Decimal number line display: Worksheets | Teachers Pay Teachers

Decimal Numbers on a Number Line

Audio:

The frog hopped 1.5 feet.

• If the frog were a math expert and jumped on a number line, where would the decimal number 1.5 be on that number line?

A number written with digits after the decimal point (.) is called a decimal number.

A decimal number is not a whole number, which means it is similar to a fraction in that it represents part of a whole.

For example, a decimal number is a number that is between two whole numbers and has a decimal point. Like 0.8 is between the whole numbers 0 and 1. Also, 2.3 is a decimal number between the whole numbers 2 and 3.

A number line can help you understand decimal numbers.

Understanding place value and how to say the decimal number is important to understand how decimals relate to fractions.

Let’s take the distance the frog hopped from before. It hopped 1.5 feet. Now, you may be tempted to say «one point five» feet, but that is incorrect.

Let’s quickly review a decimal place value chart.

	ones (whole number)	.	tenths	hundredths
	1	.	5
	1	.	0	5

 

Using the place value chart, 1.5 is said «one and five tenths.» Also, 1.05 is said as «one and five hundredths.»

• Do you see how these decimals relate to fractions?

A number line helps you visualize numbers by evenly spacing the numbers on a horizontal line. The decimal numbers increase the farther to the right you move. This will be important when it comes to comparing decimals!

On the decimal number line below, there are 10 evenly-spaced sections from 0 to 1. If you start counting at 0, you will count each tenth until you reach «ten tenths,» which is equal to one whole.

You can relate money to decimals on a number line by thinking of a dime. A dime is a tenth of a dollar, so ten dimes equal a dollar, or one whole.

Some number lines involve hundredths.

There are 10 evenly spaced sections from 0 to 0.1. If you start counting at 0, you will count each hundredth until you reach «ten hundredths,» which is equal to one tenth.

Look at the problem from the beginning of the lesson:

The frog hopped 1.5 feet. Where would the decimal number 1.5 be on a number line?

The decimal number 1.5 has a whole number of 1, which is the number to the left of the decimal. The number 1.5 is halfway between 1 and 2.

In your own words, explain out loud how a number line can help you visualize and plot decimal numbers.

In the Got It? section, you will practice locating decimal numbers on a number line.

What is Number Line? — Definition and Examples

Understanding the concept of the number line can be considered the first step of learning mathematics. The first thing taught in learning high-order math is the number line. But what is a number line? 

Let us split the word number line into two individual words, number and line. Numbers, as we all know, are 0, 1, 2, 3, 4,…….., infinity. The line indicates a length extending from one point to another having no width. Thus we can say that a number line is a line that contains numbers on it. 

Definition of a Number Line

 

A number line is the simplest graph that depicts numbers in a straight line. This line compares integers that are evenly spaced on an endless line that extends horizontally or vertically on both sides. The numbers on a horizontal number line increase as we walk to the right and decrease as we walk to the left. 

Locating Numbers on a Number Line

On a number line, arithmetic operations of numbers can be better described. First, one needs to understand how to find numbers on a number line. A number line’s midway point is zero. Positive numbers inhabit the right side of the zero on the number line, whereas negative numbers fill the left side of the zero. The value of a number falls as we travel from 0 to the left side and increases as we move from 0 to the right side. 

Like integers, fractions, and decimals, almost every number, rational, or irrational numbers can be simply expressed on a number line. The only numbers that cannot be depicted on the number line are imaginary. They are represented on a special number line known as the Argon plot. In this article, we shall only focus on numbers represented on the number line.  

Negative and Positive Number Line

As described earlier, positive and negative numbers are the two sorts of numbers represented on a number line. Let us learn more about this segment’s positive and negative number lines. 

A negative number line is the segment of the number line on the left side of zero. The area on the right side of zero, on the other hand, is known as a positive number line since it contains all positive integers. From both ends, it may be extended indefinitely. There is no boundation on how long a number line can be. It can be as short as to fit inside your notebook and as long as to cover the entire earth a million times. Below are some of the qualities of the negative and positive number lines. 

• The numbers on the right are greater in value than those on the left.
• The left-hand numerals are smaller in value than the right-hand numbers.
• The point of origin, or the midpoint of the number line, is 0 (zero).
• The numbers are spaced out evenly. This means that if you have to make a number line, you have to take numbers arranged in a proper sequence. A number line cannot be made like 1, 2, 5, 11,…….99. A number line is always made with proper spacing like 5, 10, 15, 20, ……., 100 or 1, 3, 5, 7, ….. 99, etc.

In the next section of this article, you will understand this concept in detail, where we shall learn how to make/draw a number line.

How to Draw a Number Line?

Read this step-by-step procedure created by our expert faculty, which will help you easily understand the concepts of number lines. 

• Step 1: Make a straight horizontal or vertical line (choice is yours) on a sheet of paper. Add two arrows at both ends. These arrows specify that the line extends to infinity on the left and the right side. 

• Step 2: This step is the most crucial. This step involves choosing an appropriate scale to make the markings on the number line. For instance, if you have to plot numbers from 0 to 4, then you can choose a scale of 1 or 2. For a scale of 1, the numbers on the number line shall be 0, 1, 2, 3, and 4. On a scale of 2, the numbers on the number line will be 0, 2, and 4. Thus choose your scaling accordingly with an equal number of intervals between any two numbers.  

• Step 3: Again, consider the number line from 0 to 4. You have already fixed the interval as 1, so the points will be 0, 1, 2, 3, and 4. The third step involves marking all these points. They will act as identifiers. 

• Step 4: The final step is to find the number of your choice and highlight it with the help of a circle. 

This is how you can easily draw and mark any number on the number line. We have learned concepts related to the number line with natural numbers, whole numbers, and integers. We shall know how to depict decimals and fractions on a number line in the next modules.

Number Line with Decimals

Decimal numbers contain a whole number part and a fractional part separated by the dot. For example, 13.3353, where 13 is the whole number part while 3353 is the fractional part of the decimal. But how do we represent this on the number line? Follow the steps mentioned below to completely grasp the concept of marking a decimal point on the number line. 

For instance, represent 1.8 on the number line?

• Step 1: Draw a straight line.  
• Step 2: Find the whole number and the fractional number. In this case, 1 is the whole number, and 0.8 is the fraction. According to the whole number mark numbers with equal intervals. In this case, 0, 1, and 2.
• Step 3: We know that 1.8 lies between 1 and 2, making smaller marks with a value of 0.1 each. Make ten equal intervals from 1 to 2. Thus, the numbers between 1 and 2 are 1.1, 1.2, 1.3,….., and so on.
• Step 4: Find 1.8 and mark it with a circle.

This is how you mark any decimal point on the number line. Hope you all must have understood the concept. 

Inequalities on a Number Line

We have learned how to mark numbers equal to integers or decimals until now. In this section, we shall learn how to deal with inequalities. Inequality means all the numbers are either greater or smaller than a specified number. Inequality is the concept of comparing two or more quantities together. The symbols used for depicting inequalities are less than symbol ‘<’, less than equal to symbol ‘<=’, greater than symbol ‘>’, greater than equal to symbol ‘>=’, and not equal to ‘≠’ symbol.  

To understand inequality on a number line, look at the example below and follow the steps mentioned in the example:

For instance, represent a <= 3 on the number line.

Solution: 

• Draw a straight line and mark the starting point 0 anywhere on the line. 
• It is given that a <= 3, therefore mark a point 3 on the right side of 0 by taking equal intervals of 1. Thus the points will be 0, 1, 2, and 3.
• Make a circle at three and start to shade to the left side of 3 (from 2 to 0).
• This shaded region is the inequality required in the example.

Remember these points while solving problems related to inequality:

1. ‘Greater than equal to X and ‘less than equal to X means that the number x must also be included in the shading.
2.  ‘Greater than x’ and ‘less than x’ means that the number x must not be included while shading the inequality. 
3. Negative numbers follow the rules opposite to that positive numbers. In a negative number larger the number after a negative sign, the smaller the value of that integer. For example, -10 is greater than -20. In reality, -1 is the greatest negative integer. 

Practice Questions 

On the number line, multiply 3 by -9. 

From -2 on the number line, subtract 8. 

A number line representing full numbers should be drawn. 

Use a number line to display the negative integers. 

Fun Fact 

A blank number line was first proposed as a visual model or diagram for solving addition and subtraction operations. An empty or blank number line is a visual representation of a number line that contains no numbers or markers and is primarily used to solve word problems. 

Frequently Asked Question?

1.

How to represent a number on the number line?

A number can be represented on the number line by placing it at a point that is equal to its value.

For example, if you want to represent the number 12, then you would place it at 12 units from 0 (the origin). Similarly, if you want to represent the number -3, then you would place it at -3 units from 0.

2.

What does a number line look like?

A number line is a line on which numbers are drawn. The numbers could be integers, fractions, or decimals. The number line may be drawn horizontally, vertically, or slanted. It may also contain other mathematical signs such as + and – signs. The number line is used to represent any quantity that can be measured in discrete units.

3.

What numbers are greater on a number line?

In the number line, numbers that are greater than 0 are those that are on the right side of 0. If a number is to the left of 0, it is negative and therefore smaller than 0. As you move along the number line to the right, your numbers get bigger. When you move up, they get smaller.

4.

How do you plot a number line?

Plotting a number line is a great way to visualize numbers and their relationships to one another. You can plot the numbers on a horizontal line with evenly spaced intervals between them. You can also plot them vertically on an axis with increasing values from left to right. The graph will be a linear function, which means that the y value varies directly as x increases.

5.

How to graph an equation on a number line?

Graphing an equation on a number line is a great way to visualize the relationship between numbers.

Converting numbers to different number systems

A number system is a systematic way of representing numbers with character symbols and uses a base value to conveniently group numbers in a concise form. The most common number system is decimal, which has a base value of 10, and the character set is 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. However, there are other number systems that may be more efficient for a particular goals. For example, because computers use a Boolean value for calculations and operations, the binary number system, which has a base value of 2, is used to perform calculations and operations. 0003

Microsoft Office Excel has several functions that you can use to convert numbers to and from number systems:

Number system	Base value	Character set
Binary	2	0.1
Octal	8	0, 1, 2, 3, 4, 5, 6, 7
Valid.	10	0, 1, 2, 3, 4, 5, 6, 7, 8 and 9
Hexadecimal	16	0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

Use the DW.W.E. function for this.

A B Formula Description (result) =B. E.DEC(1100100) Converts binary 1100100 to decimal (100). =B.E.DEC(1111111111) Converts binary 11111111111 to decimal (-1)

Use the DIR EX function for this.

1 2 3 4

A B Formula Description (result) =BHHH(11111011; 4) Converts binary 11111011 to hexadecimal with 4 digits (00FB) =BHHH(1110) Convert binary 1110 to hexadecimal (E) =BHHH(1111111111) Convert binary 1111111111 to hexadecimal (FFFFFFFFFF)

To do this, use the ROI function.

1 2 3 4

A B Formula Description (result) =BIN OCT(1001, 3) Converts binary number 1001 to 3-digit eighth (011). =BIN OCT(1100100) Converts the binary number 1100100 to an eighth (144). =BIN OCT(1111111111) Converts binary number 1111111111 to eighth (77777777777)

Use the DEC2BIN function for this.

A B Formula Description (result) =DEC. V.DV(9;4) Converts decimal 9 to binary with 4 digits (1001). =DEC.V.DV(-100) Converts decimal -100 to binary (1110011100)

Use the DEC2HEX function for this.

A B Formula Description (result) =DEC. HEX(100;4) Converts decimal 100 to hex. with 4 characters (0064). =DEC.HEX(-54) Converts decimal -54 to hexadecimal (FFFFFFFFCA)

Use the DEC2OCT function for this.

A B Formula Description (result) =DEC. EIGHT(58;3) Converts decimal 58 to octal (072). =DEC.OIGHT(-100) Converts decimal to octal (7777777634)

Use the HEX2BIN function for this.

1 2 3 4

A B Formula Description (result) =HEX MOTOR(«F»; 8) Converts the letter F to 8-digit binary (00001111). =HEX MOTOR(«B7») Converts hexadecimal B7 to binary (10110111) =HEX MOTOR(«FFFFFFFFFF») Converts hexadecimal FFFFFFFFFF to binary (11111111111)

Use the HEX2DEC function for this.

1 2 3 4

A B Formula Description (result) =HEX. DEC(«A5») Converts A5 hex to decimal (165). =HEX.DEC(«FFFFFFFF5B») Converts hexadecimal FFFFFFFF5B to decimal (-165). =HEX.DEC(«3DA408B9») Converts hexadecimal 3DA408B9 to decimal (1034160313).

Use the HEX2OCT function for this.

1 2 3 4

A B Formula Description (result) =HEX. TO.OCT(«F», 3) Converts the eighth F to the eighth with 3 digits (017). =HEX.TO.OCT(«3B4E») Converts the hexadecimal number 3B4E to an eighth number (35516). =HEX TO OCT(«FFFFFFFF00») Converts hexadecimal FFFFFFFF00 to eighth (777777400)

To do this, use the OCT.MOT function.

A B Formula Description (result) =OCT. H.DW(3; 3) Converts the 8th 3 to binary with 3 digits (011). =OCT.H.DW(7777777000) Converts the eighth 7777777000 to binary (10000000000)

To do this, use the OCT.CU function.

A B Formula Description (result) =OCT. DEC(54) Converts octal 54 to decimal (44). =OCT.DEC(7777777533) Converts octal 7777777533 to decimal (-165).

Use the OCT.EX function for this.

A B Formula Description (result) =OCT. HEX(100, 4) Converts the eighth of 100 to hexadecimal with 4 digits (0040). =OCT.HEX(7777777533) Converts the eighth number 7777777533 to hexadecimal (FFFFFFFF5B)

Binary code — frwiki.wiki

The word «Wikipedia» is represented by binary ASCII code, consisting of 9 bytes (72 bits).

The binary code represents text, a processor instruction, or other data using a two-character system. In the binary system, the two-character system «0» and «1» is often used. Binary code assigns a combination of binary digits, also known as bits, to each character, instruction, etc. For example, an eight-bit binary string can represent any of the 256 possible values, and thus represent a large number of different elements.

In computing and telecommunications, binary codes are used for various methods of encoding data, such as character strings, into bit strings. These methods can be used with any fixed-width or variable-width string. In fixed-width binary, each letter, digit, or other character is represented by a string of bits of the same length; this bit string, interpreted as a binary number, is usually displayed in code tables in octal, decimal, or hexadecimal notation. There are many character sets and many character encodings for them.

Bit string, interpreted as binary number, can be converted to decimal number. For example, the lowercase letter a , if represented by the bit string 01100001 (as in standard ASCII code), can also be represented by the decimal number «97».

Summary

• 1 History of binary code
• 2 Other forms of binary code
  • 2. 1 Braille
  • 2.2 Bagua
• 3 Coding systems
  • 3.1 ASCII code
  • 3.2 Binary coded decimal number
• 4 First use of binary codes
• 5 Current use of binary code
• 6 Weight of binary codes
• 7 See also
• 8 links
• 9 External links

History of the binary code

Gottfried Leibniz

The modern binary number system, the basis of the binary code, was invented by Gottfried Leibniz in 1689year and appeared in his paper An Explanation of Binary Arithmetic in 1703.

Full title — «Explanation of binary arithmetic, which uses only the symbols O and I, with remarks on its usefulness and how it conveys the meaning of the ancient Chinese figures Fohi.» The Leibniz system uses 0 and 1, just like the modern binary number system.

Leibniz discovered I Ching through the French Jesuit Joachim Bouvet and noted with admiration how its hexagrams corresponded to the binary numbers 0-111111, and concluded that this mapping was evidence of China’s major achievements in the field of philosophical mathematics, which he admired. Leibniz saw in the hexagrams a confirmation of the universality of his own religious faith.

Binary numbers were at the center of Leibniz’s theology. He believed that binary numbers symbolize the Christian idea creatio ex nihilo or creation from nothing. Leibniz was trying to find a system that would transform verbal statements from logic into a purely mathematical system. After his ideas were ignored, he discovered a classical Chinese text called I Ching or «Book of Changes» which used a type of binary code. The book confirmed his theory that life can be simplified or reduced to a series of simple statements.

He created a system consisting of strings of zeros and ones. During this period, Leibniz had not yet found a use for this system.

Binary systems before Leibniz also existed in the ancient world. Ching noted that Leibniz discovered dates from IX — ^th century BC. AD in China. The binary system of the I Ching , a divination text, is based on the duality of Yin and Yang. Binary slit drums were used to encode messages in Africa and Asia. Indian scholar Pingala (circa V — — II — . BC. ) developed a binary system for describing prosody in Chandashutram .

George Buhl

Inhabitants of the island of Mangarevu in French Polynesia using a hybrid system of decimal, binary, before 1450 XI — ^th century, the scientist and philosopher Shao Yong developed a method of organizing the hexagram, which, however unwittingly, in sequence from 0 to 63, represented in binary, where yin is 0, yang is 1, and the least significant bit is at the top. The order is also the lexicographic order of the sextuples of elements selected from a set of two elements.

In 1605, Francis Bacon discussed a system by which the letters of the alphabet could be converted into sequences of binary digits, which could then be encoded as subtle font variations in any text. Significantly in relation to the general theory of binary encoding, he added that this technique could be used with any object: “provided that these objects can only have a double difference; as with bells, trumpets, lanterns, torches, muskets, and all such instruments.» .

In 1847, George Boole published an article entitled «The Mathematical Analysis of Logic», which describes an algebraic logical system, hereinafter referred to as Boole’s algebra. Boole’s system was based on a binary yes-no approach involving three basic operations: AND, OR, and NO. This system was not used and was forgotten until MIT graduate student, Claude Shannon, realized that the Boolean algebra he had learned was like an electrical circuit. Shannon wrote her dissertation at 1937, in which her discoveries were realized. Shannon’s thesis was the starting point for using binary code in practical applications such as computers, electrical circuits, etc.

Other forms of binary code

Taoist bagua

Bit string is not the only type of binary code. In general, a binary system is one that only allows two choices, such as a switch in an electronic system, or a simple test for true or false.

Braille

Braille is a type of binary code widely used by blind people to read and write by touch, named after its creator, Louis Braille. This system consists of grids of six dots each, three per column, in which each dot has two states: high or low. Various combinations of raised and flat dots can represent any letter, number, and punctuation mark.

Bagua

In Bagua charts used in studies 9 are presented1158 feng shui , in Taoist cosmology and ching . Ba gua consists of 8 trigrams; bā means 8 and guà means the figure of divination. The same word is used for 64 guas (hexagrams). Each figure combines three lines ( yáo ) that are either interrupted ( yáo ) or not interrupted ( yáo ). Relationships between trigrams are presented in two forms: the original bagua «Old Sky» or «Fushi» and the manifested Bagua , «Later Heaven» or «King Wen» . (See also King Wen’s sequence of 64 hexagrams).

Coding systems

ASCII code

ASCII code (ascii) uses a 7-bit binary code to represent text and other characters in computers, communications equipment, and other devices. Each letter or symbol is assigned a number from 0 to 127. For example, the lowercase letter «a» is represented by 1100001 as a bit string («97» in decimal).

Decimal number with binary code

Binary Decimal (BCD) is an integer encoded in binary notation using a nibble of 4 bits to encode decimal digits. Four binary bits can encode up to 16 different values; but in DCB encoded numbers, only the first ten values ​​of each nibble are valid and encode zero to nine decimal digits. The remaining six values ​​are invalid and may cause a computer exception or undefined behavior, depending on the computer’s implementation of DCB arithmetic.

DCB arithmetic is sometimes preferred over floating point numeric formats in business and financial applications where complex rounding of floating point numbers is inappropriate.

First use of binary codes

• 1875: Emile Baudot «Adding binary strings to his encryption system», which eventually led to today’s ASCII.
• 1884: A linotype machine in which matrices are sorted into their respective channels after using a binary code guide.
• 1932: counter «balance of two» CE Wynn-Williams
• 1937: Alan Turing’s electromechanical binary multiplier
• 1937 «excess-3» code from Stiebics in complex computer
• 1937: Atanasoff Computer — Berry
• 1938: Konrad Zuse Z1

Current use of binary code

Most modern computers use binary encoding for instructions and data. CDs, DVDs, and Blu-ray discs represent sound and video digitally in binary form. Telephone calls are routed digitally over long distances and mobile networks using coded pulse modulation as well as VoIP networks.

Binary code weight

Binary code weight, as defined in the constant weight code table, is the Hamming weight of the binary words encoding the represented words or sequences.

see also

• Binary number
• Binary code list
• Binary file
• Unicode
• Gray code

Recommendations

1. ↑ Godefroy-Guillaume Leibniz, « An Explanation of Binary Arithmetic Using Only the Symbols O and I, with Remarks on Its Usefulness and the Significance of the Ancient Chinese Fohi Figures «, Mémoires de mathatique et of Physics from Royal Academy of Sciences, Royal Academy of Sciences , 1703
2. ↑ Eric J. Ayton, Leibniz: biography , Taylor & Francis, 1985 , 245–8 pp. (ISBN 978-0-85274-470-3)
3. ↑ ^{a and b} (in) Smith , Leibniz: what rationalist? : What rationalist? , Dordrecht/London, Springer 2008
4. ↑ Yuen-Ting Lai, Leibniz, Mysticism and Religion , Springer, 1998 , 149–150 pp. (ISBN 978-0-7923-5223-5, read online)
5. ↑ Gottfried Wilhelm Leibniz (1646-1716)
6. ↑ Edward Hacker, Steve Moore and Lorraine Pascoe, I Ching: An Annotated Bibliography , Routledge 2002 , 336 p. (ISBN 978-0-415-93969-0, read online)
7. ↑ ^{a and b} Jonathan Schektman, Pioneering Scientific Experiments, Inventions and Discoveries of the 18th Century , Greenwood Publishing, 2003 , 317 pp. (ISBN 978-0-313-32015-6, read online)
8. ↑ Julio Sanchez and Maria P. Canton, Microcontroller Programming: The PIC Microchip , Boca Raton, FL, CRC Press, 2007
9. ↑ W.S. Anglin and J. Lambek, The Heritage of Thales , Springer, 1995, (ISBN 0-387-94544-X)
10. ↑ Bender and Beller, « Mangarevan Invention of Binary Steps to Simplify Calculations «, Proceedings of the National Academy of Sciences , vol. 111, P ^O 4, December 16, 2013 , pp. 1322–1327 (PMID 24344278, PMCID 3

    3, DOI 10.1073 / PNAS.1309160110) ) )

11. ↑ Ryan, «» Leibniz Binary and Shao Yong «Yijing», « Philosophy of East and West , Vol. 46, p ^o 1, Jan. 1996 , pp. 59–90 (DOI 10.2307/1399337, JSTOR 139 9336)
12. ↑ ^{a and b} Bacon, « The Development of Learning «, London 1605 , pp. Chapter 1
13. ↑ What’s so big about boolean algebra?
14. ↑ Claude Shannon (1916-2001)
15. ↑ Richard Wilhelm, I Ching or Book of Changes , Princeton, NJ, Princeton University Press, 1950 , 266, 269 pp. (ISBN 978-0-691-09750-3, read online)
16. ↑ Cowlishaw, « General Decimal Arithmetic «, IBM, 2015 (as of January 2, 2016)
17. ↑ ^{a b and c} Glazer 1971
18. ↑ Table of constant weight binary codes

External links

• Binary coding
• Sir Francis Bacon’s BiLiteral Cypher system preceded the binaar system.
• (en) Eric W. Weisstein, « {{{title}}} «, on MathWorld
• Table of Common Binary Codes. General table of binary codes.
• Table of non-linear binary codes. Maintainu Simon Litsyn, E.M. Raines and NJA Sloane. Updated to 1999
• Anton Glazer , History of binary and other non-decimal numbering , Tomasz, 1971 , 218 pp.