Decimal system is a way of writing numbers. Most of the numbers people commonly use, from huge quantities to tiny fractions, can be written in the decimal system using only the 10 basic symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. The value of any of these symbols depends on the place it occupies in the number. The symbol 2, for example, has different values in the numbers 832 and 238, because the 2 is in different places in each of the numbers. Because the value of a symbol depends on where it is in a number, the decimal system is known as a place-value system.
The word decimal comes from decem, the Latin word for ten. The decimal system received its name because it is a base-ten system. The value of each place is 10 times greater than the value of the place just to its right. Thus, the symbols on the left of a number have larger values than symbols farther to the right. For example, the symbol 2 in 238 is worth much more than the symbol 2 in 832, because the 2 in 238 is farther to the left than is the 2 in 832.
The decimal system is also called the Hindu-Arabic system. It was developed by Hindu mathematicians in India more than 2,000 years ago. Arabs learned this system after conquering parts of India in the A.D. 700’s. They spread knowledge of the system throughout their empire, including the Middle East, northern Africa, and Spain.
The decimal system and number words
In the English language, special number words are used to name the value of each place in the decimal system except the ones place (the place farthest to the right). The letters “ty” at the end of the words for numbers in the second place (just left of the ones place) indicate the number of tens. For example, sixty means six tens. The word hundred is used to show the size of the third place. There is another new word for the fourth place, thousand, but after that there are only new words for every third place to the left.
A comma is placed after each third place in order to make it easier to read a decimal system number. The words ten and hundred are used with thousand and the other special words to name all the places between the special places. Each group of three numbers is read as if it had only three places, and then the name of its group is added. For example, in the United States and Canada, the number 5,246,380,901,483 is read as “five trillion, two hundred forty-six billion, three hundred eighty million, nine hundred one thousand, four hundred eighty-three.”
Large numbers in the decimal system can easily be expressed using exponents. An exponent is a symbol written to the right of and above a number. The exponent tells how many times a number is used as a factor. For example, 106 is equivalent to the expression 10 X 10 X 10 X 10 X 10 X 10, in which 10 appears as a factor six times. Because multiplying by 10 moves a number written in the decimal system over one place to the left, the exponent for ten also tells how many zeros to write when that number is written in the decimal system. Thus, 106 is written as a 1 followed by six zeros—1,000,000.
Decimals less than one
In the decimal system, as the places go to the left of the ones place, each place gets ten times larger than the last. But the places can also go to the right of the ones place. As places go to the right, the values of those places get smaller. In the first place to the right, the one is divided into ten equal parts, called tenths. In the second place to the right, each tenth is itself divided into ten parts. As a result, in this place, the one has been divided into ten times ten—or one hundred—small parts. Each of these small parts, which are called hundredths, gets divided into ten smaller parts in the third place, and so on.
The names for the places to the right are like those for the places to the left, except that the letters “th” are added to the name for each place. The letters “th” show that the one is divided into that many small parts. The names for the places to the right sound as if the values are getting bigger, but the “th” shows that the values are really getting smaller. It takes only ten tenths to make one, but it takes a million millionths to equal one.
A period, called the decimal point, is written in between the ones place and the tenths place. When a decimal system number does not include any places to the right of the ones place, a period does not have to be written. The period is usually read as “and” to show that the smaller places are starting. For example, 345.678 is read as “three hundred forty-five and six hundred seventy-eight thousandths.” To name the places to the right, read the numbers as if they were to the left of the decimal point and then add the name of the place farthest to the right. The places in the decimal system are symmetric (balanced) around the ones place, not around the decimal point:
hundreds tens ones . tenths hundredths
Addition and subtraction
of decimals smaller than one are done in the same way as addition and subtraction of whole numbers. Only numbers in the same places can be added or subtracted. One number is written beneath the other number so that matching places line up—that is, tenths are beneath tenths, hundredths beneath hundredths, and so on.
To add or subtract numbers with decimals less than one, write one number beneath the other number so that the decimal point of the bottom number is right beneath the decimal point of the top number. It does not matter if one number has numerals sticking out to the right or left of the other number. You can put in zeros in any places that are missing numerals. Then add or subtract the numerals that are just above and below each other.
Multiplication
of one whole number by another gives a number larger than the original number. But multiplication of a number by a decimal less than one gives a number smaller than the original number.
2 X 3 means two groups of three 0.1 X 3 means 0.1 group of three, or one-tenth of three, which is just part of three
The multiplication shift rule for multiplying a number by 0.1 (one-tenth) is that each digit in that number moves one place to the right—that is, it moves one place smaller.
For example, the amount $43.50 consists of four ten-dollar bills, three one-dollar bills, and five dimes. When multiplied by 0.1, each of the five dimes becomes a penny, because a penny is a tenth of a dime. Each of the three dollars becomes a dime, because a dime is one-tenth of a dollar. One-tenth of ten dollars is a dollar, so each of the 4 tens becomes a one. So the $43.50 becomes $4.35.
The multiplication shift rule for multiplying a number by 0.01 (one-hundredth) states that each digit in the number moves two places to the right. Each digit in a number multiplied by 0.001 (one-thousandth) moves three places to the right, and so on. In general, when a number is multiplied by any decimal smaller than one, the number moves as many places to the right as there are places smaller than one. Therefore, the rule for multiplying any number by a decimal number is: Multiply as usual. Then add the number of decimal places in the top number to the number of places in the bottom number and put that many decimal places in the answer.
Division
of a number by a decimal number smaller than one means finding out how many of those small decimal parts there are in that number. In problems involving the division of a whole number by a decimal smaller than one, the answer is always larger than the number being divided.
6 divided by 2 means “How many twos in six?” 6 divided by 0.1 means “How many tenths in six?”
Asking how many tenths there are in six is similar to asking “How many dimes in six dollars?” There are ten dimes in one dollar, so there are 6 X ten (60) dimes in six dollars. Therefore, 6 divided by 0.1 = 60.
The division shift rule for tenths is just the opposite of the multiplication shift rule for tenths. Each place in the number being divided shifts one place to the left (gets one place larger). When a number is divided by 0.01 (one-hundredth), each place moves two places to the left, and so on.
To divide by a number with places smaller than one, write the problem in long division form. See Division (Long division).
Decimals and fractions
In mathematics, any number that can be written in the form of a fraction—that is, as one number divided by another—is called a rational number. All rational numbers can be written in the decimal system. When rational numbers are changed to the decimal form, the result is either a repeating decimal or a terminating decimal. A repeating decimal is one that goes on repeating the same number or series of numbers, such as 0.333 … and 0.148514851485…. The dots at the end show that the same pattern repeats over and over. A terminating decimal is one in which the division at some point comes out even and so the decimal number stops.
Any repeating or terminating decimal can be written as a rational number—that is, in fraction form. But some numbers, called irrational numbers, never end and do not have any repeating patterns. Thus, they are impossible to exactly write out with decimal numbers. Two examples of irrational numbers are the square root of 2 and pi. The square root of 2 is the number which, when multiplied times itself, gives two. It is between 1.4142135 and 1.4142136. Pi is the number you get when you divide the circumference (distance around) of any circle by its diameter (the distance across it through its center). Pi is between 3.1415926 and 3.1415927. See Circle; Pi.
Changing fractions to decimals.
To change a number from the fraction form to the decimal form, just carry out the division that the fraction represents. Divide the numerator (top number) by the denominator (bottom number). This division will always give either a terminating decimal or a repeating decimal, because the remainder will eventually be 0 or will repeat an earlier remainder. A repeating decimal can be rounded off to any place.
Changing decimals to fractions.
To change a decimal to a regular fraction, write the number without any decimal point as the top of the regular fraction. For the bottom of the regular fraction, write the numeral 1 followed by as many zeros as there are places to the right of the decimal point in the decimal. This bottom number is the value of the last place in the decimal.
The exact procedure for changing a repeating decimal to a fraction varies with the form of the repeating decimal. If the repeating decimal starts in the tenths place and has no whole numbers in it, the fraction form of the repeating decimal has the repeating pattern as the top number and as many 9’s as there are places in the repeating pattern for the bottom number.
In some repeating decimals, there are nonrepeating numbers before the pattern of the decimal starts repeating. If these nonrepeating numbers are whole numbers, the top number for the new fraction is made by writing the repeating decimal up to the first repeat of the pattern and subtracting from this the whole number. The bottom number is as many 9’s as there are in the repeating pattern.
In some repeating decimals, the nonrepeating numbers are to the right of the decimal point. In such cases, the top number is made by writing the nonrepeating part followed by one repeat of the pattern. The nonrepeating part is then subtracted from this number. The bottom number is made by writing as many 9’s as there are places in the repeating pattern, followed by as many 0’s as there are nonrepeating places to the right of the decimal point.
History
Invention of the decimal system.
The decimal system was invented in India, but no one knows exactly when or where. As early as the 300’s and 200’s B.C., a base-ten number system was written in Brahmi, a script used for writing the Sanskrit language of the Hindus. The Hindu-Arabic numerals 1, 2, 3, 4, 5, 6, 7, 8, and 9 are based on the Brahmi symbols for the numbers one through nine. However, the Brahmi number system also used special symbols for ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety, one hundred, and one thousand.
By about 600 A.D., all the extra symbols had been dropped from the system. All numbers were written by using just the symbols for one through nine. The place in which a symbol was written told its value. However, there was a problem with this place-value system. If a given place was empty, some new symbol was needed to hold that place empty so that all the other symbols would stay in their correct places. The first record of the use of such a new symbol in the Brahmi system is from the 800’s. This symbol is what we now call zero. The Maya of Central America, who also invented a place-value system, used a zero before A.D. 300 (see Zero).
Spread of the decimal system.
During the 700’s, Arabs learned about the decimal system of the Hindus. During the next 300 years, they spread it throughout their empire—through the Middle East to northern Africa, and into Spain.
The system was introduced into Europe by several people, including Pope Sylvester II about 1000 and Leonardo Fibonacci, an Italian mathematician, in 1202. At that time, however, new learning in books did not reach large numbers of people, chiefly because books were copied by hand and were therefore scarce. But soon after the printing press was invented in the mid-1400’s, several arithmetic books that explained the use of the decimal system were published in England, France, Germany, the Netherlands, and other countries. Schools opened in many countries to teach decimal-system calculations, and the system was taught in universities.
The widespread interest in the decimal system was due largely to the number of advantages the system had over Roman numerals, which most people in Europe used at the time (see Roman numerals). Calculations are difficult with Roman numerals, so people used little round pieces of metal as counters. They performed their calculations with such devices as calculating boards or calculating cloths that had vertical columns drawn on them to make places for the counters. But because of the place-value nature of the decimal system, calculations could be performed with decimal numbers by using just a pen and paper. It also takes less space to write a number in the decimal system. Larger numbers can be written without new symbols. Another advantage is that numbers smaller than one can be written in the decimal system, and these numbers can be used in calculations.
Use of decimals smaller than one.
The first books written in Europe about the decimal system did not say anything about decimals smaller than one. Such decimals were used in China many centuries before they were introduced into Europe and were used by Arab astronomers by at least the early 1400’s. Some European mathematicians and astronomers had also known about decimals smaller than one, but the first evidence of their use by merchants and ordinary people appeared in a Flemish pamphlet called De Thiende, published in the Netherlands in 1585. John Napier, a Scottish baron who studied mathematics, published in 1619 an easier way to write decimals smaller than one, and we still use his method today. Such decimals gradually began to be used with the rest of the decimal system.
In the late 1700’s, France adopted a metric system of weights and measures and a new money system. Both were based on the decimal system (see Metric system). They enabled many more people to use decimals less than one. By the late 1900’s, nearly every country had converted, or planned to convert, to the metric system.
The importance of decimals smaller than one was further increased in the late 1970’s and early 1980’s by the development of inexpensive electronic calculators. Many problems that were previously solved with fractions could be done more easily with calculators that use the decimal system.
