Problem 39 - Prealgebra

Question

For each pair of numbers, place the correct symbol, \(<\) or \(>,\) between the numbers. Write the following numbers in order from smallest to largest. \(0.02 \quad 0.05 \quad 0.025 \quad 0.052 \quad 0.005 \quad 0.002\)

Step-by-Step Solution

Answer

0.002, 0.005, 0.02, 0.025, 0.05, 0.052

Step 1: Compare Two Numbers at a Time

Start by comparing the given numbers two at a time. First, consider the smallest place value: the thousandths. Compare the numbers \(0.02\), \(0.05\), \(0.025\), \(0.052\), \(0.005\), and \(0.002\). Begin by comparing \(0.002\) and \(0.005\). Since \(2 < 5\) when comparing thousandths, place the symbol: \(0.002 < 0.005\).

Step 2: Continue Pair Comparisons

Next, compare \(0.005\) and \(0.02\). Again, when comparing thousandths or hundredths, \(0.005 < 0.02\). Now compare \(0.02\) and \(0.025\). Since \(20 < 25\) when comparing hundredths/thousandths, \(0.02 < 0.025\).

Step 3: Complete Pair Comparisons

Now compare \(0.025\) and \(0.05\). Since \(25 < 50\), \(0.025 < 0.05\). Finally, compare \(0.05\) and \(0.052\). Since \(50 < 52\), we conclude \(0.05 < 0.052\).

Step 4: Order All Numbers

Using comparisons from the previous steps, arrange all numbers in order from smallest to largest. The order is: \(0.002 < 0.005 < 0.02 < 0.025 < 0.05 < 0.052\).

Key Concepts

ordering numbers

To order numbers means to arrange them from smallest to largest or vice versa. This is an important skill because it helps us understand the relative size of numbers, which is vital in many real-world scenarios. For example, when dealing with money, weight, or distances, knowing how to order numbers allows us to prioritize, compare, and make better decisions.
When ordering numbers, especially decimals, start by listing all numbers involved. Look at each digit from left to right. Start from the largest place value (which is typically the leftmost digit). For decimals, this process involves comparing tenths, hundredths, and so on. If we face similar digits, move to the next place value—continuing until you determine which number is greater or lesser.
Ordering numbers using this approach not only aids in answering mathematical questions but also builds a solid foundation for more advanced math topics.

place value

Place value is a system that gives digits in a number their respective values based on their position. This is critical when working with both whole numbers and decimals. Understanding place value will make it easier to compare, add, subtract, and handle numbers effectively.
For decimals, each place to the right of the decimal point represents tenths, hundredths, thousandths, and so on. For instance, in the number 0.052, 5 is in the hundredths place, and 2 is in the thousandths place. This is what gives 0.052 its value.
Recognizing the place value of each digit is instrumental when ordering or comparing numbers. During comparison, we start with the highest place value, and then move sequentially to the right, spotting any differences that tell us which number is larger or smaller. Just by knowing place value, you can decode a whole world of information from a seemingly simple set of numbers.

inequality symbols

Inequality symbols, such as \(<\) and \(>\), are used to show the relationship of size between two numbers. These symbols are essential in mathematics to quickly convey which number is larger or smaller, without writing out full descriptions.
Here’s how they work: The symbol \(<\) means "less than," indicating that the number on the left is smaller than the number on the right. Conversely, \(>\) means "greater than," signifying the number on the left is larger. These symbols can be understood as arrows that point to the smaller number. Thus, the open side of the symbol always faces the larger number, like "0.005 < 0.02" where 0.005 is clearly smaller.
Understanding and using inequality symbols streamlines communication and problem-solving. They offer a quick and efficient way to record and interpret relationships between numbers, which is foundational not just in basic math, but also in algebra and higher-level math challenges.