CHAPTER 1: NUMBER SENSE
12: The Power of 10
Decimals
The decimal number system is a base10 number system that uses positional placevalues. A base10 number system counts in groups of 10 and uses only 10 digits to represent any number. The 10 digits we use are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Placevalue in a decimal number tells us how many groups of ten we are dealing with. Every placevalue gets ten times bigger as we move to the left in a decimal number and every placevalue gets ten times smaller as we move to the right in a decimal number.
PlaceValue
Explore 1 – The relationship between places
Explain how the name of each placevalue shows that it is ten times the value of its adjacent placevalue to the right, and consequently onetenth the value of its adjacent value to the left.
Solution
The number 100 represents one hundred and has a 1 in the hundreds place. 100 is ten times the number 10, which has a 1 in the tens place. One thousand is represented as 1,000 with the 1 in the thousands place and is ten times 100. Tenthousand is represented by 10,000 and means 10 times one thousand. One hundredthousand is represented by 100,000 and is ten times 10,000.
Decimals after the point represent fractions of a whole. The number 0.1 is read onetenth which is 10 times smaller than 1. Equivalently, 1 is ten times larger than onetenth. 0.01 is read onehundredth since the 1 is in the hundredths place and is ten times smaller than 0.01. Likewise, 0.001 has a 1 in the thousandths place so is read onethousandth and is ten times smaller than 0.01. This pattern continues with 0.0001 representing onetenthousandth, which is ten times smaller than 0.001, etc.
Explore 2 – Sets of three
People like to add a comma for every three places to the left of the decimal point when writing large numbers of 10,000 or more. For example, the population of the United States was 335,967,423 as of January 25th, 2023.
 How do you read the number 335,967,423?
Solution
The number 335 formed by the digits before the first comma is read as three hundred thirtyfive, but its last digit is in the millions place. Consequently, the digits 335 represent three hundred thirtyfive million. The three digits 967 form the number nine hundred sixtyseven, but its last digit is in the thousands place, so 967 represents nine hundred sixtyseven thousand. The three digits 423 form the number four hundred twentythree. Putting these sets of numbers together we have, three hundred thirtyfive million, nine hundred sixtyseven thousand, four hundred twentythree.
 What is the function of the two commas in the expression 335,967,423? Explain your reasoning.
Solution
The two commas separate three different sets of placevalues. The commas are used to make reading the number easier as it splits the number into sets of ones, tens, and hundreds of the placevalue before the comma.
Explore 3 – Decimal places
In life, the measure of length in the metric system follows the base10 number system. Latin and Greek root prefixes are attached to the basic unit of meter to represent the different placevalues.
Greek roots  Base unit  Latin Roots  
Length  Kilometer  hectometer  dekameter  meter  decimeter  Centimeter  millimeter 
Placevalue  thousands  hundreds  tens  ones  tenths  hundredths  thousandths 
 One decimeter is how many times as small as one meter?
Solution
Decimeter is one place to the right of meter so one decimeter is 10 times as small as a meter.
 How many centimeters are in one meter?
Solution
Centimeter lies 2 places to the right of meter so there are 100 centimeters in one meter.
 One kilometer is how many times as large than one meter?
Solution
Kilometer lies 3 places to the left of meter so one kilometer is 1000 times as large as one meter.
 Write 1 decimeter in terms of meter as a decimal number.
Solution
Decimeter is onetenth of a meter: 0.1 meter
 Write 1 kilometer in terms of meter as a decimal number.
Solution
Kilometer lies 3 places to the left of meter so 1 kilometer = 1000 meters
 Write 23 centimeters in terms of meter as a decimal number.
Solution
Centimeter lies 2 places to the right of meter, so 23 centimeters = 0.23 meters
 Write 1 millimeter in terms of meter as a decimal number.
Solution
Millimeter lies 3 places to the right of meter so 1 millimeter = 0.001 meters
 Write 90 millimeters in terms of meter as a decimal number.
Solution
90 millimeters = 0.090 meters
Examples
1. The monetary units of dollar, dime, and cent follow the placevalue rules of the base10 number system.
a) Write 2 dollars and 57 cents as a decimal number.
b) Write two hundred fiftyseven dollars and eightyfive cents as a decimal number.
c) What does the word ‘and’ represent in a spoken decimal number?
Show/Hide Answer
a) $2.57 b) $257.85 c) The decimal point.
2. Convex lenses are used in eyeglasses for correcting farsightedness while concave lenses correct nearsightedness. Concave lenses have a positive focal length, while convex lenses have a negative focal length.
Don wears contact lenses to correct his vision. The prescription says 1.65 for his left eye and 1.75 for his right eye.
What is the placevalue of the digit 5 in either prescription?
Show/Hide Answer
The 5 sits two places after the decimal point so is in the hundredths place
 What is the numerical relationship between two adjacent digits (places) in a decimal number?
Show/Hide Answer
The digit to the left is in a place where its value is 10 times the placevalue of the digit to the right.
 Is there any number you know that cannot be represented by the base10 number system?
Show/Hide Answer
No. Any number can be represented by the base10 number system.
 What is the difference between digits to the left of the decimal point and digits to the right of the decimal point?
Show/Hide Answer
The number formed by the digits to the left of the decimal point represents an integer value. The number formed by the digits to the right of the decimal point represents a fraction of a whole.
 What is the purpose of the decimal point?
Show/Hide Answer
The decimal point separates the integer value of the number from the fractional value of the number.
Standardized Decimals
While it is common in the U.S. to use commas every 3 decimal places before the decimal point, and use a period for the decimal point, that is not the case in other parts of the world. In Europe, for example, the commas and period are reversed. So, in Europe is equivalent to in the U.S.
“In decimal numbers, the comma (French practice) or the dot (British practice) is used only to separate the integral part of numbers from the decimal part. Numbers may be divided in groups of three in order to facilitate reading; neither dots nor commas are ever inserted in the spaces between groups.”^{[1]}Indeed, in 1948, and again in 2018, the GCPM standardized the printing of decimal numbers under the metric system to use a small space rather than a comma or a period as spacevalue separators.
So, if you see a number written as do not be surprised. Notice also that a small space separator is also used after the decimal point to separate the digits into groups of three.
Exponents & The Power of 10
Exponents
Writing repeated addition is tiresome, for example, . Repeated addition of a number may be expressed by multiplication. For example, adding three 5s (5+5+5) may be expressed as . The number 5 is added to itself three times. Therefore, we say 3 “times” 5 or 3 “” 5. Likewise, .
Is there a similar way to write repeated multiplication of a number? Afterall, writing repeated multiplication is just as tiresome and prone to mistakes as writing repeated addition. Rather than writing, , we can use an exponent to indicate how many times the number is multiplied by itself. An exponent is a small number written to the top right of the number being multiplied. We write . The number that is repeatedly multiplied (in this case 5) is called the base. The exponent number 3 is the number of times the base is multiplied by itself. For example, since the base 4 is multiplied by itself a total of 5 times.
The Power of 10
Explore 4 – The power of 10 with positive exponents
One hundred dollars is 10 lots of 10 dollars: . One thousand dollars is 10 lots of one hundred dollars: . Likewise, tenthousand dollars are 10 lots of one thousand dollars: . We can continue this pattern indefinitely: One hundredthousand dollars is 10 lots of tenthousand dollars: .
Write one hundred, one thousand, tenthousand, one hundredthousand, one million, tenmillion, one hundredmillion, one billion, and one trillion dollars in exponential expressions with 10 as the base.
Solution
 One hundred dollars
 One thousand dollars
 Tenthousand dollars
 One hundredthousand dollars
 One million dollars
 Tenmillion dollars
 One hundredmillion dollars
 One billion dollars
 One trillion dollars
Did you notice that every time a power of 10 is multiplied by another 10, the exponent increases by 1? For example, . Since it is multiplied by itself just once, can be written . So, . Also, (see 9 in Explore 4). There are two patterns here. First, the number of zeros in a number that is a power of 10 is equal to the exponent. For example, has four zeros and is equal to . The second pattern is that when we multiply powers of 10 (e.g. ) we can add the exponents to get the exponent of the product: .
Explore 5 – The power of 10 with negative exponents
Just as multiplying by 10 causes an increase by 1 in the exponent on a power of 10, dividing by 10 causes a decrease by 1 in the exponent. To illustrate, one tenth of one thousand dollars is $1000 divided by 10 which equals $100: . That is : the exponent decreases by 1 from 3 to 2 . Likewise, one tenth of one hundred dollars is $100 divided by 10, which is $10. The exponent decreases by 1 again: .
 What is the exponential expression for the number that is onetenth of 10?
Solution
Onetenth of 10 means . The exponents are subtracted.
 Since , what does equal?
Solution
and , so .
 What is the exponential expression for the number that is onetenth of 1? Write this number as a decimal number.
Solution
Onetenth of 1 means . The exponent is decreased by 1. As a decimal, . So, .
 What is the exponential expression for the number that is onetenth of ? Write this number as a decimal number.
Solution
Onetenth of means . The exponent is decreased by 1. As a decimal, . So, .
 What is the exponential expression if the number decreases 10 times? Write this number as a decimal number.
Solution
Onetenth of means . The exponent is decreased by 1. As a decimal, . So, .
Did you notice the pattern that dividing a power of 10 by 10 decreases the exponent by 1? Also, a power of 10 with a negative exponent means we are dealing with a fraction so it belongs to the right of the decimal place. Indeed, the absolute value of the exponent tells us the number of places after the point where the 1 lies. For example, in , the 1 lies in the 5th place after the point where .
Properties of Powers of 10
Exponent of 1
Any number to the power 1 equals the number.
Zero Exponent
Any nonzero number to the power zero equals 1.
Product Rule
Two or more exponential expressions with the same base can be multiplied by
keeping the common base and adding the exponents.
Quotient Rule
Two exponential expressions with the same base can be divided by
keeping the common base and subtracting the exponents.
Negative Exponents
When an exponential expression has a negative exponent
the number is a fraction of a whole.
with the digit in the ^{th} place after the decimal point
Explore 6 – Multiplication by powers of 10
Precious tells Brandon a fast way to understand multiplying by a power of 10. She says, if you have 10, which is , it becomes if you multiply it by 10. If you multiply by 10, it becomes , then multiply by 10 again to get , then if you keep multiplying by 10 and so on.
 What do you get if you multiply by ? Write your answer as a power of and as a decimal.
Solution
Since , multiplying it by just adds another zero to the answer. So, . Adding one extra zero is equivalent to adding 1 to the power on the . .
 How do you write multiplied by as a decimal?
Solution
is equivalent to . This is like writing the number and attaching four zeros.
 How do you multiply by ? What happens to the decimal point?
Solution
. Using a calculator we get . The decimal point in gets moved 4 places to the right.
 Is there a rule for multiplying by a power of 10?
Solution
To multiply a number by a power of 10, we move the decimal point in the number to the right the same number of places as the power.
 What is multiplied by ?
Solution
means we move the decimal point in zero places to the right. So, . When multiplying by to the power of zero, the decimal point doesn’t move. The only number we can multiply by that doesn’t change the number is . So, .
 What is multiplied by ?
Solution
and . So, . Since there are seven zeros after the , .
 Write a rule that explains how to multiply two powers of .
Solution
To multiply two powers of , we keep the base and add the exponents.
 What is multiplied by ?
Solution
We add the powers, so .
 How is the positive power of 10 (e.g., ) related to the number of zeros to the left of the decimal point?
Show/Hide Answer
and so the exponent in the power of 10 tells us the number of zeros to write to the left of the decimal point.
 How is the positive power of 10 (e.g., ) related to the placevalue of the digit 1 to the left of the decimal point?
Show/Hide Answer
and so the placevalue of the digit 1 is one more than the positive power on the 10.
 How is the negative power of 10 (e.g., ) related to the number of zeros to the right of the decimal point?
Show/Hide Answer
In , there are 2 zeros after the point. In , there are 4 zeros after the point. So the number of zeros after the decimal point is one less than the absolute value of the negative power.
 How is the negative power of 10 (e.g., ) related to the placevalue of the digit 1 to the right of the decimal point?
Show/Hide Answer
In , the digit 1 is 3 places after the point. In , the digit 1 is 5 places after the point. So we use the absolute value of the negative power to determine the placevalue of the digit 1.
Practice Exercises
Precious tells Brandon a fast way to understand negative exponents. She says, if you have , or , it becomes if you divide it by . If you divide by again, it becomes , then , then , , if you keep dividing it by .
 What would equal as a decimal?
 What would equal as a power of ?
 What is divided by ?
 What is divided by ?
 What is divided by ?
 What is multiplied by ?
 Write a rule that governs multiplying or dividing by powers of 10?
Show/Hide Answer
 0.00001
 To multiply by a power of 10, keep the base 10 and add the exponents. To divide by a power of 10, keep the base and subtract the exponents.
National Debt
People don’t have a strong intuitive sense of how much bigger a trillion is than a billion, or a billion is than a million. As an example, a million seconds is about 11.6 days. A billion seconds is about 31.7 years. A trillion seconds is about 31,688 years.
The U.S. national debt can be found at https://www.usdebtclock.org.
 Is the current national debt measured in millions, billions, or trillions of dollars?
 Koa’s annual salary is $100,000. How many years would it take Koa to payoff the national debt?
 How much would each of the billionaires in the U.S. have to donate to be able to pay off the national debt?
In this section, we will take what we have learned and apply the concepts to new situations.
Perspectives
The table on the right shows metric prefixes in common use.
 A Youtuber bought a oneterabyte (bytes) external drive for storing the videos she made. She typically makes 10minute videos for her channel. A 10minute video takes up about 100 megabytes ()bytes of storage.
a) How many 10minute videos can be stored on a oneterabyte external drive?
b) How many 10minute videos can be stored on a 5terabyte external drive?
 The most valuable substance on earth is Botulinum toxin which is used to create Botox. It values at $1.5 trillion per kilogram. In each vial of Botox, there is just under 1 nanogram of the toxin.
a) How many vials can be made from 1 kilogram of Botulinum toxin?
b) What is the cost of 1 nanogram of the toxin?
 Nuclear bombs are discussed in megatons or kilotons. In this context, ton refers not to the weight of the bomb but to the explosive yield of a ton of TNT. The only two nuclear bombs used in war were dropped on Japan in 1945. The first bomb hit Hiroshima on August 6th and the second bomb hit Nagasaki on August 9th. The damage was so extensive and horrific that some believe it ended the war in the Pacific. The first bomb, “Little Boy,” had an explosive yield of 15 kilotons. The second bomb, “Fat Man,” had an explosive yield of 21 kilotons. As of 2020, the US had 320 gravity bombs with the following designations and yields: B617 (360 kilotons); B6111 (400 kilotons); B831 (1200 kilotons). ^{[2]}
a) How many tons of TNT would have the same explosive yield as “Little Boy”?
b) How many tons of TNT would have the same explosive yield as “Fat Man”?
c) How many times bigger is the B617 than the “Fat Man” in terms of explosive yield?
d) How many times bigger is the B831 than the “Fat Man” in terms of explosive yield?
e) Estimate the destructive power of the US arsenal of gravity bombs in terms easier to grasp than kilotons.
Show/Hide Answer

 a) Ten thousand 10minute videos may be stored on the external drive. b) Fifty thousand 10minute videos.
 a) There are 10^{12} nonograms in 1 kilogram. b) It costs $0.015 for 1 nanogram of the toxin.
 a) 15,000 tons b) 21,000 tons c) 17.1 times bigger d) 57.1 times bigger e) Total annihilation
Currency
A packet of one hundred $100 bills, currently the largest U.S. bill in circulation, is less than ½ inch thick. It would fit easily in your pocket.
 How much money is there in a stack of 100 $100 bills?
 How many of these stacks would you need to make $1 billion?
 How high would the stack be to make $1 billion?
 Compare this height to something tangible.
 How many of these stacks would you need to make $1 trillion?
 Why is the $100 bill the largest U.S. bill in circulation?
In this section, we will use what we have learned so far to practice skill problems.
Skills Practice
Evaluate the expression. Write the answer as a power of 10 and as a decimal.
Show/Hide Answer
 ; 100,000
 ; 10,000,000
 ; 0.0000001
 ; 0.00001
 ; 1
 ; 10
 ; 1
 ; 10,000,000
 ; 0.000000001
 ; 0.1
 4000
 7,000,000
 0.03
 4,500,000
 0.00028
Real numbers written using a base 10 placevalue system.
The international decimal system of weights and measures.
General Conference on Weights and Measures (CGPM, Conférence Générale des Poids et Mesures). The CGPM is the primary intergovernmental treaty organization responsible for the SI (metric system), representing nearly 50 countries.
a quantity representing the power to which a given number or expression is to be raised, usually expressed as a raised symbol beside the number or expression.