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*Thumbnail image -* *Binary numbers, using just 0’s and 1’s, are the language of computers.*^{[1]}

The “Dots and Boxes” approach to place value used in this part (and throughout this book) comes from James Tanton, and is used with his permission. See his development of these and other ideas at __http://gdaymath.com/__.

- Images and Videos on Pixabay are released under Creative Commons CC0.
__↵__

- Identifying tens and ones from 2 digit numbers
- Combining tens and ones into 2 digit numbers
- Identifying a digit's place value (tens, ones)
- Building a 2 digit number with missing addends
- Write 2 digit numbers in expanded form
- Write 2 digit numbers in normal form

- Building a 3-digit number from the parts
- Missing place values in 3-digit numbers
- Write 3-digit numbers in expanded form
- Write 3-digit numbers in normal form
- Hundreds, tens & ones - identify the underlined digit
- Comparing and ordering numbers up to 100 and 1,000

## Contents

Today, the base-10 (decimal) system, which is presumably motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past, and some continue to be used today. For example, the Babylonian numeral system, credited as the first positional numeral system, was base-60. However it lacked a real 0. Initially inferred only from context, later, by about 700 BC, zero came to be indicated by a "space" or a "punctuation symbol" (such as two slanted wedges) between numerals. [1] It was a placeholder rather than a true zero because it was not used alone. Nor was it used at the end of a number. Numbers like 2 and 120 (2×60) looked the same because the larger number lacked a final placeholder. Only context could differentiate them.

The polymath Archimedes (ca. 287–212 BC) invented a decimal positional system in his Sand Reckoner which was based on 10 8 [2] and later led the German mathematician Carl Friedrich Gauss to lament what heights science would have already reached in his days if Archimedes had fully realized the potential of his ingenious discovery. [3]

Before positional notation became standard, simple additive systems (sign-value notation) such as Roman numerals were used, and accountants in ancient Rome and during the Middle Ages used the abacus or stone counters to do arithmetic. [4]

Counting rods and most abacuses have been used to represent numbers in a positional numeral system. With counting rods or abacus to perform arithmetic operations, the writing of the starting, intermediate and final values of a calculation could easily be done with a simple additive system in each position or column. This approach required no memorization of tables (as does positional notation) and could produce practical results quickly. For four centuries (from the 13th to the 16th) there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the latter continues to be used in Japan and other Asian countries. [* citation needed *]

After the French Revolution (1789–1799), the new French government promoted the extension of the decimal system. [5] Some of those pro-decimal efforts—such as decimal time and the decimal calendar—were unsuccessful. Other French pro-decimal efforts—currency decimalisation and the metrication of weights and measures—spread widely out of France to almost the whole world.

### History of positional fractions Edit

J. Lennart Berggren notes that positional decimal fractions were used for the first time by Arab mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century. [6] The Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, but did not develop any notation to represent them. [7] The Persian mathematician Jamshīd al-Kāshī made the same discovery of decimal fractions in the 15th century. [6] Al Khwarizmi introduced fractions to Islamic countries in the early 9th century his fraction presentation was similar to the traditional Chinese mathematical fractions from Sunzi Suanjing. [8] This form of fraction with numerator on top and denominator at bottom without a horizontal bar was also used by 10th century Abu'l-Hasan al-Uqlidisi and 15th century Jamshīd al-Kāshī's work "Arithmetic Key". [8] [9]

The adoption of the decimal representation of numbers less than one, a fraction, is often credited to Simon Stevin through his textbook De Thiende [10] but both Stevin and E. J. Dijksterhuis indicate that Regiomontanus contributed to the European adoption of general decimals: [11]

European mathematicians, when taking over from the Hindus, *via* the Arabs, the idea of positional value for integers, neglected to extend this idea to fractions. For some centuries they confined themselves to using common and sexagesimal fractions. This half-heartedness has never been completely overcome, and sexagesimal fractions still form the basis of our trigonometry, astronomy and measurement of time. ¶ . Mathematicians sought to avoid fractions by taking the radius *R* equal to a number of units of length of the form 10 n and then assuming for *n* so great an integral value that all occurring quantities could be expressed with sufficient accuracy by integers. ¶ The first to apply this method was the German astronomer Regiomontanus. To the extent that he expressed goniometrical line-segments in a unit *R*/10 n , Regiomontanus may be called an anticipator of the doctrine of decimal positional fractions. [11] : 17,18

In the estimation of Dijksterhuis, "after the publication of De Thiende only a small advance was required to establish the complete system of decimal positional fractions, and this step was taken promptly by a number of writers . next to Stevin the most important figure in this development was Regiomontanus." Dijksterhuis noted that [Stevin] "gives full credit to Regiomontanus for his prior contribution, saying that the trigonometric tables of the German astronomer actually contain the whole theory of 'numbers of the tenth progress'." [11] : 19

A key argument against the positional system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing (e.g.) 100 into 5100, or 100 into 1000. Modern cheques require a natural language spelling of an amount, as well as the decimal amount itself, to prevent such fraud. For the same reason the Chinese also use natural language numerals, for example 100 is written as 壹佰, which can never be forged into 壹仟(1000) or 伍仟壹佰(5100).

Many of the advantages claimed for the metric system could be realized by any consistent positional notation. Dozenal advocates say duodecimal has several advantages over decimal, although the switching cost appears to be high.

### Base of the numeral system Edit

The highest symbol of a positional numeral system usually has the value one less than the value of the radix of that numeral system. The standard positional numeral systems differ from one another only in the base they use.

The radix is an integer that is greater than 1, since a radix of zero would not have any digits, and a radix of 1 would only have the zero digit. Negative bases are rarely used. In a system with more than | b |

It is important that the radix is finite, from which follows that the number of digits is quite low. Otherwise, the length of a numeral would not necessarily be logarithmic in its size.

(In certain non-standard positional numeral systems, including bijective numeration, the definition of the base or the allowed digits deviates from the above.)

In standard base-ten (decimal) positional notation, there are ten decimal digits and the number

In standard base-sixteen (hexadecimal), there are the sixteen hexadecimal digits (0–9 and A–F) and the number

14 B 9 h e x = ( 1 × 16 3 ) + ( 4 × 16 2 ) + ( B × 16 1 ) + ( 9 × 16 0 ) ( = 5305 d e c ) , ** imes 16^<1>)+(9 imes 16^<0>)qquad (=5305_ >),>**

** **

**where B represents the number eleven as a single symbol.**

### Notation Edit

When describing base in mathematical notation, the letter *b* is generally used as a symbol for this concept, so, for a binary system, *b* equals 2. Another common way of expressing the base is writing it as a **decimal** subscript after the number that is being represented (this notation is used in this article). 1111011_{2} implies that the number 1111011 is a base-2 number, equal to 123_{10} (a decimal notation representation), 173_{8} (octal) and 7B_{16} (hexadecimal). In books and articles, when using initially the written abbreviations of number bases, the base is not subsequently printed: it is assumed that binary 1111011 is the same as 1111011_{2}.

The base *b* may also be indicated by the phrase "base-*b*". So binary numbers are "base-2" octal numbers are "base-8" decimal numbers are "base-10" and so on.

To a given radix *b* the set of digits <0, 1, . *b*−2, *b*−1> is called the standard set of digits. Thus, binary numbers have digits <0, 1> decimal numbers have digits <0, 1, 2, . 8, 9> and so on. Therefore, the following are notational errors: 52_{2}, 2_{2}, 1A_{9}. (In all cases, one or more digits is not in the set of allowed digits for the given base.)

### Exponentiation Edit

Positional numeral systems work using exponentiation of the base. A digit's value is the digit multiplied by the value of its place. Place values are the number of the base raised to the *n*th power, where *n* is the number of other digits between a given digit and the radix point. If a given digit is on the left hand side of the radix point (i.e. its value is an integer) then *n* is positive or zero if the digit is on the right hand side of the radix point (i.e., its value is fractional) then *n* is negative.

As an example of usage, the number 465 in its respective base *b* (which must be at least base 7 because the highest digit in it is 6) is equal to:

If the number 465 was in base-10, then it would equal:

If however, the number were in base 7, then it would equal:

10_{b} = *b* for any base *b*, since 10_{b} = 1×*b* 1 + 0×*b* 0 . For example, 10_{2} = 2 10_{3} = 3 10_{16} = 16_{10}. Note that the last "16" is indicated to be in base 10. The base makes no difference for one-digit numerals.

This concept can be demonstrated using a diagram. One object represents one unit. When the number of objects is equal to or greater than the base *b*, then a group of objects is created with *b* objects. When the number of these groups exceeds *b*, then a group of these groups of objects is created with *b* groups of *b* objects and so on. Thus the same number in different bases will have different values:

The notation can be further augmented by allowing a leading minus sign. This allows the representation of negative numbers. For a given base, every representation corresponds to exactly one real number and every real number has at least one representation. The representations of rational numbers are those representations that are finite, use the bar notation, or end with an infinitely repeating cycle of digits.

### Digits and numerals Edit

A *digit* is a symbol that is used for positional notation, and a *numeral* consists of one or more digits used for representing a number with positional notation. Today's most common digits are the decimal digits "0", "1", "2", "3", "4", "5", "6", "7", "8", and "9". The distinction between a digit and a numeral is most pronounced in the context of a number base.

A non-zero *numeral* with more than one digit position will mean a different number in a different number base, but in general, the *digits* will mean the same. [12] For example, the base-8 numeral 23_{8} contains two digits, "2" and "3", and with a base number (subscripted) "8". When converted to base-10, the 23_{8} is equivalent to 19_{10}, i.e. 23_{8} = 19_{10}. In our notation here, the subscript "_{8}" of the numeral 23_{8} is part of the numeral, but this may not always be the case.

Imagine the numeral "23" as having an ambiguous base number. Then "23" could likely be any base, from base-4 up. In base-4, the "23" means 11_{10}, i.e. 23_{4} = 11_{10}. In base-60, the "23" means the number 123_{10}, i.e. 23_{60} = 123_{10}. The numeral "23" then, in this case, corresponds to the set of base-10 numbers <11, 13, 15, 17, 19, 21, **23**, . 121, 123> while its digits "2" and "3" always retain their original meaning: the "2" means "two of", and the "3" three.

In certain applications when a numeral with a fixed number of positions needs to represent a greater number, a higher number-base with more digits per position can be used. A three-digit, decimal numeral can represent only up to **999**. But if the number-base is increased to 11, say, by adding the digit "A", then the same three positions, maximized to "AAA", can represent a number as great as **1330**. We could increase the number base again and assign "B" to 11, and so on (but there is also a possible encryption between number and digit in the number-digit-numeral hierarchy). A three-digit numeral "ZZZ" in base-60 could mean **215 999** . If we use the entire collection of our alphanumerics we could ultimately serve a base-*62* numeral system, but we remove two digits, uppercase "I" and uppercase "O", to reduce confusion with digits "1" and "0". [13] We are left with a base-60, or sexagesimal numeral system utilizing 60 of the 62 standard alphanumerics. (But see *Sexagesimal system* below.) In general, the number of possible values that can be represented by a d

The common numeral systems in computer science are binary (radix 2), octal (radix 8), and hexadecimal (radix 16). In binary only digits "0" and "1" are in the numerals. In the octal numerals, are the eight digits 0–7. Hex is 0–9 A–F, where the ten numerics retain their usual meaning, and the alphabetics correspond to values 10–15, for a total of sixteen digits. The numeral "10" is binary numeral "2", octal numeral "8", or hexadecimal numeral "16".

### Radix point Edit

The notation can be extended into the negative exponents of the base *b*. Thereby the so-called radix point, mostly ».«, is used as separator of the positions with non-negative from those with negative exponent.

Numbers that are not integers use places beyond the radix point. For every position behind this point (and thus after the units digit), the exponent *n* of the power *b* *n* decreases by 1 and the power approaches 0. For example, the number 2.35 is equal to:

### Sign Edit

If the base and all the digits in the set of digits are non-negative, negative numbers cannot be expressed. To overcome this, a minus sign, here »-«, is added to the numeral system. In the usual notation it is prepended to the string of digits representing the otherwise non-negative number.

### Base conversion Edit

For example: converting A10B_{Hex} to decimal (41227):

For the fractional part, conversion can be done by taking digits after the radix point (the numerator), and dividing it by the implied denominator in the target radix. Approximation may be needed due to a possibility of non-terminating digits if the reduced fraction's denominator has a prime factor other than any of the base's prime factor(s) to convert to. For example, 0.1 in decimal (1/10) is 0b1/0b1010 in binary, by dividing this in that radix, the result is 0b0.0 0011 (because one of the prime factors of 10 is 5). For more general fractions and bases see the algorithm for positive bases.

In practice, Horner's method is more efficient than the repeated division required above [14] [* better source needed *] . A number in positional notation can be thought of as a polynomial, where each digit is a coefficient. Coefficients can be larger than one digit, so an efficient way to convert bases is to convert each digit, then evaluate the polynomial via Horner's method within the target base. Converting each digit is a simple lookup table, removing the need for expensive division or modulus operations and multiplication by x becomes right-shifting. However, other polynomial evaluation algorithms would work as well, like repeated squaring for single or sparse digits. Example:

### Terminating fractions Edit

The numbers which have a finite representation form the semiring

### Infinite representations Edit

#### Rational numbers Edit

The representation of non-integers can be extended to allow an infinite string of digits beyond the point. For example, 1.12112111211112 . base-3 represents the sum of the infinite series:

Since a complete infinite string of digits cannot be explicitly written, the trailing ellipsis (. ) designates the omitted digits, which may or may not follow a pattern of some kind. One common pattern is when a finite sequence of digits repeats infinitely. This is designated by drawing a vinculum across the repeating block:

This is the repeating decimal notation (to which there does not exist a single universally accepted notation or phrasing). For base 10 it is called a repeating decimal or recurring decimal.

An irrational number has an infinite non-repeating representation in all integer bases. Whether a rational number has a finite representation or requires an infinite repeating representation depends on the base. For example, one third can be represented by:

For integers *p* and *q* with *gcd*(*p*, *q*) = 1, the fraction *p*/*q* has a finite representation in base *b* if and only if each prime factor of *q* is also a prime factor of *b*.

For a given base, any number that can be represented by a finite number of digits (without using the bar notation) will have multiple representations, including one or two infinite representations:

#### Irrational numbers Edit

A (real) irrational number has an infinite non-repeating representation in all integer bases.

Examples are the non-solvable *n*th roots

which are transcendental. The number of transcendentals is uncountable and the sole way to write them down with a finite number of symbols is to give them a symbol or a finite sequence of symbols.

### Decimal system Edit

In the decimal (base-10) Hindu–Arabic numeral system, each position starting from the right is a higher power of 10. The first position represents 10 0 (1), the second position 10 1 (10), the third position 10 2 ( 10 × 10 or 100), the fourth position 10 3 ( 10 × 10 × 10 or 1000), and so on.

Fractional values are indicated by a separator, which can vary in different locations. Usually this separator is a period or full stop, or a comma. Digits to the right of it are multiplied by 10 raised to a negative power or exponent. The first position to the right of the separator indicates 10 −1 (0.1), the second position 10 −2 (0.01), and so on for each successive position.

As an example, the number 2674 in a base-10 numeral system is:

(2 × 10 3 ) + (6 × 10 2 ) + (7 × 10 1 ) + (4 × 10 0 )

(2 × 1000) + (6 × 100) + (7 × 10) + (4 × 1).

### Sexagesimal system Edit

The sexagesimal or base-60 system was used for the integral and fractional portions of Babylonian numerals and other mesopotamian systems, by Hellenistic astronomers using Greek numerals for the fractional portion only, and is still used for modern time and angles, but only for minutes and seconds. However, not all of these uses were positional.

Using a digit set of digits with upper and lowercase letters allows short notation for sexagesimal numbers, e.g. 10:25:59 becomes 'ARz' (by omitting I and O, but not i and o), which is useful for use in URLs, etc., but it is not very intelligible to humans.

In the 1930s, Otto Neugebauer introduced a modern notational system for Babylonian and Hellenistic numbers that substitutes modern decimal notation from 0 to 59 in each position, while using a semicolon () to separate the integral and fractional portions of the number and using a comma (,) to separate the positions within each portion. [16] For example, the mean synodic month used by both Babylonian and Hellenistic astronomers and still used in the Hebrew calendar is 2931,50,8,20 days, and the angle used in the example above would be written 1025,59,23,31,12 degrees.

### Computing Edit

In computing, the binary (base-2), octal (base-8) and hexadecimal (base-16) bases are most commonly used. Computers, at the most basic level, deal only with sequences of conventional zeroes and ones, thus it is easier in this sense to deal with powers of two. The hexadecimal system is used as "shorthand" for binary—every 4 binary digits (bits) relate to one and only one hexadecimal digit. In hexadecimal, the six digits after 9 are denoted by A, B, C, D, E, and F (and sometimes a, b, c, d, e, and f).

The octal numbering system is also used as another way to represent binary numbers. In this case the base is 8 and therefore only digits 0, 1, 2, 3, 4, 5, 6, and 7 are used. When converting from binary to octal every 3 bits relate to one and only one octal digit.

Hexadecimal, decimal, octal, and a wide variety of other bases have been used for binary-to-text encoding, implementations of arbitrary-precision arithmetic, and other applications.

*For a list of bases and their applications, see list of numeral systems.*

### Other bases in human language Edit

Base-12 systems (duodecimal or dozenal) have been popular because multiplication and division are easier than in base-10, with addition and subtraction being just as easy. Twelve is a useful base because it has many factors. It is the smallest common multiple of one, two, three, four and six. There is still a special word for "dozen" in English, and by analogy with the word for 10 2 , *hundred*, commerce developed a word for 12 2 , *gross*. The standard 12-hour clock and common use of 12 in English units emphasize the utility of the base. In addition, prior to its conversion to decimal, the old British currency Pound Sterling (GBP) *partially* used base-12 there were 12 pence (d) in a shilling (s), 20 shillings in a pound (£), and therefore 240 pence in a pound. Hence the term LSD or, more properly, £sd.

The Maya civilization and other civilizations of pre-Columbian Mesoamerica used base-20 (vigesimal), as did several North American tribes (two being in southern California). Evidence of base-20 counting systems is also found in the languages of central and western Africa.

Remnants of a Gaulish base-20 system also exist in French, as seen today in the names of the numbers from 60 through 99. For example, sixty-five is *soixante-cinq* (literally, "sixty [and] five"), while seventy-five is *soixante-quinze* (literally, "sixty [and] fifteen"). Furthermore, for any number between 80 and 99, the "tens-column" number is expressed as a multiple of twenty. For example, eighty-two is *quatre-vingt-deux* (literally, four twenty[s] [and] two), while ninety-two is *quatre-vingt-douze* (literally, four twenty[s] [and] twelve). In Old French, forty was expressed as two twenties and sixty was three twenties, so that fifty-three was expressed as two twenties [and] thirteen, and so on.

In English the same base-20 counting appears in the use of "scores". Although mostly historical, it is occasionally used colloquially. Verse 10 of Pslam 90 in the King James Version of the Bible starts: "The days of our years are threescore years and ten and if by reason of strength they be fourscore years, yet is their strength labour and sorrow". The Gettysburg Address starts: "Four score and seven years ago".

The Irish language also used base-20 in the past, twenty being *fichid*, forty *dhá fhichid*, sixty *trí fhichid* and eighty *ceithre fhichid*. A remnant of this system may be seen in the modern word for 40, *daoichead*.

The Welsh language continues to use a base-20 counting system, particularly for the age of people, dates and in common phrases. 15 is also important, with 16–19 being "one on 15", "two on 15" etc. 18 is normally "two nines". A decimal system is commonly used.

The Inuit languages use a base-20 counting system. Students from Kaktovik, Alaska invented a base-20 numberal system in 1994 [17]

Danish numerals display a similar base-20 structure.

The Māori language of New Zealand also has evidence of an underlying base-20 system as seen in the terms *Te Hokowhitu a Tu* referring to a war party (literally "the seven 20s of Tu") and *Tama-hokotahi*, referring to a great warrior ("the one man equal to 20").

The binary system was used in the Egyptian Old Kingdom, 3000 BC to 2050 BC. It was cursive by rounding off rational numbers smaller than 1 to 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 , with a 1/64 term thrown away (the system was called the Eye of Horus).

A number of Australian Aboriginal languages employ binary or binary-like counting systems. For example, in Kala Lagaw Ya, the numbers one through six are *urapon*, *ukasar*, *ukasar-urapon*, *ukasar-ukasar*, *ukasar-ukasar-urapon*, *ukasar-ukasar-ukasar*.

North and Central American natives used base-4 (quaternary) to represent the four cardinal directions. Mesoamericans tended to add a second base-5 system to create a modified base-20 system.

A base-5 system (quinary) has been used in many cultures for counting. Plainly it is based on the number of digits on a human hand. It may also be regarded as a sub-base of other bases, such as base-10, base-20, and base-60.

A base-8 system (octal) was devised by the Yuki tribe of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight. [18] There is also linguistic evidence which suggests that the Bronze Age Proto-Indo Europeans (from whom most European and Indic languages descend) might have replaced a base-8 system (or a system which could only count up to 8) with a base-10 system. The evidence is that the word for 9, *newm*, is suggested by some to derive from the word for "new", *newo-*, suggesting that the number 9 had been recently invented and called the "new number". [19]

Many ancient counting systems use five as a primary base, almost surely coming from the number of fingers on a person's hand. Often these systems are supplemented with a secondary base, sometimes ten, sometimes twenty. In some African languages the word for five is the same as "hand" or "fist" (Dyola language of Guinea-Bissau, Banda language of Central Africa). Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until the secondary base is reached. In the case of twenty, this word often means "man complete". This system is referred to as *quinquavigesimal*. It is found in many languages of the Sudan region.

The Telefol language, spoken in Papua New Guinea, is notable for possessing a base-27 numeral system.

Interesting properties exist when the base is not fixed or positive and when the digit symbol sets denote negative values. There are many more variations. These systems are of practical and theoretic value to computer scientists.

Balanced ternary [20] uses a base of 3 but the digit set is < 1 ,0,1>instead of <0,1,2>. The " 1 " has an equivalent value of −1. The negation of a number is easily formed by switching the on the 1s. This system can be used to solve the balance problem, which requires finding a minimal set of known counter-weights to determine an unknown weight. Weights of 1, 3, 9, . 3 *n* known units can be used to determine any unknown weight up to 1 + 3 + . + 3 *n* units. A weight can be used on either side of the balance or not at all. Weights used on the balance pan with the unknown weight are designated with 1 , with 1 if used on the empty pan, and with 0 if not used. If an unknown weight *W* is balanced with 3 (3 1 ) on its pan and 1 and 27 (3 0 and 3 3 ) on the other, then its weight in decimal is 25 or 10 1 1 in balanced base-3.

10 1 1_{3} = 1 × 3 3 + 0 × 3 2 − 1 × 3 1 + 1 × 3 0 = 25.

The factorial number system uses a varying radix, giving factorials as place values they are related to Chinese remainder theorem and residue number system enumerations. This system effectively enumerates permutations. A derivative of this uses the Towers of Hanoi puzzle configuration as a counting system. The configuration of the towers can be put into 1-to-1 correspondence with the decimal count of the step at which the configuration occurs and vice versa.

Decimal equivalents | −3 | −2 | −1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Balanced base 3 | 1 0 | 1 1 | 1 | 0 | 1 | 1 1 | 10 | 11 | 1 1 1 | 1 1 0 | 1 1 1 | 10 1 |

Base −2 | 1101 | 10 | 11 | 0 | 1 | 110 | 111 | 100 | 101 | 11010 | 11011 | 11000 |

Factoroid | 0 | 10 | 100 | 110 | 200 | 210 | 1000 | 1010 | 1100 |

Each position does not need to be positional itself. Babylonian sexagesimal numerals were positional, but in each position were groups of two kinds of wedges representing ones and tens (a narrow vertical wedge ( | ) and an open left pointing wedge (<))—up to 14 symbols per position (5 tens (<<<<<) and 9 ones ( ||||||||| ) grouped into one or two near squares containing up to three tiers of symbols, or a place holder () for the lack of a position). [21] Hellenistic astronomers used one or two alphabetic Greek numerals for each position (one chosen from 5 letters representing 10–50 and/or one chosen from 9 letters representing 1–9, or a zero symbol). [22]

## Place Value Worksheets

Printable worksheets on place value, reading and writing large numbers, ordering numbers, expanded form, and digit values. Choose the number of digits below and you'll be forwarded to a page with a selection of worksheets.

This page offers a large collection of place value worksheets with 2-digit numbers. Skills include finding the value of the underlined digit, expanded form, comparing numbers, ordering, and reading numbers. *(Approx. level: Kindergarten - 1st)*

This page has a set of 3-digit PV worksheets and games. Skills covered include: digit values, standard/expanded form, reading and writing numbers, ordering, comparing, and place value blocks. *(Approx. level: 1st and 2nd Grades)*

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The printables on this page can be used for teaching and reviewing 5-digit place value. Practice finding the values of the underlines digits, write numbers in expanded notation, arranging numbers from least to greatest, and placing commas correctly. *(Approx. level: 3rd and 4th Grades)*

Practice place value up to hundred-thousands with these games and worksheets. Learn about digit values, writing number names, expanded and standard notation, and comparing large numbers. *(Approx. level: 3rd and 4th Grades)*

These PV activities have large 7, 8, 9, 10, or 12-digit numbers. Includes a variety of worksheets, games, cut-and-glue projects, and lesson ideas. *(Approx. level: 4th through 6th Grades)*

Place value to with tenths, hundredths, and thousandths. Also includes dollars and cents. *(Approx. level: 4th through 6th Grades)*

Rounding to the nearest ten, hundred, and/or thousand.

Download a variety of different 100 charts and 120 charts.

This index page will link you to worksheets on counting up to 10, counting up to 20, and counting up to 30. Also includes specific number trace-and-print activities for 1 through 30.

Learn about place value with these "special number" worksheets. Available in 1-digit, 2-digit, 3-digit, 4-digit, and 5-digit numbers.

## 2: Place Value

Numbers, such as 495,784, have six digits. Each digit is a different place value.

The first digit is called the hundred thousands' place. It tells you how many sets of one hundred thousand are in the number. The number 495,784 has four hundred thousands.

The second digit is the ten thousands' place. In this number there are nine ten thousands in addition to the four hundred thousands.

The third digit is the one thousands' place which is five in this example. Therefore there are four sets of one hundred thousand, nine sets of ten thousand, and five sets of one thousand in the number 495,784.

The fourth digit is called the hundreds' place. It tells how many sets of one hundred are in the number. The number 495,784 has seven hundreds in addition to the thousands.

The next digit is the tens place. This number has are eight tens in addition to the four hundred thousands, nine ten thousands, five thousands and seven hundreds.

The last or right digit is the ones' place which is four in this example. Therefore there are four sets of one hundred thousand, nine sets of ten thousand, five sets of one thousand, seven sets of one hundred, eight sets of ten, and four ones in the number 495,784.

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## 2: Place Value

Numbers, such as 84, have two digits. Each digit is a different place value.

The left digit is the tens' place. It tells you that there are 8 tens.

The last or right digit is the ones' place which is 4 in this example. Therefore, there are 8 sets of 10, plus 4 ones in the number 84.

**The number 24 could be represented by this table:**

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

21 | 22 | 23 | 24 |

The tens' place value of 2 in the number 24 is due to the presence of two full sets of 10.

The ones' place value of 4 in the number 24 is due to 4 units that are not included in a full set of 10.

## Place Value Charts | Millions

Download our colorful posters, blank charts and practice printable worksheets to assist 4th grade and 5th grade students master place values in large numbers. Our handy reference charts make for a great instructional aid to help students recognize the place value of numbers up to 9 digits. Procure some of these worksheets for free!

Place Value Charts: Millions

Distribute these vibrantly illustrated place value chart pdfs to teach place values in millions. The charts make for a great visual aid to help students grasp the millions place value concept with ease.

Place Value Charts: Ten Millions

The position of a digit within a number determines its value. Pin up these vivid posters that contain balloons, number keys, and picket fences in the classroom to aid learners identify the place values up to ten millions.

Place Value Charts: Hundred Millions

Download and print this array of charts to enhance a student's knowledge in comprehending place values for numbers up to 9 digits. They are sure to love the enchanting illustrations displayed here.

Circulate our blank charts among learners to test their knowledge on place values for 7-digit, 8-digit, and 9-digit numbers. Available in both color and monochrome, the charts make for a great evaluation tool.

Level 1: Place Values up to Hundred Millions

Employ this series of printable worksheets split into millions, ten millions, and hundred millions to provide children of grade 4 and grade 5 with abundant practice. Get them to master the concept of place values for numbers up to 9 digits.

Level 2: Place Values up to Hundred Millions - Mixed Review

Each pdf worksheet contains a variety of 7-digit, 8-digit and 9-digit numbers. Identify the place value for each number and write them in the appropriate place value boxes.

## Place

Place utility refers primarily to making goods or services physically available or accessible to potential customers. Examples of place utility range from a retail store's location to how easy a company's website or services are to find on the internet. Companies that have effective search engine optimization or SEO strategies can improve their place utility. SEO is the process of increasing a website's availability to internet users through their searches on the web.

Increasing convenience for customers can be a key element in attracting business. A company that offers easy access to technical assistance offers an added value in comparison to a similar company that does not offer a similar service. Making a product available in a wide variety of stores and locations is considered an added value since its more convenient. For example, Apple Inc. (AAPL) sells iPhones and laptops through its retail stores, but also offers its products through other electronics retailers, including Best Buy Co. Inc. (BBY).

## Rounding Numbers

When a child understands place value, she is usually able to round numbers to a specific place. The key is understanding that rounding numbers are essentially the same as rounding digits. The general rule is that if a digit is five or greater, you round up. If a digit is four or less, you round down.

So, to round the number *387* to the nearest tens place, for example, you would look at the number in the ones column, which is *7.* Since seven is greater than five, it rounds up to 10. You can't have a 10 in the ones place, so you would leave the zero in the ones place and round the number in the tens place, *8*, up to the next digit, which is *9*. The number rounded to the nearest 10 would be *390*. If students are struggling to round in this manner, review place value as discussed previously.