• Digits after the decimal: Represent smaller parts of the whole number

Decimals are used in many areas, especially in measurement and finance, because they provide a clear way to illustrate how much of a whole something is. Whenever you deal with currency, weights, or distances, you'll likely see decimal numbers.

Integers

Integers are simply the whole numbers and their opposites, including zero. This means they are numbers without any fractional or decimal parts. Examples are -3, 0, 4, and 57. They can be positive, negative, or zero:

• Positive integers: Numbers greater than zero (e.g., 1, 2, 100)
• Negative integers: Numbers less than zero (e.g., -1, -20, -300)
• Zero: Neither positive nor negative

Integers are used every day to count items, measure distances, and even in technology to create codes. They are straightforward because there are no parts of wholes to consider—just whole numbers themselves.
When placing integers on a number line, you can easily see their values in order from left to right, providing a clear visual representation of their size and relationship to one another.

Tenths

Tenths are a specific type of decimal representing the ten equal parts between two consecutive whole numbers. If you think of a number between two integers, such as between 3 and 4, dividing this gap into 10 equal parts gives 'tenths'. Each tenth is equal to 0.1. In the decimal 3.6:

• The '6' is in the tenths place, meaning six-tenths, or 6/10

This is how decimals show more details beyond mere whole numbers. With decimals, you can specify numbers that aren't whole numbers in a very precise way.
On a number line, tenths are marked by dividing the space between two whole numbers into ten sections. This makes it easier to find more accurate positions for numbers like 3.6, since 3.6 is exactly six-tenths past the number 3.

Locating Numbers on a Number Line

Placing numbers on a number line is a useful method to visualize their value and their relationship to other numbers. Here's how you can do it:

• Start by drawing a horizontal line
• Mark evenly spaced points for integers first, like 0, 1, 2, and so on
• To locate a decimal number like 3.6, focus on the interval between two integers, here between 3 and 4
• Divide this interval into ten equal parts. Each part represents a tenth.

To locate 3.6, count 6 tenths past 3. This method provides a clear visual representation of where the decimal fits relative to whole numbers.
Number lines help us see how close a decimal is to integer values, gaining insight into its size and the size of gaps between numbers.