What does base ten mean: Base-10 system ~ A Maths Dictionary for Kids Quick Reference by Jenny Eather

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Math Clip Art—Base Ten Blocks—35

This is a collection of clip art images that you can use to model different numbers. This complete collection models two- and three-digit numbers using base 10 blocks. To see the complete collection of these clip art images, click on this link


The following review provides basic information about place value.

The Base 10 Number System

When you first learned to count, you started at zero or one and counted to nine.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

After nine, the numbers started repeating.

10, 11, 12, 13, 14, 15, 16, 17, 18, 19

After 19, the pattern of numbers continues. In fact, counting numbers involves just using the digits from 0 to 9 in different combinations.

Why is that?

Our numbering system is called a base 10 system. This means there are ten digits, which you know as 0 to 9. All numbers in a base 10 system just use these digits.

Counting in a Base 10 System

When you look at a number, each digit represents a certain place value. Take a look at this three-digit number

The digit 3 is in the ones place. The digit 2 is in the tens place. The digit 1 is in the hundreds place.

The digit 3 is in the ones place. The digit 2 is in the tens place. The digit 1 is in the hundreds place.

For whole numbers, place value increases 10-fold.


Place Value in the Base 10 System

Place value determines the size of a number. Take a look at this number:


In the ones place is the digit 3, but int he tens place is the digit 2. Don’t let the single digit fool you. The 2 in the tens place has a value of 20.

Now look at this number:


In the ones place is the digit 2. In the tens place is the digit 4. In the hundreds place is 5. If we were to separate these numbers, we would see this:




So, in going from left to right starting at the decimal point, each place value is ten times the value of the place value to the right.

If you’re given a description of a number using just place value information, you can write a number. Here’s an example. 

The digit in the ones place is 5. The digit in the tens place is 3. The digit in the hundreds place is 9. What is the number?

Using this description we write the following number: 935.

Number Systems with Other Bases

You’ve seen how the base 10 number system works, but why do we use a base 10 system? Are there other number systems?

Let’s address the first question. The reason we use a base 10 system is probably because we have ten fingers to count with. Do you think it’s a coincidence that our base 10 numbering system aligns with our 10 fingers to count?

But we have 10 fingers and 10 toes. Are there number systems that use base 20. Yes!

The Mayan culture used a base 20 system for counting. These are twenty symbols used to count.

This numbering system uses just three symbols in different combinations: dots, dashes, and the shell. The shell represents zero. The dash represents an increment of five. The dot represents a value of one. Different combinations of these symbols can model any number.

Another, even more popular system is the binary system, which is used extensively with computers. With the binary system, there are only two digits: 0 and 1. This combination of digits can be used any number  equivalent to our base 10 numbers. Here are the first 10 numbers in binary and base 10.

Base 10 Binary
0 0
1 1
2 10
3 11
4 100
5 101
6 110
7 111
8 1000
9 1001

Why do computers use a binary, or base 2, system? What is the advantage of a binary system?

Computer circuitry and connections involve a lot of on-off switching. Many computer commands are basically combinations of these on-off switches. A binary system is ideal for modeling an on-off system. If you let 0 mean off and 1 mean on, then a binary code can be used to model not only numbers but any computer state.

Each number system has its own rules for writing numbers. Once you learn the rules, then you can write any number in any number system.

Number Models: Base Ten Blocks

One way to model base ten numbers is to use number models, and one of the best ones to use is to use base ten blocks. Base ten blocks are a visual model to represent numbers. Here is what these models look like.

For each of these models, place the correct number of blocks to model the digit in that place value.

This is a model for the number 24. There are two tens blocks in the tens place and four ones blocks in the ones place.

Here is a number to the hundreds place.

To model 250, put two hundreds blocks in the hundreds place, five tens blocks in the tens place, and no blocks in the ones place.

Note: The download is a PNG file.

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Big Idea: The Base Ten System

Base-Ten Number System

By: Miriam Coyle

The National Council of Teachers of Mathematics states that “In prekindergarten through grade 2 all students should use multiple models to develop initial understandings of place value and the base-ten number system” (p. 78). It also expects that “In grades 3-5 all students should understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals” (p 148).

So, if we as teachers are expected to teach our students about the base-ten number system, what exactly is it and what is the best way to teach it for understanding?

Numeration Systems

A numeration system is a way of representing numbers using symbols. Each number must be represented by a different symbol. Otherwise, confusion as to which number was intended would be a frequent occurrence. Particular symbols used to represent numbers are known as the digits of the system. Under ideal circumstances a numeration system will: “represent a useful set of numbers (e.g. all whole numbers, integers, or real numbers), give every number represented a unique representation (or at least a standard representation), reflect the algebraic and arithmetic structure of the numbers” (Wikipedia, the free encyclopedia-numeral system).

Place-value numeration systems have limited numbers of digits which need to be repeated in order to represent larger numbers. The base of a system is the highest number to which you can count without repeating any previous digit. The number of digits in the place-value numeration system is the same number as the base of that system and must include a place-holder. The value of a number is determined by its position. The base number is how the numbers are grouped. A base of four would group by fours. The position to the right would hold up to four of the number before it would then be grouped as numbers of up to four times four, and then groups of four times four times four, and so forth.


The base-ten numeration system is the system that most of the world uses today. No one knows exactly why the number ten was chosen as the base for this system, but it is theorized that it is due to the number of fingers that humans possess. Fingers are very easily assessable as counting tools. The base-ten numeration system consists of ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and groups numbers into ten. Zero serves as the placeholder. It holds a position when there is nothing to be placed in a particular position. The value of a numeral in this system is determined by multiplication. Starting from the position directly to the right of the decimal point and going left, the value is (1 x the numeral), (10 x the numeral), (100 x the numeral), and so on. Each position to the left is ten times the positional value of the previous position. Each position to the right of the decimal point is 1/10the position to the right. For example, a four in the second spot to the left of the decimal point would be (10 x 4), or in other words, would have the value of 40. A 4 in the second position to the right of the decimal point would have the value of 4/100 or 0.04. This is known as the Multiplicative Principle. The numbers of the base-ten system can be added to determine the total value of the number. This is called the Additive Principle.

(100 x 4) + (10 x 5) + (1 x 6) = (400) + (50) + (6) = 456

Teaching for Understanding

When students do not have a solid grasp or understanding of the base-ten number system, it can prevent students from developing sophisticated counting strategies. They may still count by one, instead of by groups of ten or hundreds. It can also cause problems when students begin to use more complex mathematics, such as the traditional algorithm of long division. If a student does not understand that the 2 in 27 actually represents 20, then all sorts of problems can begin to surface in more advanced mathematics, resulting in frustration for both student and teacher.

    • Teachers need to help their students see numbers as groups of tens and ones. A student’s ability to count to a large number does not show understanding of the ability to group numbers into tens and ones. A great way to teach and give students experience with grouping in this way is through the use of ten-frames. Ten frames are a way for students to keep track and visually see what makes a ten, and then see what is left.

Notice the grouping of the tens. It is obvious to students that there are two groups of ten with some left over, in this case three. Students can readily see that 23 is made up of two tens and three ones. This also proves beneficial in helping students easily add and subtract ten. By adding a ten-frame, students would easily identify that there are now 33 counters, and by removing a ten-frame they would quickly identify 13 counters.

Students have also been shown to benefit from building a 100 chart and then finding patterns within the chart. This activity “helps to strengthen skip counting, and reinforces patterns that occur in the base-ten system” (Bahr, p. 110).

Another way to help students develop a grasp of base-ten is by the frequent use of base-ten blocks. Base-ten blocks consist of a flat (one hundred), a long (ten), and a unit (one). Students can visually see the groupings of a number when using base-ten blocks.

Students can easily see the different groupings of ten that go into a number when using base-ten blocks. In this case of the number 147, students can see that there is 1 hundred (100), 4 tens (40), and 7 ones (7). They can also see that the number 147 is made by adding all of these group values together (100+40+7=147) and should practice writing numbers in this expanded notation to aid in conceptual understanding of the base-ten numeration system.

Base-ten blocks are also great tools for helping students develop understanding of carrying and borrowing and addition and subtraction. Mathematicians prefer to refer to these operations as composing (carrying) and decomposing (borrowing) because these terms are more representative of what is actually taking place. When composing takes place, it is because there are now more than ten in a given position. In order to eliminate the over abundance, the groups of ten are then traded for one of the next order of grouping, in a one to one correspondence (one group of ten= one in the next position).

An example of using base-ten blocks to visually represent addition: If Mary had 9 pieces of gum and John gave her 6 more, then Mary now has a group of ten ones and five ones. She would then trade the ten ones for one ten-block (long), moving the numbers to the next position (composing), just like putting a one over the tens spot in the traditional algorithm.

An example of using base-ten blocks to visually represent subtraction: If Mary had 15 pieces of gum (represented by one ten-block and 5 units) and gave 7 to John, she would need to take the ten and break it apart into ones in order to give enough to John. This is comparable to when with the traditional algorithm of subtraction, borrowing takes place (decomposing).

Another great tool giving student exposure and practice with the base-ten system is through the use of The National Library of Virtual Manipulatives. Students can practice adding and subtracting with virtual base-ten blocks, and can develop better understanding adding and subtracting using the decimal system using the virtual base-ten blocks as well.

Most of students’ confusion and frustration with math in the upper elementary grades is grounded in a lack of proper base-ten understanding. By giving students proper exposure to the use of manipulatives that promote base-ten in the younger grades, much of the heartache that goes with lack of understanding can be eliminated. Students who do show lack of understanding in the upper elementary grades should be taken back to the use of base-ten manipulatives in order to make the connections that should have been made previously. All students benefit from their use and will wean themselves from them when they are developmentally ready.


Bahr, Damon L., and de Garcia, Lisa Ann. (2010). Elementary mathematics is anything but elementary. Belmont, CA: Wadsworth.

Investigations in Number, Data, and Space. (2007). Math content by strand: The base-ten number system: Place value. Retrieved April 18, 2009, from http://investigations.terc.edu/library/curric-math/placevalue_2ed.pdf

KarolYeats.Com. (2008). Place value-developing understanding of numeration. Retrieved April 18, 2009, from http://www.karolyeatts.com        /Math/Numeration%20and%20Place%20Value.pdf

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Key Curriculum Press.

Science of Everyday Things. (2009). Numeration systems. Retrieved April 18, 2009, from http://www.scienceclarified.com/Mu-Oi/Numeration-Systems.html

Wikipedia, the Free Encyclopedia. (2009). Numeral system. Retrieved April 18, 2009, from http://en.wikipedia.org/wiki/Numeral_system

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DES (DES function) — Microsoft Support

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This article describes the formula syntax and usage of formula decimal
in Microsoft Excel.


Converts the textual representation of a number with the specified base to a decimal number.



The DEC function syntax has the following arguments.


  • The length of the text argument string must not exceed 255 characters.

    953 may result in loss of accuracy.

  • The base of the number system must be greater than or equal to 2 (binary) or less than or equal to 36 (36).
    Base over 10 uses numeric values ​​0-9 and letters A-Z. For example, base 16 (hexadecimal) uses the numbers 0-9 and the letters A-F, while base 36 uses the numbers 0-9 and the letters A-Z.

  • If at least one of the arguments is out of bounds, decIMAL may return #NUM! or the #VALUE! error value.


Copy the sample data from the following table and paste it into cell A1 of a new Excel sheet. To display formula results, select them and press F2 followed by ENTER. Change the width of the columns, if necessary, to see all the data.







Converts the hexadecimal value FF (base 16) to its equivalent decimal value (base 10). The result is 255.


«F» in position 15 hexadecimal. Since all number systems start at 0, the 16th character in the hexadecimal system will be in the 15th position. The formula below shows how a number is converted to decimal.

The HEX DEC function in cell C3 confirms this result.

=HEX DEC(«ff»)




Converts the binary value 111 (base 2) to its equivalent decimal value (base 10). Result — 7.


«1» in position 1 in binary. The formula below shows how a number is converted to decimal.

The BIN DEC function in cell C6 confirms this result.



Converts base 36 «zap» to its equivalent decimal value (45745).


«Z» is in position 35, «a» is in position 10, and «p» is in position 25. The formula below shows how a number is converted to decimal.


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Logarithms. Base of the logarithm. natural logarithm. Logarithm 10.

A Brief History of the Logarithm

The logarithm has many uses in science and engineering.

The natural logarithm has a constant base and its use is widespread in discrete mathematics, especially in calculus. The binary logarithm uses a base and is prominent in computer science. Logarithms were introduced by John Napier at the beginning of the 17th century as a means of simplifying calculations. They have been readily adopted by scientists, engineers, and others to make calculations easier. The modern concept of logarithms comes from Leonhard Euler, who related them to the exponential function in the 18th century 93.\)

ODZ logarithm

ODZ (range of acceptable values) of the logarithm is the set of all real numbers for which this function is defined. For a logarithmic function with base a, the LPV is defined as follows:

x > 0 (if a > 1) or x < 0 (if 0 < a < 1)

That is, the argument of the logarithm must be positive if the base greater than 1, and negative if the base is less than 1.

Logarithm range — main:

  • Argument and base cannot be zero and negative numbers.
  • The base cannot be equal to one, since a unit to any degree is still a unit.
  • The number b can be anything.
  • ODZ of the logarithm \(log_a x = b ⇒ x > 0, a > 0, a ≠ 1\).

Types of logarithms

There are two main types of logarithms: ordinary (or decimal) logarithms and natural logarithms. 91 = e.

Ordinary and natural logarithms are related to each other by the formula:

log(y) = ln(y) / ln(10)

where ln(10) ≈ 2.3026.

There are also logarithms with other bases (for example, the base 2 logarithm), but they are less commonly used in practical calculations.

Decimal logarithms

Decimal logarithms are logarithms whose base is \(10\). Example \(log_{10}10 =1\),

Log 10 100 =2. They are written as \(lg 10 = 1\), \(lg 100 = 2.\)

Natural logarithm

Natural logarithm is a logarithm whose base is \(e\). What does \(e\) mean? This is an irrational number, an infinite non-periodic decimal number, a mathematical constant to be remembered:


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