Question

Write the numbers in increasing order. $$3 \frac{1}{2}, 3.4,4.1,-5,-5.1,-4 \frac{1}{2}$$

Short answer

The numbers in increasing order are \(-5.1\), \(-5\), \(-4 \frac{1}{2}\), \(3.4\), \(3 \frac{1}{2}\), \(4.1\).

Step-by-step solution

Organize the positive integers

Identify and separate the positive integers and fractions which are: \(3 \frac{1}{2}\), \(3.4\), and \(4.1\). In increasing order, these numbers become: \(3.4\), \(3 \frac{1}{2}\), and \(4.1\). Here, \(3 \frac{1}{2}\) is equivalent to 3.5.

Organize the negative integers and fractions

Next, identify and separate the negative fractions and integers, which are: \(-5\), \(-5.1\), and \(-4 \frac{1}{2}\). In ascending order these numbers become: \(-5.1\), \(-5\), and \(-4 \frac{1}{2}\). Here, \(-4 \frac{1}{2}\) is equivalent to -4.5

Combine all numbers

Finally, combine both lists from steps 1 and 2. Start with the list of negatives in ascending order (because negative numbers will always be lesser than any positive number) followed by the list of positive numbers: \(-5.1\), \(-5\), \(-4 \frac{1}{2}\), \(3.4\), \(3 \frac{1}{2}\), \(4.1\).

Key concepts

Positive and Negative Numbers

Understanding the basics of positive and negative numbers is essential for ordering rational numbers. In mathematics, numbers to the right of zero on the number line are positive, while numbers to the left are negative. Positive numbers represent quantities greater than zero, while negative numbers represent quantities less than zero.

When comparing these numbers, a positive number is always greater than a negative number. For example, a temperature of 5 degrees is warmer than -5 degrees. Similarly, having \(5 in your bank account is better than being \)5 in debt. It is important to recognize that as we move further to the left on the number line, negative numbers become smaller, even though the digit itself might be getting larger. Thus, in our exercise, -4.5 is greater than -5.1 because it is closer to zero.

Fractions and Decimals

Dealing with fractions and decimals can sometimes be tricky, however, they are just two different ways to represent parts of a whole. A fraction consists of a numerator (top number) and a denominator (bottom number), like \( \frac{1}{2} \). A decimal is a way of writing numbers that are not whole, represented with a decimal point, such as 0.5.

To compare fractions and decimals, you might find it easier to convert fractions to decimals. For instance, the fraction \( 3 \frac{1}{2} \) can be converted to the decimal 3.5. This conversion simplifies the process of ordering numbers, as it is usually more straightforward to compare decimals. You line them up according to their decimal places and compare digit by digit from left to right.

Ascending Order

When arranging numbers in ascending order, we list them from the smallest to the largest. For example, the sequence of whole numbers 2, 5, and 7 has an ascending order of 2, 5, 7. With rational numbers, which include both positive and negative numbers, fractions, and decimals, we start with the most negative number and work our way up towards positive numbers.

In practice, to sort numbers in ascending order, it's useful to first identify if numbers are positive or negative, then arrange each subset accordingly before combining them. In our exercise, once the negative numbers are ordered from least to greatest (-5.1, -5, and -4.5) and similarly with the positive numbers (3.4, 3.5, and 4.1), we merge the two lists while keeping the negative numbers first, followed by the positive numbers. This ensures a smooth, incremental sequence from the smallest to the largest value.