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Chapter 9: Problem 33

Order each set of numbers from greatest to least. $$-\sqrt{14},-4 \frac{1}{10},-\frac{17}{4},-3.8$$

Short Answer

Expert verified
Order from greatest to least: \(-\sqrt{14}, -3.8, -4\frac{1}{10}, -\frac{17}{4}\).

Step by step solution

01

Convert all numbers to decimals

Convert each number into a decimal to easily compare them.\(-\sqrt{14}\approx -3.74\), \(-4\frac{1}{10} = -4.1\), \(-\frac{17}{4}= -4.25\), \(-3.8\).
02

Compare the decimal values

Now, compare the decimal values. Remember that for negative numbers, a number closer to zero is greater than a number farther from zero. The decimals to compare are: \(-3.74\), \(-3.8\), \(-4.1\), \(-4.25\).
03

Arrange numbers from greatest to least

From the comparison, order the numbers as follows: \(-3.74\), \(-3.8\), \(-4.1\), \(-4.25\).
04

Convert decimals back to original forms

Finally, convert each decimal back to its original form: \(-\sqrt{14}\), \(-3.8\), \(-4\frac{1}{10}\), \(-\frac{17}{4}\).

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Key Concepts

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

Decimals
Decimals are numbers that have a whole number part and a fractional part, separated by a decimal point. They provide an easy way to express fractions or parts of a whole. For example, the number 3.8 has 3 as the whole number part and 8 as the fractional part.
  • To convert fractions to decimals, divide the numerator (top number) by the denominator (bottom number). For example, \( \frac{1}{2} \) becomes 0.5 when 1 is divided by 2.
  • Decimals are especially useful in many real-world contexts, such as money and measurements, where precise values are needed.
When ordering decimals, remember that the smaller the decimal number, the closer it is to zero. This is particularly important when you're dealing with negative decimals, where values closer to zero are greater than those further away.
Square Roots
The square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted by the radical symbol, \( \sqrt{} \). For example, \( \sqrt{9} = 3 \) because 3 * 3 = 9.
  • Square roots are significant in many areas of mathematics, including geometry and algebra, as they regularly appear in formulas and theorems.
  • To approximate square roots that are not perfect squares, you can use a calculator to find a decimal value.
  • Understanding how to approximate square roots can also help in ordering numbers; for instance, \( \sqrt{14} \) is about 3.74, which can be crucial for comparing against other numbers.
When working with square roots in ordering numbers, knowing their approximate decimal values helps to compare their magnitude effectively.
Fractions
Fractions consist of two parts: a numerator and a denominator. The numerator represents the parts of the whole, and the denominator indicates how many parts make up a whole. For example, in \( \frac{3}{4} \), 3 is the numerator and 4 is the denominator.
  • Fractions can be easily converted to decimals by dividing the numerator by the denominator, which simplifies comparison between numbers.
  • They often appear in problems that deal with parts of a whole, such as pieces of cake or segments of time.
  • When comparing fractions, converting them to the same denominator or to decimals can help establish their order.
Fractions are often more intuitive for visualizing parts of a whole, but decimals make it simpler to compare values, which is why converting tends to be helpful when ordering numbers.
Negative Numbers
Negative numbers, indicated by a minus sign (-), are less than zero and represent values below zero. They are often used in contexts such as temperatures below freezing, debts, or below-sea-level elevations.
  • Ordering negative numbers can be tricky because they behave opposite to positive numbers in that the closer a negative number is to zero, the greater its value.
  • For instance, -3 is greater than -5, even though 3 is less than 5.
  • In many real-life situations, understanding negative values is crucial, such as determining depths or losses.
When ordering negative numbers, remember that visualizing them on a number line can be very helpful in seeing which numbers are positioned closer to zero.

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Most popular questions from this chapter

Give a counterexample for each statement. What is the value of \(x\) to the nearest tenth if \(x^{2}-4^{2}=\sqrt{15^{2}} ?\)

CHALLENGE Find the values of \(x\) if the distance between \((1,2)\) and \((x, 7)\) is 13 units.

Solve each equation. Round to the nearest tenth, if necessary. $$y^{2}=25$$

Order each set of numbers from least to greatest. $$3 . \overline{7}, 3 \frac{3}{5}, \sqrt{13}, \frac{10}{3}$$

Name all of the sets of numbers to which each real number belongs. Let \(\mathbf{N}=\) natural numbers, \(\mathbf{W}=\) whole numbers, \(\mathbf{Z}=\) integers, \(\mathbf{Q}=\) rational numbers, and I = irrational numbers. $$\sqrt{12}$$
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