Problem 7 - Calculate With Confidence

Question

Arrange the following decimals from smallest to largest. 0.123, 0.1023, 1.23 ___________________________

Step-by-Step Solution

Answer

The numbers in order are 0.1023, 0.123, 1.23.

Step 1: Compare the numbers based on their integer parts

Look at the numbers 0.123, 0.1023, and 1.23. The integer parts are 0, 0, and 1 respectively. Since 0 is less than 1, the number 1.23 is the largest among these. Therefore, it should be placed at the end of the sequence.

Step 2: Compare the decimals with same integer parts

Now let's compare the numbers that have the integer part of 0: 0.123 and 0.1023. Compare the digits following the decimal point, starting from the tenths place.

Step 3: Tenths place comparison

Look at the tenths place for both numbers: 0.123 has '1' and 0.1023 also has '1'. Since both tenths places are equal, move to the next digit.

Step 4: Hundredths place comparison

Compare the hundredths place: 0.123 has '2' and 0.1023 has '0'. Since '0' is less than '2', 0.1023 is smaller than 0.123.

Step 5: Arrange the numbers

Based on the above comparisons, arrange the numbers from smallest to largest: 0.1023, 0.123, 1.23.

Key Concepts

Comparing Decimals

When comparing decimal numbers, we focus on both the integer part and the decimal part. Let's take the example of three decimal numbers: 0.123, 0.1023, and 1.23. To understand which is smaller or larger, we follow a systematic comparison:

First, examine the integer parts of the numbers. The integer part is what comes before the decimal point. For instance, 0.123 and 0.1023 have the integer part of 0, whereas 1.23 has an integer part of 1. Clearly, 1 is greater, making 1.23 the largest number.

For numbers with equal integer parts, like 0.123 and 0.1023, compare digit by digit after the decimal point. Begin with the tenths place and move right as needed. The aim is to find the first instance where the digits differ, determining their order.

Integer and Decimal Parts

Decimals are unique numbers that have two parts: the integer part and the decimal part. Understanding these parts helps us compare and order different decimal numbers.

The integer part is found to the left of the decimal point. It represents a whole number, just like integers such as -3, 0, 4, and so on. For example, in the decimal 1.23, the integer part is 1.

The decimal part is located to the right of the decimal point. This part represents the fractional component of the number. For instance, in 0.123, the decimal part is 123. By separating these two elements, we can better understand and manipulate decimal numbers.

Ordering Decimals

Ordering decimals means arranging them from smallest to largest, or vice versa. This requires careful comparison of their integer and decimal parts.

Here's how we approach ordering decimals:

• Compare integer parts: Larger integers indicate larger decimal numbers (e.g., 1 in 1.23 is larger than 0 in 0.1023).


• Examine decimal places: When integers are equal, compare each subsequent digit after the decimal point until you find a difference. For instance, in 0.123 vs. 0.1023, the hundredths place tells us 0.1023 is smaller.

Practicing this method ensures accurate ordering of any set of decimal numbers.

Place Value in Decimals

Understanding place value in decimals is crucial for comparing and arranging them correctly. Place value indicates the significance of digits based on their position relative to the decimal point.

Right after the decimal point, each digit has a specific place value: tenths, hundredths, thousandths, and so on. For example, in 0.123:

• The '1' is in the tenths place, indicating one-tenth.


• The '2' is in the hundredths place, meaning two-hundredths.


• The '3' is in the thousandths place, showing three-thousandths.

Recognizing these place values helps us see the contribution of each digit to the overall value of the decimal, aiding in accurate comparison and arrangement.