Understanding Decimal Numbers
Maya finishes the 100-meter dash in $12.45$ seconds. Her friend Jayden finishes in $12.5$ seconds. Jayden's number looks shorter, but who actually won? Decimals show up wherever whole numbers are not precise enough: race times, prices, lab measurements. In this lesson you will learn to read them, place them, compare them, and tell which kinds of decimals exist.
Place Value in Decimals
Every digit in a number has a place value. To the left of the decimal point the places get 10 times bigger. To the right of the decimal point they get 10 times smaller.
So in $42.308$ the digit $3$ means $3$ tenths, the digit $0$ means $0$ hundredths, and the digit $8$ means $8$ thousandths. We can write the number in expanded form:
$$42.308 = 40 + 2 + \tfrac{3}{10} + \tfrac{0}{100} + \tfrac{8}{1{,}000}$$Here is the same idea as a table:
| Place | tens | ones | . | tenths | hundredths | thousandths |
|---|---|---|---|---|---|---|
| Digit | 4 | 2 | . | 3 | 0 | 8 |
| Value | $40$ | $2$ | $0.3$ | $0.00$ | $0.008$ |
Indicate what place value (ones, tenths, hundredths, or thousandths) the digit $7$ has in each of the following numbers: a) $15,72$ b) $0,007$ c) $7,431$ d) $2,075$
Indicate what place value (ones, tenths, hundredths, or thousandths) the digit $7$ has in each of the following numbers:
- a) $15,72$
- b) $0,007$
- c) $7,431$
- d) $2,075$
Decompose the following decimal numbers according to their place values, as in the example: Example: $4,25 = 4$ Ones + $2$ tenths + $5$ hundredths.* a) $8,39$ b)…
Decompose the following decimal numbers according to their place values, as in the example:
Example: $4,25 = 4$ Ones + $2$ tenths + $5$ hundredths.
- a) $8,39$
- b) $12,045$
- c) $0,614$
In a science laboratory, a group of students measured the mass of a mineral sample with a precision balance. The result obtained is $42,308$ grams. Indicate what value…
In a science laboratory, a group of students measured the mass of a mineral sample with a precision balance. The result obtained is $42,308$ grams.
Indicate what value each digit represents in this number:
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The digit $4$ represents the ............................
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The digit $2$ represents the ............................
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The digit $3$ represents the ............................
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The digit $0$ represents the ............................
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The digit $8$ represents the ............................
Also write its expanded form using ones and decimal fractions.
Reading and Writing Decimals, with Money
Say the whole-number part, say "and" for the decimal point, then read the decimal part as a whole number followed by the name of its last place.
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$2.307$ is "two and three hundred seven thousandths."
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$0.45$ is "forty-five hundredths."
Money works the same way. Dollars are the ones place, and cents are hundredths, because one cent is $\$0.01$, or one hundredth of a dollar.
In a sporting goods store, the prices of three items have been written in words. Write in your notebook the decimal number corresponding to each one using digits: a)…
In a sporting goods store, the prices of three items have been written in words. Write in your notebook the decimal number corresponding to each one using digits:
- a) Swimming goggles that cost eight dollars and fifteen cents.
- b) A tennis ball that costs two dollars and eight cents.
- c) Running shorts that cost twelve dollars and fifty cents.
Regrouping Place Values
Each place is worth $10$ of the next smaller place. This lets you trade between places, just like trading a dollar for $10$ dimes.
Complete these equivalences between place values: a) $4$ ones = tenths. b) $70$ hundredths = tenths. c) $5$ tenths = thousandths. d) $300$ thousandths = hundredths.
Complete these equivalences between place values:
- a) $4$ ones = ____ tenths.
- b) $70$ hundredths = ____ tenths.
- c) $5$ tenths = ____ thousandths.
- d) $300$ thousandths = ____ hundredths.
A decimal number is composed of $4$ ones, $15$ tenths, $27$ hundredths, and $105$ thousandths. Perform the necessary groupings to write the number correctly in its…
A decimal number is composed of $4$ ones, $15$ tenths, $27$ hundredths, and $105$ thousandths. Perform the necessary groupings to write the number correctly in its standard decimal form, and then indicate how many total thousandths this number contains.
Mystery Number Puzzles
Some problems describe a number with clues about its digits. Make a blank template and fill each place as you learn it.
Find the "mystery" number following these clues: 1. It has three decimal places. 2. Its whole number part is the smallest even prime number. 3. The hundredths digit is…
Find the "mystery" number following these clues:
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It has three decimal places.
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Its whole number part is the smallest even prime number.
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The hundredths digit is double the tenths digit.
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The thousandths digit is the sum of the tenths and hundredths.
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The tenths digit is $3$.
Write the resulting number as a decimal fraction.
Find the number that meets the following conditions: 1. Its whole number part is the smallest even prime number. 2. It has four decimal places that do not repeat. 3. The…
Find the number that meets the following conditions:
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Its whole number part is the smallest even prime number.
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It has four decimal places that do not repeat.
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The thousandths digit is three times the tenths digit.
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The sum of all its digits is $15$.
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The ten-thousandths digit is half the hundredths digit.
Decimals on the Number Line
To place a decimal on a number line, split the space between two whole numbers into $10$ equal parts (tenths). To get more precise, zoom in on one part and split it into $10$ again (hundredths).
Observe the following number line and write which decimal number corresponds to each of the letters marked with an orange dot:
Observe the following number line and write which decimal number corresponds to each of the letters marked with an orange dot:
Observe the following number line where several points have been represented. Identify which terminating decimal number corresponds to each of the marked letters ($A$,…
Observe the following number line where several points have been represented. Identify which terminating decimal number corresponds to each of the marked letters ($A$, $B$, and $C$).
Finding Decimals Between Two Decimals
Between two whole numbers there are only a few other whole numbers, or none. But between any two decimals there are always more decimals, no matter how close they are. Zoom in once more and there is room for new numbers.
One of the most curious properties of decimal numbers is that we can always find another number between two given ones, no matter how close they are. Represent…
One of the most curious properties of decimal numbers is that we can always find another number between two given ones, no matter how close they are.
Represent graphically on a number line the interval between $0,56$ and $0,57$ and place on it three decimal numbers located between them.
Represent on a number line the interval between $5,67$ and $5,68$. Divide this interval into ten equal parts and mark the exact position of the numbers $5,674$ and…
Represent on a number line the interval between $5,67$ and $5,68$. Divide this interval into ten equal parts and mark the exact position of the numbers $5,674$ and $5,679$. Then, find a decimal number that has five decimal places and is located between those two marked points.
Find a decimal number $x$ that is at the exact same distance from $3,14$ and $3,15$. Express it both in decimal form and as a fraction in simplest form. What type of…
Find a decimal number $x$ that is at the exact same distance from $3,14$ and $3,15$. Express it both in decimal form and as a fraction in simplest form. What type of decimal number is the result?
Comparing and Ordering Decimals
To compare decimals, line up the decimal points. Compare the whole parts first. If they match, compare tenths, then hundredths, then thousandths. The first place where the digits differ decides the winner.
Adding zeros at the end of a decimal does not change its value: $0.5 = 0.50 = 0.500$. This makes it easy to compare numbers with different lengths.
Back to the start: Maya ran $12.45$ s and Jayden ran $12.5$ s. Write $12.5$ as $12.50$. Then $12.45 < 12.50$, so Maya has the lower time and she won.
Copy into your notebook and use the signs $<$ or $>$ to compare these pairs of decimal numbers: a) $5,4$ $5,39$ b) $0,25$ $0,3$ c) $12,07$ $12,071$ d) $6,1$ $6,099$
Copy into your notebook and use the signs $<$ or $>$ to compare these pairs of decimal numbers:
- a) $5,4$ ____ $5,39$
- b) $0,25$ ____ $0,3$
- c) $12,07$ ____ $12,071$
- d) $6,1$ ____ $6,099$
In the school Olympics, the times recorded in the final of the $100$ meters sprint were very close. The judges recorded the following results in seconds: $12,45$ —…
In the school Olympics, the times recorded in the final of the $100$ meters sprint were very close. The judges recorded the following results in seconds:
$12,45$ — $12,408$ — $12,5$ — $12,099$ — $12,41$
Order the runners from the fastest (lowest time) to the slowest (highest time) using the symbol $<$ and write how the decimal number corresponding to the gold medal winner is read.
Ashley is organizing her backpack for a trip. She bought a bottle of water for $\$1,25$, a bag of nuts for $\$3,050$, and a cap for $\$9,9…
Ashley is organizing her backpack for a trip. She bought a bottle of water for $\$1,25$, a bag of nuts for $\$3,050$, and a cap for $\$9,9$.
- a) Write all the prices so that they have the same number of decimal places (adding zeros if necessary).
- b) Convert the price of the cap into a fraction in simplest form.
- c) How many cents does the bag of nuts cost?
Decimals and Decimal Fractions
Every decimal that ends can be written as a fraction with denominator $10$, $100$, $1{,}000$, and so on. These are called decimal fractions. The number of decimal places tells you how many zeros the denominator has.
Match each decimal number with its corresponding decimal fraction: Numbers: $0,7$ | $0,07$ | $7,7$ | $0,77$ Fractions: $\frac{77}{100}$ | $\frac{7}{10}$ |…
Match each decimal number with its corresponding decimal fraction:
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Numbers: $0,7$ | $0,07$ | $7,7$ | $0,77$
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Fractions: $\frac{77}{100}$ | $\frac{7}{10}$ | $\frac{7}{100}$ | $\frac{77}{10}$
If we divide a square into $100$ equal parts, each part is one hundredth ($0,01$). Observe the following image and write what decimal number the orange shaded area…
If we divide a square into $100$ equal parts, each part is one hundredth ($0,01$). Observe the following image and write what decimal number the orange shaded area represents:
Powers of 10 and Metric Prefixes
Place value is built on powers of 10: $10^2 = 100$, $10^3 = 1{,}000$, $10^6 = 1{,}000{,}000$, and $10^9$ is one billion. The exponent tells how many zeros follow the $1$.
Science uses metric prefixes that match these powers: milli- is a thousandth, micro- is a millionth, and nano- is a billionth.
Tiny numbers: negative exponents
Scientists need a short way to write numbers like $0.000000001$. They use a negative exponent: a $1$ in the $n$th decimal place is written $10^{-n}$.
| Decimal | Fraction | Power of 10 |
|---|---|---|
| $0.1$ | $\frac{1}{10}$ | $10^{-1}$ |
| $0.01$ | $\frac{1}{100}$ | $10^{-2}$ |
| $0.001$ | $\frac{1}{1{,}000}$ | $10^{-3}$ |
| $0.000001$ | $\frac{1}{1{,}000{,}000}$ | $10^{-6}$ |
| $0.000000001$ | $\frac{1}{1{,}000{,}000{,}000}$ | $10^{-9}$ |
So $10^{-n} = \frac{1}{10^n}$: the exponent counts how many places the $1$ sits to the right of the decimal point. This matches the metric prefixes: $1$ mm $= 10^{-3}$ m, $1$ µm $= 10^{-6}$ m, and $1$ nm $= 10^{-9}$ m.
One nanometer ($nm$) is equivalent to $0,000000001$ meters. If the thickness of a human hair is approximately $0,08$ millimeters, express this thickness in meters using…
One nanometer ($nm$) is equivalent to $0,000000001$ meters. If the thickness of a human hair is approximately $0,08$ millimeters, express this thickness in meters using powers of $10$ and determine how many nanometers the hair measures.
Terminating and Repeating Decimals
Not every fraction turns into a decimal that stops. There are two families.
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Terminating decimal: the digits end, like $5.08$.
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Repeating decimal: a block of digits repeats forever. We draw a bar over the repeating block: $0.\overline{3} = 0.333\ldots$. If the repeating block starts right after the decimal point, it is purely repeating, like $2.\overline{3}$ or $0.\overline{27}$. If some non-repeating digits come first, it is mixed repeating, like $0.1\overline{6} = 0.1666\ldots$ or $14.2\overline{5}$.
Some problems put a hat over the repeating digits instead, like $0.\widehat{7}$. It means exactly the same thing as the bar.
Classify the following decimal numbers indicating whether they are terminating decimals, pure repeating, or mixed repeating. For the repeating ones, clearly indicate…
Classify the following decimal numbers indicating whether they are terminating decimals, pure repeating, or mixed repeating. For the repeating ones, clearly indicate which is the repeating part and, if there is one, the non-repeating part:
- a) $5,08$
- b) $2,333...$
- c) $14,252525...$
- d) $0,1666...$
- e) $3,141592...$ (is this last one a repeating decimal?)
Sometimes fractions help us to better understand the type of decimal number we have. Without performing long division, classify the decimal expression of the following…
Sometimes fractions help us to better understand the type of decimal number we have. Without performing long division, classify the decimal expression of the following fractions as terminating or repeating:
- a) $\frac{7}{20}$
- b) $\frac{4}{9}$
- c) $\frac{13}{6}$
- d) $\frac{1}{25}$
Hint: Look at the prime factors of the denominators.
Classify the following fractions according to the type of decimal expression they generate (Terminating, Pure repeating, or Mixed repeating) without performing the…
Classify the following fractions according to the type of decimal expression they generate (Terminating, Pure repeating, or Mixed repeating) without performing the division, based solely on the prime factorization of their denominators:
- a) $\frac{7}{40}$
- b) $\frac{5}{18}$
- c) $\frac{11}{21}$
- d) $\frac{9}{250}$
Turning Repeating Decimals into Fractions
A repeating decimal goes on forever, but it is still a fraction. The trick is to multiply by a power of $10$ so the repeating parts line up, and then subtract to make them disappear. Some problems call the fraction you get the generating fraction.
To order numbers that mix fractions and decimals, convert everything to decimals with at least four places. Fractions become decimals by dividing the numerator by the denominator.
Write the generating fraction in simplest form of the following repeating decimals, showing the step-by-step process: a) $2,1\widehat{36}$ b) $0,00\widehat{5}$
Write the generating fraction in simplest form of the following repeating decimals, showing the step-by-step process:
- a) $2,1\widehat{36}$
- b) $0,00\widehat{5}$
Order the following decimal numbers mixed with fractions from least to greatest. To compare them, convert all the numbers into their decimal form with at least four…
Order the following decimal numbers mixed with fractions from least to greatest. To compare them, convert all the numbers into their decimal form with at least four decimal places:
$1,2\widehat{5}$ ; $\frac{5}{4}$ ; $1,2525...$ ; $1,25$ ; $\frac{11}{9}$