# Decimal Expansions

## Decimal Expansions – Introduction

Whenever we deal with numbers in mathematics or in daily life, we often encounter figures that aren't whole numbers. These numbers may have parts or pieces that are less than one whole, and representing these parts accurately is essential for precision and clarity. This is the role of decimal expansion, a concept that is fundamental in various fields, including finance, science, engineering, and everyday commerce.

## Understanding Decimal Expansion

Decimal expansion is the process of expressing numbers using a decimal point to separate the whole number part from the fractional part. It allows us to precisely represent numbers less than one, as well as numbers with fractional parts that are not whole numbers.

For instance, in the decimal expansion of 3.14, the number 3 represents the whole number part, while the digits 1 and 4 after the decimal point represent the fractional parts, specifically the tenths and hundredths respectively.

To fully grasp decimal expansions, one must become familiar with place value, such as tenths, hundredths, and thousandths, which indicate the size of the fractional part that each digit represents.

What is the place value of the digit 2 in the decimal 0.526?
Write the number 1 and three-quarters as a decimal.
Which decimal represents the smallest value: 0.9, 0.09, or 0.009?

## How to Convert Fractions to Decimal Expansions? – Method

To better understand how to work with decimal expansions, let's convert fractions to decimals:

1. Fraction to Decimal Conversion: Take the fraction $\frac{1}{2}$. To convert this to a decimal, divide the numerator (1) by the denominator (2) to get 0.5. This is the decimal expansion of $\frac{1}{2}$.

2. Repeating Decimals: Now, take the fraction $\frac{1}{3}$. When you divide 1 by 3, the decimal goes on forever as 0.3333..., which we represent with a repeating symbol over the 3 (0.\overline{3}).

## Rounding and Truncating Decimal Expansions – Guided Practice

Rounding and truncating numbers are two techniques used to simplify decimal expansions:

1. Rounding: Consider the number 0.6579. To round this to the nearest hundredth, look at the third decimal place (which is 7). Since 7 is greater than 5, we round up the second decimal place to 66. So, 0.6579 rounded to the nearest hundredth is 0.66.

2. Truncating: To truncate the number 5.987 to the nearest tenth, simply remove the digits beyond the first decimal place without rounding. The result is 5.9.

Rounding to the nearest tenth, what is 2.345?
How would you truncate 4.567 to the nearest hundredth?

## Decimal Expansion in Real-Life Applications – Application

Decimals are not just theoretical constructs; they have practical implications in daily life:

• Money: Prices, such as $\$$3.99, are expressed in decimals. • Measurement: Precise measurements in construction or DIY projects often require decimal representation. • Science: Researchers use decimals for accurate data representation. Understanding how to use decimals correctly can help you in budgeting, measuring, calculating, and analyzing data more effectively. ## Decimals Expansions – Summary Key Learnings from this Text: • Decimal expansion allows for the precise representation of fractional parts of numbers. • Understanding place values is crucial for interpreting and working with decimals. • Rounding and truncating are methods used to simplify decimals. • Decimals are widely used in real-world contexts like finance, measurement, and scientific data. • Practicing decimal conversions and calculations can make these concepts more familiar and easier to use. Continue to explore and practice these concepts with our interactive tools and resources designed to strengthen your numerical literacy and proficiency. ## Decimals Expansions – Frequently Asked Questions What is a decimal expansion? How do you convert a fraction to a decimal? What is the difference between rounding and truncating a decimal? Why are decimal expansions important in real life? How do you find the decimal expansion of a repeating decimal? Can all fractions be converted to finite decimal expansions? What do you mean by place value in decimals? How do you round a decimal to the nearest thousandth? When truncating a decimal, do you ever change the remaining numbers? What is a non-terminating decimal? ## Decimal Expansions exercise Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the learning text Decimal Expansions . • ### What is the place value of the digit$4$in each of the numbers? Hints Place value charts are helpful when deciding the value of a digit. For example,$162.57$has$1$hundred,$6$tens,$2$ones,$5$tenths and$7$hundredths. There are three numbers in each group. Solution$4$ones •$4.8$•$4.93$•$24.74$tenths •$0.421$•$5.47$•$8.44$hundredths •$0.84$•$8.04$•$22.142$• ### Round each number to the given accuracy. Hints Rounding to the nearest tenth means the number is written up to the tenths place value. We have to decide whether to round up or down each time. If the digit in the hundredths place is greater than$5$we round up. If the digit is less than$5$it stays the same. For example, what is$3.86$to the nearest tenth. We know that it is between$3.8$and$3.9$, which is it nearest to? First, look at the digit in the tenths place, this is an$8$. Next, look at the digit in the hundredths place, this is$6$. As$6$is greater than$5$so we round up to get$3.9$. Solution$2.43$to the nearest tenth is$2.4$.$0.238$to the nearest hundredth is$0.242.43$to the nearest whole number is$224.281$to the nearest tenth is$24.30.241$to the nearest hundredth is$0.24$• ### Convert each fraction to its decimal expansion. Hints To convert a fraction to a decimal we divide the numerator by the denominator. For example,$\frac{2}{5} = 2 \div 5 = 0.4$For numbers greater than$1$, work out the fraction first. For example, for$3 \frac{1}{8}$.$\frac{1}{8} = 0.125$so$3 \frac{1}{8} = 3.125$. Solution$\frac{1}{4} = 0.25$because$1 \div 4 = 0.25$.$1\frac{1}{5} =1.2$because$1 \div 5 = 0.2$.$\frac{3}{4} = 0.75$because$3 \div 4 = 0.75$.$2\frac{1}{2} = 2.5$because$1 \div 2 = 0.5$. • ### Convert$\frac{2}{3}$to a decimal. Hints To convert a fraction to a decimal, divide the numerator by the denominator. Terminating decimals have an end point:$0.5, 3.47, 6.7032$etc. Recurring decimals continue forever:$0.222..., 8.4545...$etc. Recurring decimals are written with a dot over the repeating part:$0.\dot 2$,$8.\dot 4 \dot 5$Our calculators will usually display a recurring decimal with the repeating digits repeated, for example,$\frac {1}{9} = 0.11111111....$. When a question asks for the full calculator display this is how we write it. The three dots at the end indicate that it continues forever. Solution Divide$\bf{2}$by$\bf{3}$. The result is$\bf{0.66666.....}$, a decimal that goes on forever. We represent this with a repeating symbol over the$\bf{6}$. Written as a recurring decimal$\frac{2}{3} =\bf{0.\dot 6}$• ### Determine the smallest number. Hints Write the numbers in a place value chart to help you. The digits in a place value chart increase in value as we move to the left and decrease as we move to the right. For example, comparing$5$and$0.5$in a place value chart, we can see that$0.5$is less than$5$. Solution$0.006$is the smallest number. It has a value of$6$thousandths. • ### Order the decimals from smallest to largest. Hints To start, convert the fractions to decimals. Be careful with recurring decimals, remember the digits are repeated. So$0.\dot4$can be written as$0.444444...$, this helps place it in the right position. It is helpful to write all the numbers to the same number of decimal places. Use zero as placeholders to do this, so$0.2$can be written as$0.20$. Write all the numbers in a place value chart to help you compare them. Solution The correct order is, •$0.13$•$0.31$•$\frac{1}{3}$•$1.03$•$1\frac{3}{5}$The image shows the conversion of the fractions to decimals. Then if we compare all the values to two decimal places we will have: •$0.13$•$0.31$•$0.33$•$1.03$•$1.60\$
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