Problem 37 - Algebra 1: Concepts and Skills

Question

Compare using \(<,>\) or \(=\) $$ 0.01 ? 0.0001 $$

Step-by-Step Solution

Answer

0.01 > 0.0001

Step 1: Understanding the Numbers

Let's look at the numbers. We have 0.01 and 0.0001. Here we can see that 0.01 has two digits after the decimal point whereas 0.0001 has four digits after the decimal point. The more the digits after the decimal point, the smaller the value. Therefore, we can conclude that 0.01 is larger than 0.0001.

Step 2: Using the Proper Symbol

As per the understanding gained in Step 1, 0.01 is larger than 0.0001. Hence, we use the greater than (>) symbol. So, 0.01 > 0.0001.

Key Concepts

Decimal Numbers

Decimal numbers are an important part of mathematics that enable us to express fractions accurately. They are written with a decimal point, which separates the whole number part from the fractional part. Each digit after the decimal point represents a fraction of a power of ten. Understanding decimal numbers involves knowing:

• The decimal point: This is used to divide the whole numbers from their fractional counterparts.
• Digits after the decimal point: These represent tenths, hundredths, thousandths, and further parts of ten, depending on how far they are from the decimal point.

For example, the number 0.01 means 1 in the hundredths place, while 0.0001 means 1 in the ten-thousandths place. This is why even with more zeros, 0.0001 is actually smaller than 0.01. Take time to understand this concept by remembering that more digits after the decimal typically imply a smaller number.

Numerical Order

Numerical order involves arranging numbers from the smallest to the largest, which is essential when comparing numbers like decimals. Ordering decimal numbers means observing the digits to the right of each decimal point. Start with the largest place value and move sequentially rightwards until a difference is observed. Here are the steps to determine numerical order:

• Look at the whole number (to the left of the decimal). Larger whole numbers mean larger numbers.
• If the whole numbers are the same, compare the tenths digit, then hundredths, and so forth.
• Zeroes can indicate that the numbers aren't whole; however, their placement is crucial for comparison.

A useful tip is to align the numbers vertically by the decimal for clarity, especially when comparing numbers like 0.01 and 0.0001. When aligned, it becomes evident that 0.01 is greater as the significant digits are compared.

Place Value

Place value is a fundamental concept when it comes to understanding and working with decimal numbers. It tells us the value of where a digit is located in a number. With each move to the right of the decimal, the place value becomes ten times smaller. Here’s how to interpret place value for decimals:

• The first place after the decimal point is the tenths place.
• The second is the hundredths place.
• It continues with thousandths, ten-thousandths, etc.

The importance of place value can be demonstrated through decimals like 0.01 and 0.0001. In 0.01, the '1' is in the hundredths place. In 0.0001, the '1' is in the ten-thousandths place. This indicates that 0.01 is larger because a hundredth is greater than a ten-thousandth. Understanding place value helps in easily deciding numerical order.