Ten base: What is Base Ten Numerals?

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What is Base Ten Numerals?

What Are Base-Ten Numerals?

The base of a number system tells us how many unique digits are used to form a number in that system. In the base ten system, ten unique digits (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) are used to represent any number. Number is an arithmetic value that we use for the purpose of counting. 

The ten digits used in the base 10 number system to form the base ten numerals are as follows:

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Base Ten Numerals Definition

Base ten numerals are written with a combination of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, such that each position has a value in powers of ten.

Base Ten Numerals Example: 150 and “One hundred and fifty” are both base ten numerals. The first is written using the digits. The latter is written in the word form.

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Base-Ten Place Value Chart

The place value of a digit tells us about the value of that digit, based on the position of the digit in the number. A place value table or chart helps to name each place value. The different groups of place values are called periods. Periods help us to determine the place values easily. The place value table is given below:

Consider the base ten numeral 5,219,037.

On placing the number on the place value table, we get 

We read it using the place value chart as “Five million two hundred nineteen thousand thirty-seven.”

How to Find the Place Value of a Digit?

Let’s find the place value of all the digits in 345,162 using the following steps.

  • Step 1: Place the digits in the place value chart. In this case, we get:
  • Step 2: Find the place where each digit is located. 
  • Step 3: Multiply the digit by its matching place value.

Place Value of 2 is $2 \times 1 = 2$

Place Value of 6 is $6 \times 10 = 60$

Place Value of 1 is $1 \times 100 = 100$

Place Value of 5 is $5 \times 1000 = 5000$

Place Value of 4 is $4 \times 10000 = 40000$

Place Value of 3 is $3 \times 100000 = 300000$

Expanded Form

Expanded form of a number is a way of expressing the number using its place values.  

Example: 

  1. Write the expanded form of 531,264.

Place value of $4 = 4$

Place value of $6 = 6 \times 10 = 60$

Place value of $2 = 2 \times 100 = 200$

Place value of $1 = 1 \times 1000 = 1000$

Place value of $3 = 3 \times 10000 = 30000$

Place value of $5 = 5 \times 100000 = 500000$

Expanded form of  $531,264 = 500000 + 30000 + 1000 + 200 + 6 $

  1. The number 4531 can be written in expanded form as

$4531 = 4000 + 500 + 30 + 1$

  1. Expanded form of 2836 

$2836 = (2 \times 1000) + (8 \times 100) + (3 \times 10) + (6 \times 1)$

         $= 2000 + 800 + 30 + 6$

Place Values in Decimals

Decimal numbers are the numerals whose whole part and fractional part is separated by a decimal point. 

Example 1: Consider the decimal 125.473

The place value chart shows the place values of the different digits in a decimal.

Example 2: Suppose we have a decimal number 345. 27. 

Place value of 7 will be $7 \times \frac{1}{100} = \frac{7}{100}$

Place value of 2 will be $2 \times \frac{1}{10} = \frac{2}{10}$

Place value of 5 will be $5 \times  1 = 5$

Place value of 7 will be $4 \times 10 = 40$

Place value of 7 will be $3 \times 100 = 300$

Expanded form of $345.27 = 300 + 40 + 5 + \frac{2}{10} + \frac{7}{100}$

$= 3 \times 100 + 4 \times 10 + 5 \times 1 + 2 \times \frac{1}{10} + 7 \times \frac{1}{100}$

$= 3 \text{hundred} + 4 \text{tens} + 5 \text{ones} + 2 \text{tenths} + 7 \text{hundredths}$

Fun Fact!

Base ten numerals form the basis for our counting system and thus, our monetary system. We use them in our daily lives for counting, measurements, and calculations.

Let’s Do It!

Instead of handing out place value worksheets to your child, ask them to use base ten blocks to show the place value. This will help them to not only help them to understand how base ten numerals are written, but also their place value.  

Conclusion

In this article, we learned about the base ten numeral form. Base ten numeral form is the expansion of a number using the place value. To read more such informative articles on other concepts, do visit our website. We, at SplashLearn, are on a mission to make learning fun and interactive for all students.

Solved Examples

1. Write 4326.18 in its expanded form. 

Solution: 

$4326.18 = (4 \times 1000) + (3 \times 100) + (2 \times 10) + (6 \times 1) + (1 \times \frac{1}{10}) + (8 \times \frac{1}{100})$

$4326.189 = 4000 + 300 + 20 + 6 + 0.1 + 0.08$

2. What is the place value of 6 in 75683?

Solution:

In the given base ten numeral, the digit 6 lies at hundreds place.

The place value of 6 in $75683 = 6 \times 100 = 600$

3. Give the place values of the digits 2 and 5 in the number 75.1824

Solution:

Place value of 2 in $75. 1824 = 2 \times 0.001 = 0.002$

Place Value of 5 in $75.1824 = 5 \times 1 = 5$

4. Write the place values of the digit 9 in the number 948.319?

Solution: 

Hundreds Tens Ones DecimalPoint Tenths Hundredths Thousandths
9 4 8 . 3 1 9

Place value of the digit 9 at hundreds place$ = 9 \times 100 = 900$

Place value of the digit 9 at thousandths place $= 9 \times 10000 = 90000$

5. Write the base ten numeral for the given expanded form. 

$4 \times 10000 + 5 \times 1000 + 7 \times 100 + 9 \times 10 + 2 \times 1 =$ ?

Solution: 

$4 \times 10000 + 5 \times 1000 + 7 \times 100 + 9 \times 10 + 2 \times 1$

$= 40000 + 5000 + 700 + 90 + 2$

$= 45792$

Practice Problems

1

Which of the following is the correct expanded form for 432.

987?

$400 + 30 + 2 + 9 + 8 + 0.007$

$400 + 30 + 2 + 0.9 + 8 + 0.007$

$400 + 3 + 2 + 0.9 + 0.8 + 0.007$

$400 + 30 + 2 + 0.9 + 0.08 + 0.007$

Correct answer is: $400 + 30 + 2 + 0.9 + 0.08 + 0.007$
The correct base ten numeral form for $432.987 = 400 + 30 + 2 + 0.9 + 0.08 + 0.007$

2

What is the place value of 0 in 7082?

100

1000

1

Correct answer is: 0
0 lies in the hundreds place.
Place value of $0 = 0 \times 100 = 0$

3

Find the difference between the place value of 2 in 652.729.

0.98

1.8

1.98

2.02

Correct answer is: 1.98
Place value of $2 = 2$ and $0.02$
Difference $= 2$ $–$ $0.02 = 1.98$

4

Write the base ten numeral. 4 hundreds $+ 2$ tens $+ 5$ ones $+ 6$ tenths $+ 7$ hundredths $+ 1$ thousandths

425.671

445671

42.5671

4256.71

Correct answer is: 425.671
4 hundreds $+ 2$ tens $+ 5$ ones $+ 6$ tenths $+ 7$ hundredths $+ 1$ thousandths
$= 400 + 20 + 5 + 0. 6 + 0.07 + 0.001$
$= 425.671$

5

What’s the base ten numeral represented by $9000 + 600 + 40 + 8$?

96.48

9648

9.648

0.9648

Correct answer is: 9648
$9000 + 600 + 40 + 8 = 9684$

Frequently Asked Questions

What is the binary number system?

The binary number system is simply the base-2 number system that uses only 2 digits (0 and 1) to form all the numbers.

What is the base ten block in math?

Base ten blocks are simply the visual representation of base ten place values that represent different place values (ones, tens, hundreds, etc) using blocks.

What is another name for the base-ten system?

The base ten system is also known as the decimal number system since the place values of digits are determined based on their position with reference to the decimal point.

Do all numbers use the base ten system?

Yes! Almost all countries across the world use the base ten number system.

    How to Use Base Ten Blocks

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    Exploring Base Ten Blocks

    Base Ten Blocks provide a spatial model of our base ten number system. Base Ten Blocks typically consist of four different concrete representations that are introduced in elementary math and utilized well into middle school.

    • Units = Ones; Measure 1 cm x 1 cm x 1 cm
    • Rods = Tens; Measure 1 cm x 1 cm x 10 cm
    • Flats = Hundreds; Measure 1 cm x 10 cm x 10 cm
    • Cubes = Thousands; Measure 10 cm x 10 cm x 10 cm

    Providing names based on the shape rather than on the value allows for the pieces to be renamed when necessary. For example, when studying decimals, a class can use the flat to represent a unit and establish the value of the other pieces from there.

    The size relationships among the blocks make them ideal for the investigation of number concepts. Initially students should explore independently with Base Ten Blocks before engaging in structured activities. As they move the blocks around to create designs and build structures, they may be able to discover on their own that it takes 10 smaller blocks to make one of the next larger blocks. Students’ designs and structures also lead them to employ spatial visualization and to work intuitively with the geometric concepts of shape, perimeter, area, and volume.

    Base Ten Blocks are especially useful in providing students with ways to physically represent the concepts of place value and addition, subtraction, multiplication, and division of whole numbers. By building number combinations with Base Ten Blocks, students ease into the concept of regrouping, or trading, and can see the logical development of each operation. The blocks provide a visual foundation and understanding of the algorithms students use when doing paper-and-pencil computation. Older students can transfer their understanding of whole numbers and whole-number operations to an understanding of decimals and decimal operations.

     

    Working with Base Ten Blocks

    Place-value mats, available in pads of 50, provide a means for students to organize their work as they explore the relationships among Base Ten Blocks and determine how groups of blocks can be used to represent numbers. Students may begin by placing unit blocks, one at a time, in the ones column on a mat. For each unit they place, they record the number corresponding to the total number of units placed (1, 2, 3…). They continue this process until they have accumulated 10 units, at which point they match their 10 units to 1 rod and trade those units for the rod, which they place in the tens column. Students continue in the same way, adding units one at a time to the ones column and recording the totals (11, 12, 13…) until it is time to trade for a second rod, which they place in the tens column (20). When they finally come to 99, there are 9 units and 9 rods on the mat. Adding one more unit forces two trades: first 10 units for another rod and then 10 rods for a flat (100). Then it is time to continue adding and recording units and making trades as needed as students work their way through the hundreds and up to the thousands. Combining the placing and trading of rods with the act of recording the corresponding numbers provides students with a connection between concrete and symbolic representations of numbers.

    Base Ten Blocks can be used to develop an understanding of the meaning of addition, subtraction, multiplication, and division. Modeling addition on a place-value mat provides students with a visual basis for the concept of regrouping.

    Subtraction with regrouping involves trading some of the blocks for smaller blocks of equal value so that the «taking away» can be accomplished. For example, to subtract 15 from 32, a student would trade one of the rods that represent 32 (3 rods and 2 units) for 10 units to form an equivalent representation of 32 (2 rods and 12 units). Then the student would take away 15 (1 rod and 5 units) and be left with a difference of 17 (1 rod and 7 units).

    Multiplication can be modeled as repeated addition or with rectangular arrays. Using rectangular arrays can help in understanding the derivation of the partial products, the sum of which is the total product.

    Division can be done as repeated subtraction or through building and analyzing rectangular arrays.

    By letting the cube, flat, rod, and unit represent 1, 0.1, 0.01, and 0.001, respectively, older students can explore and develop decimal concepts, compare decimals, and perform basic operations with decimal numbers.

    The squares along each face make the blocks excellent tools for visualizing and internalizing the concepts of perimeter and surface area of structures. Counting unit blocks in a structure can form the basis for understanding and finding volume.

    Assessing Students’ Understanding

    Base Ten Blocks provide a perfect opportunity for authentic assessment. Watching students work with the blocks gives you a sense of how they approach a mathematical problem. Their thinking can be «seen» through their positioning of the blocks. When a class breaks up into small working groups, you can circulate, listen, and raise questions, all the while focusing on how individuals are thinking.

    The challenges that students encounter when working with Base Ten Blocks often elicit unexpected abilities from those whose performance in more symbolic, number-oriented tasks may be weak. On the other hand, some students with good memories for numerical relationships have difficulty with spatial challenges and can more readily learn from freely exploring with Base Ten Blocks. Thus, by observing students’ free exploration, you can get a sense of individual styles and intellectual strengths.

    Having students describe their creations and share their strategies and thinking with the whole class gives you another opportunity for observational assessment. Furthermore, you may want to gather student’s recorded work or invite them to choose pieces to add to their math portfolios.

    2021-06-03 21:30:00

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    Table of degrees 🆕

    Let’s help you understand and love mathematics

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    After learning the basics of arithmetic in the form of addition, subtraction, division and multiplication, mathematics and its big world open up. In this material, we will consider such a concept as a degree.

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    Basic concepts

    The power of the number with a natural indicator is the result of multiplying the number by itself. The number itself is called the base of the degree, and the number of multiplication operations is called the exponent.

    • a n = a × a × … × a, where a is the base of the power,
    • n is the natural exponent.

    The entry is read as «a» to the power of «n».

    Here is an example to illustrate:

    • 3 5 = 3 × 3 × 3 × 3 × 3 = 243

    This entry can be read in three ways:

    • 3 to the 5th power;
    • fifth power of three;
    • Raise the number three to the fifth power.

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    Degree properties

    Degree properties are usually used to shorten or simplify complex examples. It is convenient to use together with the power table and the multiplication table.

    a 1 = a

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    (a ≠ 0)

    a −n = 1 : a n

    a m × a n = a m+n

    a m : a n = a m-n

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    Table of degrees from 1 to 10

    Table of degrees is a list of numbers from 1 to 10 raised to a power from 1 to 10. Below are two types of tables: choose the one that is more convenient for you, download on your phone or print and put in your textbook.

    How to find the required values ​​in this table:

    • In the first column we find a number that indicates the degree. Let’s remember this line number.
    • In the first line we find the exponent. Let’s remember the found column.
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    And if you need an answer as quickly as possible, you can use the online degree calculator.

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    Task 1. Simplify and solve the expression 5 2 × 5 3 .

    How to solve:

    5 2 × 5 3 = 5 2+3 = 5 5 = 3125

    Task 2. Simplify and solve the expression 2 4 × 3 3 × 2 5 .

    How to solve:

    2 4 × 3 3 × 2 5 = 2 4+5 × 3 3 = 2 9 × 3 3 = 512 × 27 = 13 824

    Task 3. Find 36 4 .

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    Author: Pimenova Olga Rushanovna.
    Nomination: Basic general education.
    Competition: Leader of innovative teaching staff in Russia.

    Lesson scenario.

    Bosova L.L., Bosova A.Yu. Computer science . 8th grade. GEF.

    Date __________________________________

    Lesson 8-15. Converting numbers from the 10th number system to the number system with base q .

    Grade: 8 ion into a number system with base q, be able to translate numbers from different number systems.

  • meta-subject — organization of the learning space, independent implementation of learning activities
  • personal — skills of concentration, development of independence and initiative.
  • Learning tasks to be solved:

    • repetition of the concept of the number system and the alphabets of the number system
    • transfer of numbers to the 10th SCH
    • study of transfer from 10th SS to SS with other base
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