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Chapter 2: Problem 57

Place the correct symbol, \(<\) or \(>,\) between the two numbers. $$0.07046 \quad 0.07036$$

Short Answer

Expert verified
0.07046 > 0.07036

Step by step solution

01

Compare the fourth decimals

The first step is to compare the fourth decimals of both numbers. For 0.07046, the fourth decimal is 4, whereas for 0.07036, the fourth decimal is 3. Since 4 is greater than 3, we can conclude that 0.07046 is greater than 0.07036.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Place Value
Place value is the value of a digit in a number based on its position. Each digit in a decimal number has a place value that starts from the left of the decimal point, moving rightward. In whole numbers, digits represent units, tens, hundreds, and so on.

In decimal numbers, this pattern continues after the decimal point but in reverse: tenths, hundredths, thousandths, ten-thousandths, etc. Understanding this sequence is crucial for comparing decimals.

For the number 0.07046:
  • 0 is in the tenths place.
  • 7 is in the hundredths place.
  • 0 is in the thousandths place.
  • 4 is in the ten-thousandths place.
  • 6 is in the hundred-thousandths place.
Knowing how each digit's place influences its value helps in determining which number is larger or smaller.
Number Comparison
When comparing two decimal numbers, the process involves evaluating digits place by place starting from the leftmost, which is integral for accuracy. Begin with the tenths place and move towards the right until a difference is found.

If digits are identical at each position, move to the next place value. The first instance of a digit difference indicates which number is greater.

Let's examine 0.07046 and 0.07036:
  • Tenths place: Both have 0.
  • Hundredths place: Both have 7.
  • Thousandths place: Both have 0.
  • Ten-thousandths place: Here is the first difference where we have 4 in 0.07046 and 3 in 0.07036.
Since 4 is greater than 3, 0.07046 is greater than 0.07036.
Fourth Decimal
The fourth decimal place, also known as the ten-thousandths place, is crucial in understanding and comparing decimal numbers with precision. It denotes how many ten-thousandths are present and significantly influences the number's value.

For example, in the decimal 0.07046, the 4 represents four ten-thousandths. Comparing the fourth decimal is often the point where distinctions between closely similar decimal numbers emerge.

In our case study, we noticed the fourth decimal places were key: 0.07046 has a 4, while 0.07036 has a 3. This tiny difference of one ten-thousandth can determine which number is larger when previous places are identical.
Greater Than Symbol
The greater than symbol, represented as \(>\), is a mathematical tool used to show that one number is larger than another. It's part of inequality symbols crucial in arithmetic and algebra for expressing the size of different quantities.

To use \(>\) properly, place it between two numbers to show that the one to the left has more value than the one to the right. Using our previous example, with 0.07046 and 0.07036, when compared, the fourth decimal dictates the outcome.
  • Since 4 (fourth decimal in 0.07046) is greater than 3 (fourth decimal in 0.07036), write the comparison as:

    0.07046 \(>\) 0.07036
Mastery of using the greater than symbol enables clear communication of which amounts hold more value in a comparison.

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Most popular questions from this chapter

For Exercises, give an example of a problem that meets the described condition. The fractions in your examples must be proper fractions with different denominators. If it is not possible to write a problem that meets the given condition, write "not possible." A proper fraction is subtracted from an improper fraction and the result is an improper fraction.

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