Discovery or invention?

Mathematics is the language mankind use to
communicate mathematical ideas. It has been a widely debated topic whether
mathematics was discovered or an invention. My first thought when I was asked
this question was mathematics was discovered as the mathematics discovered in
nature such as the golden ratio or tessellations were not invented by mankind;
instead they were already there long before the existence of mankind. Even if
humans didn’t discover these mathematical ideas, they would have still existed.
Humans simpily identified the concept behind these ideas and gave it a name.

However, as it thought more about this question, I realize there are aspects of
mathematics which were created by humans: algebra, number systems. If humans
disappeared, these mathematical aspects would also follow. As a result of this,
I came to a conclusion that mathematics is both discovered and invented.

#exploremaths

This message is intended for the addressee named and may contain privileged information
or confidential information or both. If you are not the intended recipient
please delete it and notify the sender.

Video Proposal (By Alex Ho, Eric Sun, Kevin Xu, Terrence Wong)

When deciding with what mathematical concept to explain for
a 90 to 180 second video, we were having difficulties deciding on something
that we would all agree on. Then the teacher made the suggestion of doing “Clock
Arithmetic” which we settled on. We still haven’t quite worked out on how we
will present the video and still need to fill in a member with what we have
chosen…

#exploremaths

Number Systems

Hexadecimal

What
is the purpose of this number system, and what are its distinctive features?

Hexadecimal
is a number system with the base of 16. It is used as a short hand for binary
numbers, allowing binary system to be expressed in fewer characters. The
distinctive features of hexadecimal system are that it uses sixteen symbols. In
most cases, the values zero to nine is represented by the symbols 0-9 and the
values ten to fifteen is represented by the symbols A,B,C,D,E,F.

Explain
where and how this number system is used today.

The
hexadecimal number system is now broadly used by computer systems designers and
programmers. In order to represent these hexadecimal constants in computing
languages, different notations are used. In addition to this, this number
system is also used in tradition Chinese units for weight.

What
are the advantages and disadvantages of this number system?

The
advantage of this number system is that it is very easy to convert between
binary number and hexadecimal numbers and it allows binary numbers to be
expressed in fewer symbols. The disadvantage of this number system is since it
is not a base of 10 system, it causes errors to be made as it is a system with
a base of 16.

 

Egyptian
Number System

What
is the purpose of this number system, and what are its distinctive features?

The
Egyptian Number System was used for a wide variety of occasions, theses
included, measuring time, straight lines, the level of the Nile floodings,
calculating areas of land, counting money, working out taxes and cooking. In
this system, it uses a base of 10 and the numbers are changed into hieroglyphic
writings. This allowed numbers up to 1,000,000 to be expressed this way. In
addition, there were also special signs for every power of 10. For example, the
symbol for 1000 is a lotus flower.

Explain
where and how this number system is used today.

This
ancient number is no longer in use today.

What
are the advantages and disadvantages of this number system?

The
advantage of this system was it allowed the Egyptians to be able to express
their numbers easier to add, subtract, divide or multiply. The disadvantage of
this system is that it can only go up to 1,000,000 and it required a lot of
space and time to write out numbers.

#exploremaths

This message is intended for the addressee named and may contain privileged information
or confidential information or both. If you are not the intended recipient
please delete it and notify the sender.

Video Proposal (by Erik Willison and Sandeep Darapuneni)

#exploremaths

 

For our video, Erik and I have a few topics in mind. We
though about doing it on the nature of mathematical paradoxes and we were going
to explain a few but then we developed the idea of limits and how this topic
would be ideal for a 90 to 180 second video. As for presentation, we have
thought of presenting it in ‘minutephysics-esque’ fashion by drawing out what
we are trying to explain on a whiteboard and also by simply narrating over the
images that we display on screen.

This message is intended for the addressee named and may contain privileged information
or confidential information or both. If you are not the intended recipient
please delete it and notify the sender.

Number Systems (Alex Ho)

ASCII

What is the purpose of
this number system, and what are its distinctive features?

ASCII (American Standard Code for Information interchange)
encodes the English alphabet, and some other punctuation characters into a
form, that a computer would be able to read, called binary.

ASCII is very noticeable as the way to convert English
letters into binary, is through the use of a chart. Each character has its set
of 1’s and 0’s. Connecting the binary from the left and then the top, gives the
binary that will work with the computer.

Explain where and how
this number system is used today.

Originally developed from telegraphic codes, ASCII is everywhere
within this modern day world of technology. It represents every single text
that technology needs to output to the user.

What are the
advantages and disadvantages of this number system?

The use of this system is that it provides an easy way to
transfer English characters into computer language. Most modern character-encoding
systems are based on ASCII. Without the use of the chart to aid with coding, it
would be quite difficult to input the correct binary code to output the desired
character.

#exploremaths

Number Systems – Babylonian and Mayan [Adam Tan]

Babylonian Number System
As it is with most other number systems, the main purpose of the Babylonian numerals was to count. A base-60 number system proved much easier for calculations, as, previously, each number up to ten had a different symbol. The distinctive features of Babylonian numerals include:

– Only two digits are used

– The value of the digit is determined using the digit itself and the positioning of it

– Number of tens written on the left, number of units on the right

Modern usage of the Babylonian number system can be found in time: sixty seconds in a minute, sixty minutes in an hour, and 360 degrees in a circle.

One big advantage of a base-60 number system is that it is a highly composite number.

Mayan Number System

Once again, this number system is used to count. The distinctive features are:

– Base-20, as the Mayans counted using their fingers AND their toes

– Three different number symbols – zero is a shell, one is a dot and five is a line

– Sets of five (lines) are placed horizontally at the bottom and the ones (dots) are placed above it

Mayan mathematics are used today in our calendar. In Mayan numerals, numbers over nineteen were expressed with a dot representing a multiple of twenty (ie. twenty-three would be expressed with a dot over three dots). The same principal was used with the calendar, however, the upmost dot represented a multiple of eighteen. For the Mayans, there were twenty days in a month and eighteen months in a year, 360 (the five days were considered bad luck).

An advantage of using this number system is that large numbers are easier to express, and simple arithmetic is easily accomplished.

#exploremaths

The Dozenal System (Josh Luong)

Dozenal System (base 12)

1.      What is the purpose
of this number system, and what are its distinctive features?

The main purpose
for the dozenal system is to count however the main distinctive features that
this counting system has over the base 10 counting system is the number 10 we
know is written like an upside-down 2 and 11 is written as a backwards 3. The
number 12 becomes our number 10.

2.      Explain where and
how this number system is used today

.The dozenal system
is rarely used today but languages such as Janji, Gbiri-Niragu, Piti were known
to have used this counting system. Units of time are seen in the dozenal system
where there are 12 months in a year, 12 signs of zodiacs and the Babylonians
had 12 hours in a day.

3.      What are the
advantages and disadvantages of this number system?

People have
considered other counting systems such as the base sixteen because it would
bring humans closer to how computers calculate. Counting in the dozenal system provides
better mental problem solving using factors of 12.

#exploremaths

Number Systems (By Joyce Liu)

#exploremaths

The decimal number system has a base of ten and its numbers can be either terminating or recurring where the recurring decimal can have a repeating sequence or infinitely un-repeating. Decimal notation is used in numbers that have a base ten numeral system such as the Roman numerals, Hindu-Arabic numerals, Chinese numerals, etc. The decimal system is also used in many different aspects of the modern day, for example; finding a part of an object. The advantage of this is that it is easily linked to our natural numbers – making it easy to calculate with. A disadvantage would be the inaccuracy, as mentioned before some decimals are forever recurring, which means you have to round them off.

The scientific notation system is special, as it represents
very large or very small numbers in a shorter form. Scientific notation forms
are always written as:

a x 10b

b = an integer

a = the coefficient of any real number

The number system is often used when numbers are too small
or too big to be written in their normal form with devices such as calculators
using this form. An advantage of this system is that it shortens the time to write
the actual number, which is found to be very convenient. A disadvantage would
be that the numbers can often be misread, as there may be confusion between big
and small numbers, especially if rushing.

This message is intended for the addressee named and may contain privileged information
or confidential information or both. If you are not the intended recipient
please delete it and notify the sender.

Re: Discovery or Invention? [Adam Tan]

Discovered:

Science

History

Invented

Legal system

History

The concept of maths is invented, but the validity of it has always been there. Things such as the golden ratio and fractals are in nature, but humans just gave a name to it.
It’s difficult to give it a single label – discovery or invention – as both relate to each other. Formulae and theories etc. have been invented by man, but that’s only to discover what is already there. The circumference of a circle already exists, but the formula: C = 2πr has been invented to figure it out.

Currently, I don’t think I’m at the point where I can make an accurate judgment, so I’d have to conclude with – both.

#exploremaths