Problem 29 - Algebra 1

Question

Write the numbers in increasing order. $$4.8,-2.6,0,-3, \frac{1}{2},-\frac{1}{2}$$

Step-by-Step Solution

Answer

The numbers in ascending order are: -3, -2.6, -\( \frac{1}{2} \), 0, \( \frac{1}{2} \), 4.8

Step 1 - Understand Number Types

We can see that this exercise involves a mix of different types of numbers including negative integers (-3), negative decimals (-2.6), positive integers (0), positive decimals (4.8), and fractions (\( \frac{1}{2}, -\frac{1}{2} \)) . The first step in arranging numbers is to correctly interpret each type.

Step 2 - Arranging Negative Numbers with Zero and Fractions

Remember that any negative number will be less than zero and fractions. So, let's arrange them first: -3, -2.6, -\(\frac{1}{2}\), 0, \( \frac{1}{2} \). Here, -3 is the smallest of all the numbers given. Then comes -2.6 which is a decimal greater than -3 but less than 0. -\(\frac{1}{2}\) is greater than -2.6 but less than 0. Lastly, 0 and \( \frac{1}{2} \) as it's positive and the smallest among the positive numbers.

Step 3 - Arranging Rest of the Numbers

The remaining number yet to be arranged is the positive decimal 4.8. Since this number is greater than zero and \( \frac{1}{2}\), it will be the last number in the sequence.

Key Concepts

Negative Numbers

A negative number is any number that is less than zero. It is represented with a minus sign ( - ) before the number. Negative numbers appear on the left side of zero on a number line. Here are some key points about negative numbers:

• They represent values below zero and are often used to express debt or loss.
• In a sequence of numbers, negative numbers will always come before zero or any positive numbers.
• The smaller the absolute value of the negative number, the closer it is to zero. For example, -3 is smaller than -2.6.

This concept is crucial when ordering numbers, as understanding how negative values relate helps in correctly placing them in order.

Integers

Integers are whole numbers that include the set of all positive numbers, negative numbers, and zero. They do not include fractions or decimals. Here are some essential features of integers:

• They are represented as {..., -3, -2, -1, 0, 1, 2, 3, ...}.
• The integers are considered a complete number system and can be used in basic arithmetic operations like addition, subtraction, multiplication, and division.
• When ordering integers, negative integers will be placed before zero, and positive integers will follow zero.

Understanding integers is essential because they form the basic building blocks for more complex numbers like fractions and decimals.

Decimals

Decimals are numbers that have a fractional component expressed in the decimal system. They are often used to represent fractions with denominators that are powers of ten. Here are some details about decimals:

• Examples include numbers like 4.8 or -2.6, where the number to the right of the decimal point indicates the fractional part.
• Decimals enable more precise representation of quantities, especially in measurement and currency.
• When comparing decimals, start from the left-most digit and compare each successive digit until a difference is found.

Decimals are particularly useful in ordering numbers of mixed types, enabling comparison with both fractions and whole numbers.

Fractions

Fractions consist of two numbers: a numerator and a denominator, represented as \(\frac{a}{b}\). The numerator represents a part of the whole, while the denominator indicates the total number of equal parts. Important aspects of fractions include:

• Fractions can be positive or negative, based on the signs of the numerator and denominator.
• When comparing fractions, they must be converted to a common denominator to see which is greater.
• Fractions are key in expressing numbers that lie between integers on the number line.

In ordering exercises that involve fractions, understanding how they measure up against whole numbers, decimals, and each other is critical for accurate arrangement.