SlideShare a Scribd company logo
Zero a Important
Discovery
Discovery Of Zero
Zero was first discovered
by Indian Mathematician
Aryabhata around 520AD.
By the Middle Ages, around
the early 1200’s, this
concept had come to
Western society.
Before Zero
• Numbers were used for thousands of years before they
used zero
• Historical records show different path towards the concept
• Zero made appearances only to vanish again
• Mathematicians were searching for it ,yet did not recognize
its fundamental significance
Egyptians
Egyptian Number system were
used as early as 3500 B.C.E.
Egypt Number system did not
have or need a zero. Even
without zero, Egyptians were
masters of Mathematics. Their
Number system contained icons
to represent certain numbers and
that icon was repeatedly used to
represent the bigger number like
4200 is represented by
Greeks
The Greeks came up with a new
and better number system and
brought mathematics to its highest
point in ancient times around 500
BC. It was much more
sophisticated system. It was made
it so that letter was not repeated.
Greeks also didn’t had zero and
used a different symbol to
represent each number.
Romans
The Romans also had a number system
and it is still used in modern life. It was
earlier less sophisticated than Egyptian’s
system. In Roman’s system they use
repeated number like 4505 requires 10
words to represent the number in Roman
number system MMMMCCCCCV.
Even these number systems were sophisticated without the
zero but still they weren’t advanced and people were lacking
arithmetic and accounting behind because of lack of zero. So
the India mathematician Aryabhata finded out the importance
and need of zero at first it was represented by a dot and was
called empty By 500 C.E., the Hindus use a small circle to
represent Zero!
This circle was recognized as a numeral
Invention of zero
• Arabic people recognized the value of the Hindu
system
• They adapted the numerals and computation
• Then spread the ideas in their travels
• The zero was named with the Arabic word sifz
• The actual word “zero” came from Italy
Hindu-Arabic Numerals
Europeans resisted Hindu-Arabic numerals
It seemed strange to them
The numerals, including zero, were not accepted
Florence, Italy, passed a law prohibiting the use of the
numerals
0 could be changed to look like 6 or 9
Slowly, the numbers became accepted
Zero Slowly got
accepted
Spreading The News
Leonardo of Pisa,
known as Fibonacci
• Born to a merchant family
living in North Africa
• Learned Hindu-Arabic
numerals from his Arabic
tutors
• He brought the news of zero
and new computational
methods to Europe in his
book
Importance of Zero
• It plays a central role in math.
i. As a additive identity
ii. in integers ,real nos. and other algebraic structures
iii. As a place holder in the place value system.
iv.It has been called a natural no and has a special role in measuring of
physical quantities.
• In the real-number system, 0 is the only number that is neither
negative nor positive.
It represents the boundary between the negative and the positive
numbers.
This property makes the natural starting point, or origin, on many
scales, as on the coordinate axes and on thermometers.
Zero is special
• If we add 0 to any number, the sum is the original
number
• Same is true for subtraction
• If you multiply any number by 0, the product is 0
• If you raise any nonzero number to the power of 0, the
resulting number is 1
• If you divide 0 by any nonzero, the quotient is 0
• Any number divided by 0 is undefined
Made By Anshul
Sawhney
Zero Importance for class 9

More Related Content

What's hot

evolution of number system
evolution of number systemevolution of number system
evolution of number system
Aparna
 
Origin, history and application of Zero.
Origin, history and application of Zero.Origin, history and application of Zero.
Origin, history and application of Zero.
Vishwanath Foundation
 
History of Zero - Mathematics
History of  Zero - MathematicsHistory of  Zero - Mathematics
History of Zero - Mathematics
divya rajput
 
Zero
ZeroZero
Zzzzz
ZzzzzZzzzz
Zzzzz
Prat Ami
 
History of pi
History of piHistory of pi
History of pi
Kunal Yadav
 
Real numbers
Real numbersReal numbers
Real numbers
Farhan Alam
 
JOURNEY OF MATHS OVER A PERIOD OF TIME..................................
JOURNEY OF MATHS OVER A PERIOD OF TIME..................................JOURNEY OF MATHS OVER A PERIOD OF TIME..................................
JOURNEY OF MATHS OVER A PERIOD OF TIME..................................
Pratik Sidhu
 
Pi Ppt
Pi PptPi Ppt
Pi Ppt
cakell01
 
number system
number system number system
number system
murugesh waran
 
natural numbers ppt for class 9 th
natural numbers ppt for class 9 thnatural numbers ppt for class 9 th
natural numbers ppt for class 9 th
akhilprasadkerla
 
History of Math
History of MathHistory of Math
History of Math
Gordana Divic
 
Story of pi
Story of piStory of pi
Story of pi
Zamaan Khan Lodhi
 
Ancient math
Ancient mathAncient math
Ancient math
Sheena Joy Patoc
 
History of Maths
History of MathsHistory of Maths
History of Maths
Judson Jude
 
number system ppt
number system ppt number system ppt
number system ppt
Akash dixit
 

What's hot (20)

Zero
ZeroZero
Zero
 
History and mystery of zero
History and mystery of zeroHistory and mystery of zero
History and mystery of zero
 
evolution of number system
evolution of number systemevolution of number system
evolution of number system
 
Origin, history and application of Zero.
Origin, history and application of Zero.Origin, history and application of Zero.
Origin, history and application of Zero.
 
History of Zero - Mathematics
History of  Zero - MathematicsHistory of  Zero - Mathematics
History of Zero - Mathematics
 
Zero
ZeroZero
Zero
 
Zzzzz
ZzzzzZzzzz
Zzzzz
 
History of pi
History of piHistory of pi
History of pi
 
Real numbers
Real numbersReal numbers
Real numbers
 
JOURNEY OF MATHS OVER A PERIOD OF TIME..................................
JOURNEY OF MATHS OVER A PERIOD OF TIME..................................JOURNEY OF MATHS OVER A PERIOD OF TIME..................................
JOURNEY OF MATHS OVER A PERIOD OF TIME..................................
 
Pi Ppt
Pi PptPi Ppt
Pi Ppt
 
number system
number system number system
number system
 
natural numbers ppt for class 9 th
natural numbers ppt for class 9 thnatural numbers ppt for class 9 th
natural numbers ppt for class 9 th
 
The beauty of mathematics
The beauty of mathematicsThe beauty of mathematics
The beauty of mathematics
 
History of pi
History of piHistory of pi
History of pi
 
History of Math
History of MathHistory of Math
History of Math
 
Story of pi
Story of piStory of pi
Story of pi
 
Ancient math
Ancient mathAncient math
Ancient math
 
History of Maths
History of MathsHistory of Maths
History of Maths
 
number system ppt
number system ppt number system ppt
number system ppt
 

Similar to Zero Importance for class 9

ZERO.ppt
ZERO.pptZERO.ppt
ZERO.ppt
JayVardhanBagh1
 
Math term paper
Math term paperMath term paper
Math term paper
Zulfikar Dipto
 
History of computer
History of computerHistory of computer
History of computer
Jesus Obenita Jr.
 
Numbers ppt
Numbers pptNumbers ppt
numerals.pptx
numerals.pptxnumerals.pptx
numerals.pptx
AsadbekAbdumannopov
 
Numbers
NumbersNumbers
Numbers
Battula Pavan
 
Number Systems
Number SystemsNumber Systems
Number Systems
美晴 福田
 
Ancient Indian Mathematics - Report-1
Ancient Indian Mathematics - Report-1Ancient Indian Mathematics - Report-1
Ancient Indian Mathematics - Report-1Muhammed Sobhy
 
The History Of Algebra
The History Of AlgebraThe History Of Algebra
Article
ArticleArticle
Article
deepuplr
 
leonardofibonacci-140918135347-phpapp02.pdf
leonardofibonacci-140918135347-phpapp02.pdfleonardofibonacci-140918135347-phpapp02.pdf
leonardofibonacci-140918135347-phpapp02.pdf
JayArRodriguez2
 
Leonardo fibonacci
Leonardo fibonacciLeonardo fibonacci
Leonardo fibonacci
Athira RL
 
Leonardo de pisa (Santiago y José Manuel)
Leonardo de pisa (Santiago y José Manuel)Leonardo de pisa (Santiago y José Manuel)
Leonardo de pisa (Santiago y José Manuel)jmuceda
 
History of hindu arabic numerals
History of hindu arabic numeralsHistory of hindu arabic numerals
History of hindu arabic numeralsAlfiramita Hertanti
 

Similar to Zero Importance for class 9 (14)

ZERO.ppt
ZERO.pptZERO.ppt
ZERO.ppt
 
Math term paper
Math term paperMath term paper
Math term paper
 
History of computer
History of computerHistory of computer
History of computer
 
Numbers ppt
Numbers pptNumbers ppt
Numbers ppt
 
numerals.pptx
numerals.pptxnumerals.pptx
numerals.pptx
 
Numbers
NumbersNumbers
Numbers
 
Number Systems
Number SystemsNumber Systems
Number Systems
 
Ancient Indian Mathematics - Report-1
Ancient Indian Mathematics - Report-1Ancient Indian Mathematics - Report-1
Ancient Indian Mathematics - Report-1
 
The History Of Algebra
The History Of AlgebraThe History Of Algebra
The History Of Algebra
 
Article
ArticleArticle
Article
 
leonardofibonacci-140918135347-phpapp02.pdf
leonardofibonacci-140918135347-phpapp02.pdfleonardofibonacci-140918135347-phpapp02.pdf
leonardofibonacci-140918135347-phpapp02.pdf
 
Leonardo fibonacci
Leonardo fibonacciLeonardo fibonacci
Leonardo fibonacci
 
Leonardo de pisa (Santiago y José Manuel)
Leonardo de pisa (Santiago y José Manuel)Leonardo de pisa (Santiago y José Manuel)
Leonardo de pisa (Santiago y José Manuel)
 
History of hindu arabic numerals
History of hindu arabic numeralsHistory of hindu arabic numerals
History of hindu arabic numerals
 

Recently uploaded

The Impact of Artificial Intelligence on Modern Society.pdf
The Impact of Artificial Intelligence on Modern Society.pdfThe Impact of Artificial Intelligence on Modern Society.pdf
The Impact of Artificial Intelligence on Modern Society.pdf
ssuser3e63fc
 
欧洲杯买球平台-欧洲杯买球平台推荐-欧洲杯买球平台| 立即访问【ac123.net】
欧洲杯买球平台-欧洲杯买球平台推荐-欧洲杯买球平台| 立即访问【ac123.net】欧洲杯买球平台-欧洲杯买球平台推荐-欧洲杯买球平台| 立即访问【ac123.net】
欧洲杯买球平台-欧洲杯买球平台推荐-欧洲杯买球平台| 立即访问【ac123.net】
foismail170
 
133. Reviewer Certificate in Advances in Research
133. Reviewer Certificate in Advances in Research133. Reviewer Certificate in Advances in Research
133. Reviewer Certificate in Advances in Research
Manu Mitra
 
太阳城娱乐-太阳城娱乐推荐-太阳城娱乐官方网站| 立即访问【ac123.net】
太阳城娱乐-太阳城娱乐推荐-太阳城娱乐官方网站| 立即访问【ac123.net】太阳城娱乐-太阳城娱乐推荐-太阳城娱乐官方网站| 立即访问【ac123.net】
太阳城娱乐-太阳城娱乐推荐-太阳城娱乐官方网站| 立即访问【ac123.net】
foismail170
 
Interactive Dictionary AIDS-B.pptx aaaaaaaaaaaaaaaaaaaaaaaaaa
Interactive Dictionary AIDS-B.pptx aaaaaaaaaaaaaaaaaaaaaaaaaaInteractive Dictionary AIDS-B.pptx aaaaaaaaaaaaaaaaaaaaaaaaaa
Interactive Dictionary AIDS-B.pptx aaaaaaaaaaaaaaaaaaaaaaaaaa
23211a7274
 
DIGITAL MARKETING COURSE IN CHENNAI.pptx
DIGITAL MARKETING COURSE IN CHENNAI.pptxDIGITAL MARKETING COURSE IN CHENNAI.pptx
DIGITAL MARKETING COURSE IN CHENNAI.pptx
FarzanaRbcomcs
 
Operating system. short answes and Interview questions .pdf
Operating system. short answes and Interview questions .pdfOperating system. short answes and Interview questions .pdf
Operating system. short answes and Interview questions .pdf
harikrishnahari6276
 
DOC-20240602-WA0001..pdf DOC-20240602-WA0001..pdf
DOC-20240602-WA0001..pdf DOC-20240602-WA0001..pdfDOC-20240602-WA0001..pdf DOC-20240602-WA0001..pdf
DOC-20240602-WA0001..pdf DOC-20240602-WA0001..pdf
Pushpendra Kumar
 
My Story of Getting into Tech By Gertrude Chilufya Westrin
My Story of Getting into Tech By Gertrude Chilufya WestrinMy Story of Getting into Tech By Gertrude Chilufya Westrin
My Story of Getting into Tech By Gertrude Chilufya Westrin
AlinaseFaith
 
Midterm Contract Law and Adminstration.pptx
Midterm Contract Law and Adminstration.pptxMidterm Contract Law and Adminstration.pptx
Midterm Contract Law and Adminstration.pptx
Sheldon Byron
 
How to Master LinkedIn for Career and Business
How to Master LinkedIn for Career and BusinessHow to Master LinkedIn for Career and Business
How to Master LinkedIn for Career and Business
ideatoipo
 
欧洲杯投注app-欧洲杯投注app推荐-欧洲杯投注app| 立即访问【ac123.net】
欧洲杯投注app-欧洲杯投注app推荐-欧洲杯投注app| 立即访问【ac123.net】欧洲杯投注app-欧洲杯投注app推荐-欧洲杯投注app| 立即访问【ac123.net】
欧洲杯投注app-欧洲杯投注app推荐-欧洲杯投注app| 立即访问【ac123.net】
foismail170
 
131. Reviewer Certificate in BP International
131. Reviewer Certificate in BP International131. Reviewer Certificate in BP International
131. Reviewer Certificate in BP International
Manu Mitra
 
Andrea Kate Portfolio Presentation.pdf
Andrea Kate  Portfolio  Presentation.pdfAndrea Kate  Portfolio  Presentation.pdf
Andrea Kate Portfolio Presentation.pdf
andreakaterasco
 
Chapters 3 Contracts.pptx Chapters 3 Contracts.pptx
Chapters 3  Contracts.pptx Chapters 3  Contracts.pptxChapters 3  Contracts.pptx Chapters 3  Contracts.pptx
Chapters 3 Contracts.pptx Chapters 3 Contracts.pptx
Sheldon Byron
 
Brand Identity For A Sportscaster Project and Portfolio I
Brand Identity For A Sportscaster Project and Portfolio IBrand Identity For A Sportscaster Project and Portfolio I
Brand Identity For A Sportscaster Project and Portfolio I
thomasaolson2000
 
皇冠体育- 皇冠体育官方网站- CROWN SPORTS| 立即访问【ac123.net】
皇冠体育- 皇冠体育官方网站- CROWN SPORTS| 立即访问【ac123.net】皇冠体育- 皇冠体育官方网站- CROWN SPORTS| 立即访问【ac123.net】
皇冠体育- 皇冠体育官方网站- CROWN SPORTS| 立即访问【ac123.net】
larisashrestha558
 
Personal Brand exploration KE.pdf for assignment
Personal Brand exploration KE.pdf for assignmentPersonal Brand exploration KE.pdf for assignment
Personal Brand exploration KE.pdf for assignment
ragingokie
 
han han widi kembar tapi beda han han dan widi kembar tapi sama
han han widi kembar tapi beda han han dan widi kembar tapi samahan han widi kembar tapi beda han han dan widi kembar tapi sama
han han widi kembar tapi beda han han dan widi kembar tapi sama
IrlanMalik
 
Dr. Nazrul Islam, Northern University Bangladesh - CV (29.5.2024).pdf
Dr. Nazrul Islam, Northern University Bangladesh - CV (29.5.2024).pdfDr. Nazrul Islam, Northern University Bangladesh - CV (29.5.2024).pdf
Dr. Nazrul Islam, Northern University Bangladesh - CV (29.5.2024).pdf
Dr. Nazrul Islam
 

Recently uploaded (20)

The Impact of Artificial Intelligence on Modern Society.pdf
The Impact of Artificial Intelligence on Modern Society.pdfThe Impact of Artificial Intelligence on Modern Society.pdf
The Impact of Artificial Intelligence on Modern Society.pdf
 
欧洲杯买球平台-欧洲杯买球平台推荐-欧洲杯买球平台| 立即访问【ac123.net】
欧洲杯买球平台-欧洲杯买球平台推荐-欧洲杯买球平台| 立即访问【ac123.net】欧洲杯买球平台-欧洲杯买球平台推荐-欧洲杯买球平台| 立即访问【ac123.net】
欧洲杯买球平台-欧洲杯买球平台推荐-欧洲杯买球平台| 立即访问【ac123.net】
 
133. Reviewer Certificate in Advances in Research
133. Reviewer Certificate in Advances in Research133. Reviewer Certificate in Advances in Research
133. Reviewer Certificate in Advances in Research
 
太阳城娱乐-太阳城娱乐推荐-太阳城娱乐官方网站| 立即访问【ac123.net】
太阳城娱乐-太阳城娱乐推荐-太阳城娱乐官方网站| 立即访问【ac123.net】太阳城娱乐-太阳城娱乐推荐-太阳城娱乐官方网站| 立即访问【ac123.net】
太阳城娱乐-太阳城娱乐推荐-太阳城娱乐官方网站| 立即访问【ac123.net】
 
Interactive Dictionary AIDS-B.pptx aaaaaaaaaaaaaaaaaaaaaaaaaa
Interactive Dictionary AIDS-B.pptx aaaaaaaaaaaaaaaaaaaaaaaaaaInteractive Dictionary AIDS-B.pptx aaaaaaaaaaaaaaaaaaaaaaaaaa
Interactive Dictionary AIDS-B.pptx aaaaaaaaaaaaaaaaaaaaaaaaaa
 
DIGITAL MARKETING COURSE IN CHENNAI.pptx
DIGITAL MARKETING COURSE IN CHENNAI.pptxDIGITAL MARKETING COURSE IN CHENNAI.pptx
DIGITAL MARKETING COURSE IN CHENNAI.pptx
 
Operating system. short answes and Interview questions .pdf
Operating system. short answes and Interview questions .pdfOperating system. short answes and Interview questions .pdf
Operating system. short answes and Interview questions .pdf
 
DOC-20240602-WA0001..pdf DOC-20240602-WA0001..pdf
DOC-20240602-WA0001..pdf DOC-20240602-WA0001..pdfDOC-20240602-WA0001..pdf DOC-20240602-WA0001..pdf
DOC-20240602-WA0001..pdf DOC-20240602-WA0001..pdf
 
My Story of Getting into Tech By Gertrude Chilufya Westrin
My Story of Getting into Tech By Gertrude Chilufya WestrinMy Story of Getting into Tech By Gertrude Chilufya Westrin
My Story of Getting into Tech By Gertrude Chilufya Westrin
 
Midterm Contract Law and Adminstration.pptx
Midterm Contract Law and Adminstration.pptxMidterm Contract Law and Adminstration.pptx
Midterm Contract Law and Adminstration.pptx
 
How to Master LinkedIn for Career and Business
How to Master LinkedIn for Career and BusinessHow to Master LinkedIn for Career and Business
How to Master LinkedIn for Career and Business
 
欧洲杯投注app-欧洲杯投注app推荐-欧洲杯投注app| 立即访问【ac123.net】
欧洲杯投注app-欧洲杯投注app推荐-欧洲杯投注app| 立即访问【ac123.net】欧洲杯投注app-欧洲杯投注app推荐-欧洲杯投注app| 立即访问【ac123.net】
欧洲杯投注app-欧洲杯投注app推荐-欧洲杯投注app| 立即访问【ac123.net】
 
131. Reviewer Certificate in BP International
131. Reviewer Certificate in BP International131. Reviewer Certificate in BP International
131. Reviewer Certificate in BP International
 
Andrea Kate Portfolio Presentation.pdf
Andrea Kate  Portfolio  Presentation.pdfAndrea Kate  Portfolio  Presentation.pdf
Andrea Kate Portfolio Presentation.pdf
 
Chapters 3 Contracts.pptx Chapters 3 Contracts.pptx
Chapters 3  Contracts.pptx Chapters 3  Contracts.pptxChapters 3  Contracts.pptx Chapters 3  Contracts.pptx
Chapters 3 Contracts.pptx Chapters 3 Contracts.pptx
 
Brand Identity For A Sportscaster Project and Portfolio I
Brand Identity For A Sportscaster Project and Portfolio IBrand Identity For A Sportscaster Project and Portfolio I
Brand Identity For A Sportscaster Project and Portfolio I
 
皇冠体育- 皇冠体育官方网站- CROWN SPORTS| 立即访问【ac123.net】
皇冠体育- 皇冠体育官方网站- CROWN SPORTS| 立即访问【ac123.net】皇冠体育- 皇冠体育官方网站- CROWN SPORTS| 立即访问【ac123.net】
皇冠体育- 皇冠体育官方网站- CROWN SPORTS| 立即访问【ac123.net】
 
Personal Brand exploration KE.pdf for assignment
Personal Brand exploration KE.pdf for assignmentPersonal Brand exploration KE.pdf for assignment
Personal Brand exploration KE.pdf for assignment
 
han han widi kembar tapi beda han han dan widi kembar tapi sama
han han widi kembar tapi beda han han dan widi kembar tapi samahan han widi kembar tapi beda han han dan widi kembar tapi sama
han han widi kembar tapi beda han han dan widi kembar tapi sama
 
Dr. Nazrul Islam, Northern University Bangladesh - CV (29.5.2024).pdf
Dr. Nazrul Islam, Northern University Bangladesh - CV (29.5.2024).pdfDr. Nazrul Islam, Northern University Bangladesh - CV (29.5.2024).pdf
Dr. Nazrul Islam, Northern University Bangladesh - CV (29.5.2024).pdf
 

Zero Importance for class 9

  • 2. Discovery Of Zero Zero was first discovered by Indian Mathematician Aryabhata around 520AD. By the Middle Ages, around the early 1200’s, this concept had come to Western society.
  • 3. Before Zero • Numbers were used for thousands of years before they used zero • Historical records show different path towards the concept • Zero made appearances only to vanish again • Mathematicians were searching for it ,yet did not recognize its fundamental significance
  • 4. Egyptians Egyptian Number system were used as early as 3500 B.C.E. Egypt Number system did not have or need a zero. Even without zero, Egyptians were masters of Mathematics. Their Number system contained icons to represent certain numbers and that icon was repeatedly used to represent the bigger number like 4200 is represented by
  • 5. Greeks The Greeks came up with a new and better number system and brought mathematics to its highest point in ancient times around 500 BC. It was much more sophisticated system. It was made it so that letter was not repeated. Greeks also didn’t had zero and used a different symbol to represent each number.
  • 6. Romans The Romans also had a number system and it is still used in modern life. It was earlier less sophisticated than Egyptian’s system. In Roman’s system they use repeated number like 4505 requires 10 words to represent the number in Roman number system MMMMCCCCCV.
  • 7. Even these number systems were sophisticated without the zero but still they weren’t advanced and people were lacking arithmetic and accounting behind because of lack of zero. So the India mathematician Aryabhata finded out the importance and need of zero at first it was represented by a dot and was called empty By 500 C.E., the Hindus use a small circle to represent Zero! This circle was recognized as a numeral Invention of zero
  • 8. • Arabic people recognized the value of the Hindu system • They adapted the numerals and computation • Then spread the ideas in their travels • The zero was named with the Arabic word sifz • The actual word “zero” came from Italy Hindu-Arabic Numerals
  • 9. Europeans resisted Hindu-Arabic numerals It seemed strange to them The numerals, including zero, were not accepted Florence, Italy, passed a law prohibiting the use of the numerals 0 could be changed to look like 6 or 9 Slowly, the numbers became accepted Zero Slowly got accepted
  • 10. Spreading The News Leonardo of Pisa, known as Fibonacci • Born to a merchant family living in North Africa • Learned Hindu-Arabic numerals from his Arabic tutors • He brought the news of zero and new computational methods to Europe in his book
  • 11. Importance of Zero • It plays a central role in math. i. As a additive identity ii. in integers ,real nos. and other algebraic structures iii. As a place holder in the place value system. iv.It has been called a natural no and has a special role in measuring of physical quantities. • In the real-number system, 0 is the only number that is neither negative nor positive. It represents the boundary between the negative and the positive numbers. This property makes the natural starting point, or origin, on many scales, as on the coordinate axes and on thermometers.
  • 12. Zero is special • If we add 0 to any number, the sum is the original number • Same is true for subtraction • If you multiply any number by 0, the product is 0 • If you raise any nonzero number to the power of 0, the resulting number is 1 • If you divide 0 by any nonzero, the quotient is 0 • Any number divided by 0 is undefined