## Intro to Decimals: Decimal Form, Examples and Meaning

So, it’s time to work with decimals, huh? Gone are the days of simple, whole numbers — but that’s okay! We have to keep life interesting, right?

Pro tip: If you find yourself having trouble navigating this page on decimals, it might help to read this page about fractions to shed some more basic light on the fundamental concepts of non-whole numbers.

## What is a decimal?

A decimal is a representation of a number that is more (or less) than a whole number. We’ll get into decimal form in a little bit, but essentially, a decimal is a number comprised of three pieces: a whole number piece, a fractional piece, and a dot (a.k.a. the decimal!).

## Decimal definition

How do we describe decimal numbers at Photomath? Like this:

A decimal number is a number consisting of a whole number part, the decimal separator, and a fractional part.

## Decimals vs fractions

You may have noticed that we’re referring to a “fractional part” of decimal numbers. That doesn’t mean that we have an actual fraction in our number — it just means that the decimal number includes a quantity that is less than one whole, just like in fractions .

In the case of decimals, the “fractional part” — or the part less than one whole — is the piece to the right of the decimal point.

BTW: You can convert decimals to fractions (and fractions to decimals ) using denominators that are multiples of ten. In other words, the decimal part of a number is actually equal to a fraction! The difference is really the format, and you can decide which makes the most sense to use in any given situation.

## Decimal meaning

So, what does working with decimal numbers really mean?

As you’ve seen, it means you’re dealing with a quantity that is less than or greater than a whole. But why is it important that we can work with numbers like this?

Well, what if you need to weigh something that’s not just a whole number? Let’s say you need $$5.5$$ lbs of bird feed to keep all the little birdies in your backyard coming back! You find a bag of $$7.3$$ lbs of bird feed — you’ll need to know your decimals in order to see if you have enough (and calculate how much you might have left over!).

## Decimal form

How do we know when we’re working with a decimal? We can tell by the form of the number!

Some people will refer to the way we write decimal numbers as “decimal notation,” but no matter how you spin it, you’ll know a decimal when you see one.

The main tip-off should be the “.” In this case, it’s a decimal point, not a period. You’ll see a number (or a few) to the left of the decimal point, and then at least one other number to the right of the decimal point, like this:

Let’s dive a little deeper into the foundation of why we write decimal numbers the way that we do:

## Decimal base

The decimal system — or “decimal numeral system,” if you’re feeling fancy — uses $$10$$ as its base. That’s why it’s also called a “base-ten” (or “base-10”) positional numeral system.

Whoah, whoah, whoah. That was a lot. Let’s break it down:

- Base-10: There are 10 different numerals (including $$0$$) used to represent numbers. The digits are $$0, 1, 2, 3, 4, 5, 6, 7, 8$$, and $$9$$.
- Positional numeral system: In our case, this means that the position of digits (and how we reference them) is related to their distance from the decimal point.

Because the decimal system is a base-10 system, the names of place values are all based on the number $$10$$ and its multiples.

You’ll notice that we differentiate the left side of the decimal from the right side by adding “th” to the end when referencing the fractional part. For example, the “thousands place” is to the left of the decimal, but the “thousandths place” is to the right.

## Where is the tenths place in a decimal?

The tenths place is one place to the right of the decimal point, actually right next to it on the right side. We know it’s to the right of the decimal because “tenths” ends in “ths.”

When working with place values to the right of the decimal, the names are still based on $$10$$ and its multiples; for place values to the right of the decimal, you can use the number of zeroes in the name to tell how many places away from the decimal it is.

For instance, there’s one $$0$$ in $$10$$, so the tenths place is one place to the right!

## Where is the hundreds place in a decimal?

The hundreds place is three places to the left of the decimal point. This place value doesn’t have “ths” at the end, so we know it’s on the left side of the decimal.

Think of how you would write three-hundred: $$300$$. We say it as “three-hundred” because the $$3$$ is in the hundreds place!

Generally, when working to the left of the decimal point, the number of digits of the place value name will tell you how many places to move to the left (NOT the number of zeroes like after the decimal). That’s because the place immediately to the left of the decimal is the “ones” place.

In the example of the “hundreds” place, we can tell that’s based on $$100$$, which has three digits total; so, the hundreds place is three to the left.

## Place values in numbers with decimals

We already know that place values to the right of our decimal — in other words, place values in the fractional part of our decimal number — will end in “ths,” like “hundredths.”

We also know that the naming of place values is slightly different depending on which side of the decimal point you’re on.

So, to sum it all up, let’s think of it this way:

- Ex: The thousands place —> 1000 —> four digits —> four places to the left of the decimal
- Ex: The hundredths place —> 100 —> two zeroes —> two places to the right of the decimal

## Reading a scale with decimal intervals

Picture a ruler: It has evenly spaces markings to help us measure more precisely. We see the whole numbers labeled very clearly, but the little lines in-between are not.

Those little lines can be used to find the fractional part of a decimal number when measuring!

Because they’re evenly spaced and based on tenths, we can refer to the distance between the scaled markings as “decimal intervals.”

For example, let’s say our ruler measures in inches, and there are 10 decimal intervals (spaces) between the whole numbers. If we measure a piece of cardboard whose edge lines up with the fourth tick after the whole number $$6$$, we know that length is $$6.4$$ inches.

## Decimal example problems

We know it’s hard to just read about decimals, so let’s put all those words into action and try some examples of real math!

- Round the decimal number $$352.081$$ to the nearest hundredth.
- What number is in the ones place in $$36.5$$?
- What number is in the tens place in $$4,732.98$$?
- Convert $$4.25$$ into a fraction.
- Rewrite $$.01$$ as a percentage.

Want to keep playing around with decimals? Scan a decimal number with your Photomath app to see how to convert, round, and more!

Here’s how we solve the first problem in the app:

FAQ What are the 3 types of decimals?

Three types of decimals are non-recurring (non-repeating or terminating, like $$4.32$$); recurring (repeating or non-terminating, like $$3.14159…$$); and decimal fractions (a.k.a. converting a decimal into a fraction whose denominator is a power of ten).

How do you explain decimals to children?

We can explain decimals as representing parts of a whole, with the left side representing how many “wholes” we have, and the numbers after the decimal point telling us how many pieces of a whole we have. It might help to start with fractions and then relate decimals accordingly!

What are the four rules of decimals?

The four “rules” of decimals are just the four arithmetic operations : addition, subtraction, multiplication, and division. Heads-up: in addition and subtraction, you’ll need to line up the decimal points in order to perform the operation correctly!

## Related Topics

- Math Explained: Arithmetic

## Related Articles

## Decimals to improper fractions

## Decimals to fractions

## Simplifying expressions with fractions

## Fractions to decimals

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## decimal representation

Quick reference.

Any real number a between 0 and 1 has a decimal representation, written. d 1 d 2 d 3 …, where each d i is one of the digits 0, 1, 2,…, 9; this means that

a = d 1 ×10 −1 + d 2 ×10 −2 + d 3 ×10 −3 +⋯.

This notation can be extended to enable any positive real number to be written as

c n c n− 1 … c 1 c 0 . d 1 d 2 d 3 …

using, for the integer part, the normal representation c n c n− 1 … c 1 c 0 to base 10 (see base). If, from some stage on, the representation consists of the repetition of a string of one or more digits, it is called a recurring or repeating decimal. For example, the recurring decimal .12748748748…can be written .̇12748̇, where the dots above indicate the beginning and end of the repeating string. The repeating string may consist of just one digit, and then, for example, .16666…is written .16̇. If the repeating string consists of a single zero, this is generally omitted and the representation may be called a terminating decimal.

The decimal representation of any real number is unique except that, if a number can be expressed as a terminating decimal, it can also be expressed as a decimal with a recurring 9. Thus .25 and .249̇ are representations of the same number. The numbers that can be expressed as recurring (including terminating) decimals are precisely the rational numbers.

From: decimal representation in The Concise Oxford Dictionary of Mathematics »

Subjects: Science and technology — Mathematics and Computer Science

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## Decimal – Definition, Types, FAQs, Examples

What is a decimal, how to read a decimal, types of decimals, solved examples on decimal, practice problems on decimal, frequently asked questions on decimal, decimal – introduction.

A decimal is a number that consists of a whole and a fractional part. Decimal numbers lie between integers and represent numerical value for quantities that are whole plus some part of a whole.

For example, in the given image, we have one whole pizza and a half of another pizza. This can be represented in two ways:

Fractional form: In fraction form, we can write that there is one and one-half of a pizza. That is $1\frac{1}{2}$ pizza.

Decimal Form: In decimal form, we will write this as 1.5 pizzas. Here, the dot represents the decimal point and the number before the dot, i.e., “1” represents one whole pizza and the number behind the decimal point represents the half pizza or the fractional part.

You might have seen decimal numbers like these when you go grocery shopping or on a weighing machine or even a game of baseball!

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We get decimals when we break a whole into smaller parts. A decimal number then has two components: a whole number part and a fractional part. The decimal place value system for the whole part of a decimal number is the same as the whole number value system. However, we get the fractional part of the decimal number as we move toward the right after the decimal point. The given image shows the decimal place value chart:

Note that as we go from left to right in the decimal place value system, each values is $\frac{1}{10}$ times smaller than the value to its left.

The first place after the decimal point is called the “ tenths ”, which represents a place value of $\frac{1}{10}$ of the whole or one-tenth of the whole. In decimal form, this fraction is written as “0.1”.

Such fractions whose denominator is 10 or a positive power of 10 is called a decimal fraction.

The second place is called the “ hundredths ”, which represents a place value of $\frac{1}{100}$ of the whole or one-hundredth of the whole. In numerical form, this decimal fraction is written as “0.01”.

And the third place is called the “ thousandths ”, which represents a place value of $\frac{1}{1000}$ of the whole or one-thousandth of the whole. In numerical form, this decimal fraction is written as “0.001”.

Here’s an example of a decimal number 17.48, in which 17 is the whole number, while 48 is the decimal part.

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An informal way to read a decimal is by reading the whole part of the decimal number as you would read any whole number and then reading the decimal dot as “point” and then reading each digit of the rational part individually.

For example, the number 17.48 would be read as “Seventeen point four eight”.

However, a more formal way to read decimals is to read the whole part as a whole number, then the decimal dot as “and” and then reading the fractional part altogether but using the place value of the last digit with it.

For example, take a look at the given number

Here, the whole part is 25 and the place value of the last digit, 8, is thousandths. So we will read this number as “Twenty-five and six hundred seventy-eight thousandths”.

Based on the number of digits after the decimal point, the decimal numbers can be divided into two categories:

Like decimals: Two decimal numbers are said to be “like” decimals if they have the same number of digits after the decimal point. For example, 6.34 and 2.67 both have two digits after the decimal point so they are Like decimals.

Unlike decimals: Two decimal numbers are said to be “unlike” decimals if they have different number of digits after the decimal point. For example, 5.3 and 6.873 both have different number of digits after the decimal point so they are unlike decimals.

Example 1: In which place does the underlined digit lie in 56.78 2 ? Solution:

The underlined digit is in the third place after the decimal point. The first place after the decimal point is tenths, the second place is hundredths and the third place is called thousandths. Hence, the undersigned digit is in the thousandths place.

Example 2: What is the place value of the underlined digit in 45. 8 2?

The underlined digit, 8, is in the tenths place. Hence, its place value is 0.8.

Example 3: Give an example of like decimals.

Two decimal numbers are said to be like decimals if they have the same number of digits after the decimal point. Example: 5.99 and 3.65

## Decimal – Definition with Examples

Attend this quiz & Test your knowledge.

## Which of the following pairs of decimals are like decimals?

In what place does the underlined digit in $6.8\underline{9}3$ lie, which number is in the tenths place in 34.987, what is the place value of the underlined digit in 124.64$\underline{8}$.

What are some applications of decimals in daily life?

A common application of the concept of decimals is in monetary transactions. A dollar is made up of 100 cents, and the prices of objects are written in decimals, such as USD 2.79. Other applications are mainly concerned with the precision required in the field, such as scientific research, sports, jewelry, and vehicle gas meters.

What is the difference between decimal numbers and integers?

All integers can be written as decimal numbers. For example, the integer 5 can be written as 5.00. But not all decimal numbers can be written as integers since integers are whole and do not have a fractional part, but decimal numbers have a fractional part and hence can only be written as fractions.

How do you read a decimal number?

A decimal number is read by saying the whole part of it as a whole number and by individually reading out every number after the decimal point. For example, 27.69 would be read as twenty-seven point six nine.

## RELATED POSTS

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## Numeral Systems

Numeral system.

- Binary Numeral System
- Octal Numeral System
- Decimal Numeral System
- Hex Numeral System
- Numeral System Conversion Table

b - numeral system base

d n - the n-th digit

n - can start from negative number if the number has a fraction part.

N +1 - the number of digits

## Binary Numeral System - Base-2

Binary numbers uses only 0 and 1 digits.

B denotes binary prefix.

10101 2 = 10101B = 1×2 4 +0×2 3 +1×2 2 +0×2 1 +1×2 0 = 16+4+1= 21

10111 2 = 10111B = 1×2 4 +0×2 3 +1×2 2 +1×2 1 +1×2 0 = 16+4+2+1= 23

100011 2 = 100011B = 1×2 5 +0×2 4 +0×2 3 +0×2 2 +1×2 1 +1×2 0 =32+2+1= 35

## Octal Numeral System - Base-8

Octal numbers uses digits from 0..7.

27 8 = 2×8 1 +7×8 0 = 16+7 = 23

30 8 = 3×8 1 +0×8 0 = 24

4307 8 = 4×8 3 +3×8 2 +0×8 1 +7×8 0 = 2247

## Decimal Numeral System - Base-10

Decimal numbers uses digits from 0..9.

These are the regular numbers that we use.

2538 10 = 2×10 3 +5×10 2 +3×10 1 +8×10 0

## Hexadecimal Numeral System - Base-16

Hex numbers uses digits from 0..9 and A..F.

H denotes hex prefix.

28 16 = 28H = 2×16 1 +8×16 0 = 40

2F 16 = 2FH = 2×16 1 +15×16 0 = 47

BC12 16 = BC12H = 11×16 3 +12×16 2 +1×16 1 +2×16 0 = 48146

## Numeral systems conversion table

Decimal Base-10 | Binary Base-2 | Octal Base-8 | Hexadecimal Base-16 |
---|---|---|---|

0 | 0 | 0 | 0 |

1 | 1 | 1 | 1 |

2 | 10 | 2 | 2 |

3 | 11 | 3 | 3 |

4 | 100 | 4 | 4 |

5 | 101 | 5 | 5 |

6 | 110 | 6 | 6 |

7 | 111 | 7 | 7 |

8 | 1000 | 10 | 8 |

9 | 1001 | 11 | 9 |

10 | 1010 | 12 | A |

11 | 1011 | 13 | B |

12 | 1100 | 14 | C |

13 | 1101 | 15 | D |

14 | 1110 | 16 | E |

15 | 1111 | 17 | F |

16 | 10000 | 20 | 10 |

17 | 10001 | 21 | 11 |

18 | 10010 | 22 | 12 |

19 | 10011 | 23 | 13 |

20 | 10100 | 24 | 14 |

21 | 10101 | 25 | 15 |

22 | 10110 | 26 | 16 |

23 | 10111 | 27 | 17 |

24 | 11000 | 30 | 18 |

25 | 11001 | 31 | 19 |

26 | 11010 | 32 | 1A |

27 | 11011 | 33 | 1B |

28 | 11100 | 34 | 1C |

29 | 11101 | 35 | 1D |

30 | 11110 | 36 | 1E |

31 | 11111 | 37 | 1F |

32 | 100000 | 40 | 20 |

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## Numbers Representation Systems – Decimal, Binary, Octal and Hexadecimal

In this article we’ll discuss about different number representation systems, where they are used and why they are useful. Briefly we’ll go through decimal , binary , octal and hexadecimal number representation.

## Decimal (base 10)

The most common system for number representation is the decimal . Everybody is using it. It’s so common than most people must believe that is the only one. It’s used in finances, engineering and biology, almost everywhere we see and use numbers.

If someone is asking you to think at a number for sure you’ll think at a decimal number. If you think at a binary or hexadecimal one, you must have an extreme passion for arithmetic or software/programming.

As the name is saying the decimal number system is using 10 symbols/characters. In Latin language 10 is “decem” so decimal might be linked to the Latin word.

Decimal Symbols | |||||||||

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

As you can see there are 10 symbols from 0 to 9 . With these symbols we can construct all the numbers in the decimal system.

All the numbers in the decimal system can be constructed by using the above mentioned symbols ( 0 … 9 ) multiplied with the power of 10. The power of ten gives us ones, tens, hundreds, thousands and so on.

10 | … | 10 | 10 | 10 | 10 | 10 | 10 |

N | … | 100000 | 10000 | 1000 | 100 | 10 | 1 |

The example below breaks down the decimal number 67049 into powers of 10 multiplied with numbers between 0 and 9 . This is just to show that any number in the decimal system can be decomposed into a sum of terms made of from the product of the power of 10 and the symbols 0 … 9 .

67049 | |||||||

10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 |

0 | 0 | 0 | 6 | 7 | 0 | 4 | 9 |

67049 = 6 ⋅ 10 + 7 ⋅ 10 + 0 ⋅ 10 + 4 ⋅ 10 + 9 ⋅ 10 = 60000 + 7000 + 0 + 40 + 9 |

The same technique is going to be applied to the binary, octal and hexadecimal systems, being in fact a method of converting a number from decimal system in another format (base).

We can keep in mind these characteristics of the decimal numbers system:

- it’s using 10 symbols
- can be decomposed in factors containing powers of 10
- it’s the most common number representation system

## Binary (base 2)

Let’s step into the geek side now.

Another number representation system is the binary one. As the name suggests and by analogy with the decimal system we can say that the binary system is using only 2 symbols/characters:

Binary Symbols | |

0 | 1 |

In the binary representation we only use 0 (zeros) and 1 (ones) to represent numbers.

The binary system is used wherever you want to store information in electronic format. All the computers that you know, intelligent devices, everything that has to do with electronics and microcontrollers use the binary system.

In electronics (digital) all the operations are done using two levels of voltage: high and low. Each level of voltage is assigned to a value/symbol: HIGH for 1 and LOW for 0. For a microcontroller which is supplied with +5V the 1 (high) will be represented by +5 V and the 0 (low) by 0 V.

Roughly we can say that the binary system is used because it can be translated in electronic signal.

All the decimal numbers we can think of can be represented into binary symbols. We do this by using a sum between terms of the power of 2 multiplied with 0 or 1.

2 | … | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

N | … | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |

As example we’ll use the number 149 (decimal representation) and transform it into binary representation. We could use any number but if it’s too big it would end up into a long string of zeros and ones.

149 | |||||||

2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 |

149 = 1 ⋅ 2 + 0 ⋅ 2 + 0 ⋅ 2 + 1 ⋅ 2 + 0 ⋅ 2 + 1 ⋅ 2 + 0 ⋅ 2 + 1 ⋅ 2 = 128 + 0 + 0 + 16 + 0 + 4 + 0 + 1 |

As you can see the decimal number 149 is represented in binary system by a series of zeros and ones ( 10010101 ). Usually to distinguish between a decimal or binary number we must specify the base to which we are referring to. The base is described as a subscript after the last character of the number

Decimal (base 10) | Binary (base 2) |

149 | 10010101 |

By specifying the base of the number we eliminate the probability of confusion, because the same representation (e.g. 11) can mean different things for different bases.

11 ≠ 11 |

Another way to avoid confusion is to use a special notation (prefix) for binary numbers. This is because 1100 can represent eleven hundreds in decimal system or the decimal twelve represented in binary system. So if want to specify a binary number we use the prefix 0b . Example: 0b1100 .

Briefly the characteristics of a binary system are:

- it’s using 2 symbols
- can be decomposed in factors containing powers of 2
- it’s used in computers, microcontrollers

## Octal (base 8)

All the numbers in the octal system are represented using 8 symbols/characters, from 0 to 7 . The reason of using the octal system instead of the decimal one can be various. One of them is that instead of using our fingers for counting, we use the spaces between fingers.

Humans have 4 spaces between the fingers of one hand; in total we’ll have 8 spaces, for both hands. In this case it makes sense to use an octal number representation system instead of a decimal one. The drawback is that higher numbers will require more characters compared to the decimal one.

Octal Symbols | |||||||

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

To transform a decimal represented number into an octal system we split it into terms containing the power of 8:

8 | … | 8 | 8 | 8 | 8 | 8 | 8 |

N | … | 32768 | 4096 | 512 | 64 | 8 | 1 |

As an example we are going to represent the decimal number 67049 in octal base:

67049 | |||||

8 | 8 | 8 | 8 | 8 | 8 |

2 | 0 | 2 | 7 | 5 | 1 |

67049 = 2 ⋅ 8 + 0 ⋅ 8 + 2 ⋅ 8 + 7 ⋅ 8 + 5 ⋅ 8 + 1 ⋅ 8 = 65535 + 0 + 1024 + 448 + 40 + 1 |

## Hexadecimal (base 16)

The hexadecimal number representation system is using 16 symbols/characters to define numbers. It’s used in computer science mostly because can represent bigger decimal numbers with fewer characters.

Hexadecimal Symbols | |||||||||||||||

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F |

Compared with decimal system it’s also using numeric symbols from 0 to 9. Additionally it’s using alphanumeric characters from A to F for values between 10 and 15.

16 | … | 16 | 16 | 16 | 16 | 16 | 16 |

N | … | 1048576 | 65535 | 4096 | 256 | 16 | 1 |

To represent a decimal number in hexadecimal format we split the decimal number into a sum of terms. Each term is a product between a hexadecimal symbol and a power of 16.

67049 | ||||

16 | 16 | 16 | 16 | 16 |

1 | 0 | 5 | E | 9 |

67049 = 1 ⋅ 16 + 0 ⋅ 16 + 5 ⋅ 16 + E ⋅ 16 + 9 ⋅ 16 = 65536 + 0 + 1280 + 224 + 9 |

The representation of the decimal number 67049 in hexadecimal format is 105E9 . Similar to the binary system a common practice is to use the prefix “ 0x ” in order to distinguish from the decimal notation. Example: 0x105E9 .

Briefly the characteristics of a hexadecimal number representation system are:

- it’s using 16 symbols
- can be decomposed in factors containing powers of 16

The table below summaries the characteristics of the above mentioned number representation systems.

Decimal | 10 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 | None | 147 |

Binary | 2 | 0, 1 | 0b | 0b10010011 |

Hexadecimal | 16 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A,B, C, D, E, F | 0x | 0x93 |

Octal | 8 | 0, 1, 2, 3, 4, 5, 6, 7 | None | 41 |

Both the octal and hexadecimal number representation systems are linked to the computer system, mainly with the kind of processors and microcontrollers. For example, if the microprocessor is using 8 bit data then the octal system is suitable to interface data. If the microprocessor is on 16 bits then the hexadecimal system is appropriate to represent data.

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## Decimal Expansion

The following table summarizes the decimal expansions of the first few unit fractions . As usual, the repeating portion of a decimal expansion is conventionally denoted with a vinculum .

decimal expansion | fraction | decimal expansion | |

1 | 1 | ||

0.5 | |||

0.25 | |||

0.2 | |||

0.0625 | |||

0.125 | |||

0.1 | 0.05 |

Factoring possible common multiples gives

To find denominators with short periods, note that

primes | |

3 | |

11 | |

37 | |

101 | |

41, 271 | |

7, 13 | |

239, 4649 | |

73, 137 | |

9 | 333667 |

9091 | |

21649, 513239 | |

9901 | |

53, 79, 265371653 | |

909091 | |

15 | 31, 2906161 |

17, 5882353 | |

2071723, 5363222357 | |

19, 52579 | |

1111111111111111111 | |

3541, 27961 |

1 | 31 | 15 | 67 | 33 | |

6 | 37 | 3 | 71 | 35 | |

11 | 2 | 41 | 5 | 73 | 8 |

6 | 43 | 21 | 79 | 13 | |

16 | 47 | 46 | 83 | 41 | |

18 | 53 | 13 | 89 | 44 | |

23 | 22 | 59 | 58 | 97 | 96 |

28 | 61 | 60 | 101 | 4 |

Portions of this entry contributed by Christopher Stover

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## Referenced on Wolfram|Alpha

Cite this as:.

Stover, Christopher and Weisstein, Eric W. "Decimal Expansion." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/DecimalExpansion.html

## Subject classifications

- Math Article
- Decimal Representation

## Representation Of Decimals On Number Line

Decimals are used in situations where more precision is required in comparison to the whole numbers. For example, when we have to divide 3 apples among 4 kids, we cannot use whole numbers to denote the result of the division, as the fraction of share that is 0.75 lies between 0 and 1. In order to deal with similar other systems, the concept of decimal was introduced.

## What are Decimal Numbers?

A decimal number is a number where the integer part is separated from the fractional part with the help of a decimal point. The digits are placed to the left and to the right of the decimal to represent numbers greater than or less than one.

There are certain rules to be followed while reading a decimal number. For e.g. 1.23 is read as one point two three and not one point twenty-three.

## What is Decimal Representation?

In Number system, each and every real number can be represented in the form of a decimal. The decimal representation of any non-negative real numbers “r” is an expression which is in the form of series and is traditionally written as:

where a i is a non-negative integer

a 1 , a 2 , …are the integers satisfying the condition 0≤ a i ≤ 9

For example, if we take the rational number 1/2, it can also be written as 0.5. In the given example,

“0” represents the whole number part

“.” represents a decimal point

“5” represents the decimal part.

For any rational number which is the ratio of any two numbers, can have either a terminating decimal representation or a repeating representation. Similarly, for any fraction of number, we can represent the number using the decimal value. Some of the examples of the recurring decimal representations are as follows:

1/3 = 0.33333..

1/7 = 0.142857142857…

In order to understand the concept of decimals, let us make a square table with 1 row and 10 columns, as shown in the figure.

Let us fill four of these blocks, as shown below

The fraction of colored blocks to the total blocks can be written as 4/10. Another representation for the same can be given in terms of decimals, like 0.4. Here 0.4 = 4 * or can be written as 4 tenths.

Similarly, if we take a square of 10 rows and 10 columns, we get 100 small squares. If we colour 27 of these blocks, the fractional representation can be written as 27/100.

Here 27/100 = 27*1/100 = 27 hundredth which is represented in the decimal form as 0.27.

So when we have a number 2.34, it is equivalent to 2 + 3 tenths and 4 hundredths.

## How to Represent the Decimals on the Number Line?

We know how to represent whole numbers on the number line . Let us consider the image shown below. Here, the digits 0 and 1 are represented on the number line.

If we divide the number line into two equal parts, as shown in the figure below, what value does mid-point hold? We will say half of what the graduation between 0 and 1 holds.

So, the point represents (1-0)/2 = 0.5

In order to represent decimals on the number line, we divide the section between two whole numbers as per the places after decimal present in the number to be represented.

## Decimal Representation on Number Line Examples

Represent 0.8 on the number line.

As we know, the number 0.8 is equivalent to 8 tenths, so we divide the section between 0 and 1 into 10 equal parts. Now, stepping 8 points from 0 towards 1 gives us 0.8.

Represent 8.6 on the number line.

The number 8.6 = 8 + 0.6

We start from the number 8 on the number line and divide the section between 8 and 9 into 10 equal parts. Now, taking 6 steps from 8 towards 9 gives us the representation of 8.6

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## Decimal Number System

A number system can be considered as a mathematical notation of numbers using a set of digits or symbols. In simpler words the number system is a method of representing numbers. Every number system is identified with the help of its base or radix. For example, Binary, Octal, Decimal and Hexadecimal Number systems are used in microprocessor programming.

Decimal Number System : If the Base value of a number system is 10. This is also known as base-10 number system which has 10 symbols, these are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Position of every digit has a weight which is a power of 10. then it is called the Decimal number system which has the most important role in the development of science and technology. This is the weighted (or positional) number representation, where value of each digit is determined by its position (or their weight) in a number.

Each position in the decimal system is 10 times more significant than the previous position, that means the numeric value of a decimal number is determined by multiplying each digit of the number by the value of the position in which the digit appears and then adding the products.

Example-1: The number 2020 is interpreted as:-

Example-2: The number 2020.50 is interpreted as:-

Note : Right most bit is the least significant bit (LSB) and left most bit is the most significant bit (MSB).

In general, a number expressed in the base-r system has coefficients multiplied by power of r. The coefficient aj ranges from 0 to (r-1). Representing real number in base-r is as following below:

Where, a 0 , a 1 , … a (n-1) and an are integer part digits, n is the total number of integer digits. a -1 , a -2 , … and a -m are fractional part digits, m is the total number of fractional digits.

Advantages and disadvantages of Decimal Number System :

- Advantages – easy readability, used by humans, and easy to manipulate.
- Disadvantages – wastage of space and time.

9’s and 10’s Complement of Decimal (Base-10) Number :

- Simply, 9’s Complement of a decimal number is the subtraction of it’s each digits from 9. For example, 9’s complement of decimal number 2005 is 9999 – 2005 = 7994.
- 10’s Complement of decimal number is 9’s complement of given number plus 1 to the least significant bit (LSB). For example 10’s complement of decimal number 2005 is (9999 – 2005) + 1 = 7995.

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The decimal representation represents the infinite sum: Every nonnegative real number has at least one such representation; it has two such representations (with if ) if and only if one has a trailing infinite sequence of 0, and the other has a trailing infinite sequence of 9. For having a one-to-one correspondence between nonnegative real ...

• A power programmer must know number systems and data representation to fully understand C's . primitive data types. Primitive values and. the operations on them. ... Finite representation of rational (floating -point) numbers. 3. 4. The Decimal Number System. Name • "decem" (Latin) ⇒ten. Characteristics • Ten symbols ...

Decimal is a numerical representation that uses a dot, which we call a decimal point, to separate the whole number part from its fractional part. The decimal numeral system is used as the standard system that is used to distinguish integer and non-integer numbers. In this article, we will understand what decimals are, the place value of decimals, and how to round decimals along with some ...

In the decimal number system, the numbers are represented with base 10. The way of denoting the decimal numbers with base 10 is also termed as decimal notation. This number system is widely used in computer applications. It is also called the base-10 number system which consists of 10 digits, such as, 0,1,2,3,4,5,6,7,8,9.

Example: 81.75 = 8175/100. 32.425 = 32425/1000. Converting the Decimal Number into Decimal Fraction: For the decimal point place "1" in the denominator and remove the decimal point. "1" is followed by a number of zeros equal to the number of digits following the decimal point. For Example: 8 1 . 7 5.

It follows that a number is a decimal fraction if and only if it has a finite decimal representation. Expressed as fully reduced fractions, the decimal numbers are those whose denominator is a product of a power of 2 and a power of 5. Thus the smallest denominators of decimal numbers are

A decimal is a representation of a number that is more (or less) than a whole number. We'll get into decimal form in a little bit, but essentially, a decimal is a number comprised of three pieces: a whole number piece, a fractional piece, and a dot (a.k.a. the decimal!). ... Round the decimal number $$352.081$$ to the nearest hundredth. What ...

• Finite representation of rational numbers (if time) Why? • A power programmer must know number systems and data ... Octal to decimal: expand using positional notation Decimal to octal: use the shortcut 17 37 O = (3*81) + (7*80) = 24 + 7 = 31 31 / 8 = 3 R 7

Search for: 'decimal representation' in Oxford Reference ». Any real number a between 0 and 1 has a decimal representation, written. d 1 d 2 d 3…, where each d i is one of the digits 0, 1, 2,…, 9; this means thata=d 1×10−1+d 2×10−2+d 3×10−3+⋯.This notation can be extended to enable any positive real number to be written asc n c ...

Types of Decimals. Based on the number of digits after the decimal point, the decimal numbers can be divided into two categories: Like decimals: Two decimal numbers are said to be "like" decimals if they have the same number of digits after the decimal point. For example, 6.34 and 2.67 both have two digits after the decimal point so they ...

Hex Numeral System. Numeral System Conversion Table. Numeral System. b - numeral system base. dn - the n-th digit. n - can start from negative number if the number has a fraction part. N+1 - the number of digits. Binary Numeral System - Base-2. Binary numbers uses only 0 and 1 digits.

Therefore, the required octal number is (25)8. Example 4: Convert hexadecimal 2C to decimal number. Solution: We need to convert 2C16 into binary numbers first. 2C → 00101100. Now convert 001011002 into a decimal number. 101100 = 1 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 0 × 2 1 + 0 × 2 0.

The representation of the decimal number 67049 in hexadecimal format is 105E9. Similar to the binary system a common practice is to use the prefix "0x" in order to distinguish from the decimal notation. Example: 0x105E9. Briefly the characteristics of a hexadecimal number representation system are: it's using 16 symbols

For positive (unsigned) integers, there is a 1-to-1 relationship between the decimal representation of a number and its binary representation. If you have a 4-bit number, there are 16 possible combinations, and the unsigned numbers go from 0 to 15: 0b0000 = 0 0b0001 = 1 0b0010 = 2 0b0011 = 3

This means the base 6 number 3,024 is equal to the base 10 number 664. From now on, if we are using a base 6 number, we will follow it with the subscript 6, like the following: 3,024 6 means the number is in base 6. A base 10 number gets no subscript (it's the standard). So, 3,024 is a base 10 number. A base 13 number would be 4,672 13.

999.dvi. Real numbers and decimal representations. 1. An informal introduction. It is likely that the reason real numbers were introduced was to make possible a numerical description of the ratios of the lengths of line segments, a task whose accomplishment seems to have escaped Greek mathematicians. It is, basically, representations of numbers ...

The decimal expansion of a number is its representation in base-10 (i.e., in the decimal system). In this system, each "decimal place" consists of a digit 0-9 arranged such that each digit is multiplied by a power of 10, decreasing from left to right, and with a decimal place indicating the 10^0=1s place. For example, the number with decimal expansion 1234.56 is defined as 1234.56 = 1×10^3+2 ...

The numeral system Represents a useful set of numbers, reflects the arithmetic and algebraic structure of a number, and provides standard representation. In the decimal number system, digits from 0 to 9 can be used to form all the numbers. With these digits, anyone can create infinite numbers. For example, 156,3907, 3456, 1298, 784859, etc.

Find all numbers less than n, which are palindromic in base 10 as well as base 2. Examples: 33 is Palindrome in its decimal representation. 100001(binary equivalent of 33) in Binary is a Palindrome. 313 is Palindrome in its decimal representation. 100111001 (binary equivalent of 313) in Binary is a Palindrome. Brute Force: We check all the numbers

The image on the left shows the decimal number in red, and the equivalent binary number in blue. For example, the number 2 dec (on the right side of the circle) is the same as 010 2. Note: Since there are n=3 bits, there are 2 n =2 3 =8 unique numbers that can be represented. The decimal numbers go from 0 to 2 n-1=2 3 −1=7 (i.e., 0→7).

Decimal Representation on Number Line Examples. Example 1: Represent 0.8 on the number line. Solution: As we know, the number 0.8 is equivalent to 8 tenths, so we divide the section between 0 and 1 into 10 equal parts. Now, stepping 8 points from 0 towards 1 gives us 0.8. Example 2: Represent 8.6 on the number line.

In computing, signed number representations are required to encode negative numbers in binary number systems. ... For example, the decimal number −125 with its sign-magnitude representation 11111101 can be represented in ones' complement form as 10000010. Two's complement. Eight-bit two's complement Binary value Two's complement ...

9's and 10's Complement of Decimal (Base-10) Number : Simply, 9's Complement of a decimal number is the subtraction of it's each digits from 9. For example, 9's complement of decimal number 2005 is 9999 - 2005 = 7994. 10's Complement of decimal number is 9's complement of given number plus 1 to the least significant bit (LSB ...

The decimal number 0.1 is represented in binary as e = −4; s = 110011001100110011001101, which is ... For this reason, financial software tends not to use a binary floating-point number representation. [61] The "decimal" data type of the C# and Python programming languages, ...