THE DEFINITIONS OF MATHEMATICS
Mathematics is
a field of study that discovers and organizes methods, theories and theorems that
are developed and proved for the needs of empirical sciences and
mathematics itself.
Mathematics is
the science and study of quality, structure, space, and change. Mathematicians
seek out patterns, formulate new conjectures, and establish truth by rigorous
deduction from appropriately chosen axioms and definitions.
Mathematics,
the science of
structure, order, and relation that has evolved from elemental practices of
counting, measuring, and describing the shapes of objects.
Mathematics
is the science that deals with the logic of shape, quantity and arrangement.
Math is all around us, in everything we do. It is the building block for
everything in our daily lives, including mobile devices, computers, software,
architecture (ancient and modern), art, money, engineering and even sports.
Mathematics
is one of the most important subjects. Mathematics is a subject of numbers,
shapes, data, measurements and also logical activities. It has a huge scope in
every field of our life, such as medicine, engineering, finance, natural
science, economics, etc.
………….
Base Ten Numeration
Numbers are represented by symbols called numerals. For example, numeral for the number ten is 10. Numeral for the number hundred is 100 and so on.
The symbols which represent numbers are called digits. For example the number 521 has three (3) digits which are 5, 2 and 1. There are only tendigits which are used to represent any number. These digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
The Place Value in each Digit in Base Ten Numeration
Identify the place value in each digit in base ten numeration
When we write a number, for example 521, each digit has a different value called place value. The 1 on the right means 1 ones which can be written as 1 × 1, the next number which is 2 means tens which can be written as 2 × 10 and the last number which is 5 means 5 hundreds which can be written as 5 × 100. Therefore the number 521 was found by adding the numbers 5 × 100 + 2 × 10 + 1× 1 = 521.
Note that when writing numbers in words, if there is zero between numbers we use word ‘and’
Example 4
Write the following numbers in words:
- 7 008
- 99 827 213
- 59 000
Solution
- 7 008 = Seven thousand and eight.
- 99 827 213 = Ninety nine millions eight hundred twenty seven thousand two hundred thirteen.
- 59 000 = Fifty nine thousand.
Example 5
Write the numbers bellow in expanded form.
- 732.
- 1 205.
Solution
- 732 = 7 x 100 + 3 x 10 + 2 x 1
- 1 205 = 1 x 1000 + 2 x 100 + 0 x 10 + 5 x 1
Example 6
Write in numerals for each of the following:
- 9 x 100 + 8 x 10 + 0 x 1
- Nine hundred fifty five thousand and five.
Solution
- 9 x 100 + 8 x 10 + 0 x 1 = 980
- Nine hundred fifty five thousand and five = 955 005.
Example 7
For each of the following numbers write the place value of the digit in brackets.
- 89 705 361 (8)
- 57 341 (7)
Solution
- 8 is in the place value of ten millions.
- 7 is in the place value of thousands.
Numbers in Base Ten Numeration
Read numbers in base ten numeration
Base Ten Numeration is a system of writing numbers using ten symbols i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Base Ten Numeration is also called decimal system of Numeration.
Numbers in Base Ten Numeration up to One Billion
Write numbers in base ten numeration up to one billion
Consider the table below showing place values of numbers up to one Billion.
If you are given numerals for a number having more than three digits, you have to write it by grouping the digits into groups of three digits from right. For example 7892939 is written as 7 892 939.
When we are writing numbers in words we consider their place values. For example; if we are told to write 725 in words, we first need to know the place value of each digit. Starting from right side 5 is in the place value of ones, 2 is in the place value of tens and seven is in the place value of hundreds. Therefore our numeral will be read as seven hundred twenty five.
Numbers in Daily Life
Apply numbers in daily life
Numbers play an important role in our lives. Almost all the things we do involve numbers and Mathematics. Whether we like it or not, our life revolves in numbers since the day we were born. There are numerous numbers directly or indirectly connected to our lives.
The following are some uses of numbers in our daily life:
- Calling a member of a family or a friend using mobile phone.
- Calculating your daily budget for your food, transportation, and other expenses.
- Cooking, or anything that involves the idea of proportion and percentage.
- Weighing fruits, vegetables, meat, chicken, and others in market.
- Using elevators to go places or floors in the building.
- Looking at the price of discounted items in a shopping mall.
- Looking for the number of people who liked your post on Facebook.
- Switching the channels of your favorite TV shows.
- Telling time you spent on work or school.
- Computing the interest you gained on your business.
We have four operations which are: addition (+), subtraction (-), multiplication ( X) and division (÷).
Addition of Whole Numbers
Add whole numbers
When adding numbers we add the corresponding digits in their corresponding place values and we start adding from the right side i.e. from the place value of ones to the next.
We can add numbers horizontally or vertically.
Horizontal addition
Example 5
- 972 + 18=
- 23 750 + 250 =
Solution
- 972 + 18 = 990
- 23 750 + 250 = 24 000
Vertical addition
Example 7
Subtraction of Whole Numbers
Subtract whole numbers
Subtraction is denoted by the sign (-). It is sometimes called minus. Subtraction is the opposite of addition. Subtraction also means reduce a number from certain number and the answer that is obtained is called difference..
Subtraction is done in similar way like addition. We subtract the corresponding digits in their corresponding place value. For example; 505 – 13. We first subtract ones, which are 5 and 3. Subtract3 from 5 gives 2. Followed by tens which are 0 and 1. Subtract 1 from 0 is not possible. In order to make it easy, take 1 from 5 (hundreds). When 1 is added to 0 it has to be changed to be tens since it is added to a place of tens. So, when 1 comes into a place of tens it becomes 10. So add 10 to 0. We get 10. Now, subtract 1 from 10. We get 9. We are left with 4 in a place of hundreds since we took 1. There for our answer will be 492.
Note that similar manner will be used when subtracting.
Example 5
Multiplication of Whole Numbers
Multiply whole numbers
Multiplication means adding repeatedly depending on the times number given. For example; 25 6 means add 25, repeat adding 6 times i.e. 25 + 25 + 25 + 25 +25 + 25 = 150. The answer obtained after multiplying two or more numbers is called product. The number being multiplied is called a multiplicand while the number used in multiplying is called a multiplier. Referring our example, 25 is multiplicand and 6 is multiplier.
Example 3
Division of Whole Numbers
Divide whole numbers
Division is the same as subtraction. You subtract divisor (the number used to divide another number) from dividend (the number which is to be divided), we repeat subtracting divisor to the answer obtained until we get zero. The answer is how many times you repeat subtraction.
For example; 27 ÷9, we take 27 we subtract 9, we get 18. Again we take 18 we subtract 9, we get 9. We take 9 we subtract 9 we get 0. We repeat subtraction three times. Therefore the answer is 3. The answer obtained is called quotient. Referring to our example; 27 is dividend, 9 is divisor and 3 is quotient. If a number can’t be divided exactly, what remains or left over is called a remainder.
Example 4
The Four Operations in Solving Word Problems
Use the four operations in solving word problems
Sometimes you may be given a question with mixed operations +, -, xand ÷ . We do multiplication and division first then addition and subtraction.
Example 3
- 12 ÷ 4 + 3 x 5
- 14 x2 ÷ 7 – 3 + 6
Solution
- 12 x 4 + 3 x 5 =3 + 15 (do division and multiplication fist) =18
- 14 x 2 ÷ 7 – 3 + 6 =28 ÷ 7 – 3 + 6 (multiply first) =4 – 3 + 6 (then divide) =10 – 3 (add then subtract) =7
We may use brackets to separate x,÷ , + and – if they are mixed in the same problem and use what is called BODMAS . BODMAS is the short form of the following:
B for Brackets, O for Open, D for Division. M for Multiplication, A for Addition and S for Subtraction
Therefore, with mixed operations, we first do the operation inside the brackets; we say that we open the brackets. Then we do division followed by multiplication, addition and lastly subtraction.
Example 7
Word problems on whole numbers
Example 9
In a school library there are 6 shelves each with 30 books. How many books are there?
Solution
Each shelf has 30 books
6 shelves have 30 × 6 = 180 books.
Therefore, there are 180 books.
Example 10
Juma’s mother has a garden with Tomatoes, Cabbages and Water Lemons. There are 4 rows of Tomato each with 30 in it. 6 rows of Cabbages with 25 in each and 3 rows of Water Lemo
Solution
There are 30 Tomatoes in each row
4 rows will have 30 × 4 = 120 Tomatoes
Each row has 25 Cabbages
6 rows have 25 × 6 = 150 Cabbages
Each row has 15 Water Lemons
3 rows have 15 × 3 = 45 Water Lemons
In total there 120 + 150 + 45 = 315 plants.
Therefore in Juma’s mother garden there are 315 plants.
Example 11
A school shop collects sh 90 000 from customers each day. If sh 380 000 from the collection of 6 days was used to buy books. How much money was left?
Each day the collection is sh 90 000
6 days collection is sh 90 000 × 6 = sh 540 000
The money left will be = Total collection – Money used
= sh 540 000 – sh 380 000 = sh 160 000
Therefore the money left was sh 160 000
Exercise 1
- For each of the following numbers write the place value of a digit in a bracket.
- 899 482 (4)
- 1 940 (0)
- 9 123 476
- Write the numerals for each of the following problems.
- Ten thousand and fifty one.
- Nine hundred thirty millions one hundred twenty five thousand three hundred seventy four.
- 6 x 100 + 1 x 10 + 7 x 1
- 5 x 10 000 + 4 x 1 000 + 2 x 100 + 7 x 10 + 8 x 1
- Write the following numerals in words
- 952 817
- 98 802 750
- Write down even, odd and prime numbers between 90 and 100.
- Compute:
- 25 940 + 72 115 – 5 750 =
- 892 x 12=
- 14 670 ÷ 15 =
- Calculate: (75 ÷3) + 7 – 14 + (13 x2) =
- There are 23 streams at Mamboleo primary school. If each stream has 60 pupils except 4 streams which they have 75 pupils each. How many pupils are there at Mamboleo primary school?
- Hamis uses 200 shillings every day for transport. How much money will he use for 35 days?
The numbers from 0 to the right are called positive numbers and the numbers from 0 to the left with minus (-) sign are called negative numbers. Therefore all numbers with positive (+) or negative (-) sign are called integers and they are denoted by Ζ. Numbers with positive sign are written without showing the positive sign. For example +1, +2, +3, … they are written simply as 1, 2, 3, … . But negative numbers must carry negative sign (-). Therefore integers are all positive and negative numbers including zero (0). Zero is neither positive nor negative number. It is neutral.
The numbers from zero to the right increases their values as the increase. While the numbers from zero to the left decrease their values as they increase. Consider a number line below.
If you take the numbers 2 and 3, 3 is to the right of 2, so 3 is greater than 2. We use the symbol ‘>’ to show that the number is greater than i. e. 3 >2(three is greater than two). And since 2 is to the left of 3, we say that 2 is smaller than 3 i.e. 2<3. The symbol ‘<’ is use to show that the number is less than.
Consider numbers to the left of 0. For example if you take -5 and -3. -5 is to the left of -3, therefore -5 is smaller than -3. -3 is to the right of -5, therefore -3 is greater than -5.
Generally, the number which is to the right of the other number is greater than the number which is to the left of it.
If two numbers are not equal to each to each other, we use the symbol ‘≠’ to show that the two numbers are not equal. The not equal to ‘≠’ is the opposite of is equal to ‘=’.
Example 10
Represent the following integers Ζ on a number line
- 0 is greater than Ζ and Ζ is greater than -4
- -2 is less than Ζ and Ζ is less than or equal to 1.
Solution
- 0 is greater than Ζ means the integers to the left of zero and Ζ is greater than -4 means integers to the left of -4. These numbers are -1, -2 and -3. Consider number line below
- -2 is less than Ζ means integers to the right of -2 and Ζ is less than or equal to 1 means integers to the left of 1 including 1. These integers are -1, 0 and 1. Consider the number line below
Example 11
Put the signs ‘is greater than’ (>), ‘is less than’ (<), ‘is equal to’ (=) to make a true statement.
Addition of Integers
Add integers
Example 2
2 + 3
Show a picture of 2 and 3 on a number line.
When drawing integers on a number line, the arrows for the positive numbers goes to the right while the arrows for the negative numbers goes to the left. Consider an illustration bellow.
The distance from 0 to 3 is the same as the distance from 0 to -3, only the directions of their arrows differ. The arrow for positive 3 goes to the right while the arrow for the negative 3 goes to the left.
Example 5
-3 + 6
Solution
Subtraction of Integers
Subtract integers
Since subtraction is the opposite of addition, if for example you are given 5-4 is the same as 5 + (-4). So if we have to subtract 4 from 5 we can use a number line in the same way as we did in addition. Therefore 5-4 on a number line will be:
Take five steps from 0 to the right and then four steps to the left from 5. The result is 1.
Multiplication of Integers
Multiply integers
Multiplication of integers tutorial
Example 3
2×6 is the same as add 2 six times i.e. 2×6 = 2 + 2 + 2 + 2 + 2 +2 = 12. On a number line will be:
Multiplication of a negative integer by a negative integer cannot be shown on a number line but the product of these two negative integers is a positive integer.
From the above examples we note that multiplication of two positive integers is a positive integer. And multiplication of a positive integer by a negative integer is a negative integer. In summary:
(+)×,(+) = (+)
(-)×,(-) = (+)
(+)×,(-) = (-)
(-)×,(+) = (-)
Division of Integers
Divide integers
Example 2
6÷3 is the same as saying that, which number when you multiply it by 3 you will get 6, that number is 2, so, 6÷3 = 2.
Therefore division is the opposite of multiplication. From our example 2×3 = 6 and 6÷3 = 2. Thus multiplication and division are opposite to each other.
Dividing two integers which are both positive the quotient (answer) is a positive integer. If they are both negative also the quotient is positive. If one of the integer is positive and the other is negative then the quotient is negative. In summary:
(+)÷(+) = (+)
(-)÷(-) = (+)
(+)÷(-) = (-)
(-)÷(+) = (-)
Mixed Operations on Integers
Perform mixed operations on integers
You may be given more than one operation on the same problem. Do multiplication and division first and then the rest of the signs. If there are brackets, we first open the brackets and then we do division followed by multiplication, addition and lastly subtraction. In short we call it BODMAS. The same as the one we did on operations on whole numbers.
Example 3
9÷3 + 3×2 -1 =
Solution
9÷3 + 3×2 -1
=3 + 6 -1 (first divide and multiply)
=8 (add and then subtract)
Example 4
(12÷4 -2) + 4 – 7=
Solution
(12÷4 -2) + 4 – 7
=1 + 4 – 7 (do operations inside the brackets and divide first)
=5 – 7 (add)
= -2
……………….
History of Mathematics
The history of Mathematics is long and magnificent, which can be traced back to the start of humanity itself. Be it the early man’s notched bones, the settled agriculture in Egypt and Mesopotamia or the revolutionary developments of the Hellenistic empire of ancient Greece, every event speaks of mathematics’s highly innovative and charming history. Great innovations were made in different parts of the world, with The East contributing an array of significant developments, mainly in India, China and the medieval Islamic empire. Europe saw some marvelous advances and innovations during the late middle age. From the 16th century, a series of discoveries have been taking place till date, forming the great history of Mathematics.
What is Mathematics?
- Counting, measuring and identifying the shapes of objects evolved the science of order, relation and structure, which began to be known as Mathematics. It has been a fundamental base for engineering, science and philosophy developments.
- The application of imagination, logic and abstraction has led to more complex developments in the discipline over the years. Moreover, it has broadened the scope of Mathematics which now covers logical reasoning, quantitative calculation, number theory, geometry, algebra, calculus, probability, statistics and many more fields of study or specialized areas.
Role of Mathematicians:
There have been great mathematicians over the centuries who were great thinkers. They have gifted the world with inspiring moments by innovating and devoting their lives to Mathematics. Today, we will unfold the history of mathematics and give an insight into the innovations as to how, when, and by whom they were discovered. It is vital to know the math importance and its rich history that has shaped the world into what it is today.
Prehistoric Mathematics or The Origin:
Basic Number System:
- Mathematics owes its foundation to counting, which humanity began almost 40,000 years ago, and its creation was not as simple as it may seem. Though mankind had an idea of the difference between “one” and “two”, with an understanding regarding amounts, it took several ages to invent a word or symbol for the abstract knowledge of “two”.
- The patterns in the environment led to mathematical thoughts of shapes. The hunter-gatherer societies knew no word for numbers greater than two. In the absence of agriculture and trade, there was no need for a standard number system.
Prime Numbers and Shapes:
- The Ishango bone demonstrations represent the earliest sequence of prime numbers. It is argued that prime numbers evolved after the concept of division which dates back to 10,000 BC.
- Geometrical shapes and designs are claimed to have been developed during the 5th millennium BC by the pictorial representation done by the Predynastic Egyptians. Various shapes, like circles, Pythagorean triples and ellipses, were found on the megalithic monuments located in England and Scotland during the 3rd millennium BC. Though these represented mathematical designs, they were mere art and decorations.
Civilisation and Math Importance:
- With the onset of agricultural trade amongst civilisations (Sumerian and Babylonian of Mesopotamia {Iraq today} being the pioneers), the need for measurement of land, taxation policies, the price of trade etc., increased and paved the way for the proper development of Mathematics. Several ancient mathematical sources bear evidence of basic geometric and arithmetic systems.
Mathematics in Sumer and Babylon (Ancient Mesopotamia) [1st and 2nd Millennium BC]:
Sumer, modern Iraq, is known as the Cradle of Civilization for its pioneering activities like writing, irrigation, agriculture, the plough and many more. They used wedge-shaped symbols inscribed on the clay tablets, which were baked under the sun, known as a cuneiform script which marks the innovation of a pictographic writing system. Increased agricultural and trade activities led to the development of a proper arithmetic system. They described large specific numbers through symbols and developed their lunar calendar. Later these symbols were replaced by cuneiform equivalents.
Arithmetic System:
- They followed a base 60 or sexagesimal number system, which could be calculated with 12 knuckles on the one hand and five fingers on the other. In addition, they used an actual place-value system where larger values were represented by digits written in the left column. The number representation was similar to Roman numerals.
- The modern-age time system of 60 seconds in one minute, 60 minutes in one hour and 360 degrees in a full circle are all inspired by the Babylonian system.
- The fact that 60 has multiple divisors like 1, 2, 3, 4, 5, 6, 10 etc., made it the base of their arithmetic system. Similarly, the number 12 with multiple factors like 1, 2, 3, 4 and 6 have been used extensively, such as 12 inches, 12 months, 2*12 hours etc.
The Famous Clay Tablets:
- Many Babylonian clay tablets have been discovered that cover various topics such as multiplication, division, square tables and roots, cube roots, algebra, linear, quadratic and cubic equations and much more.
- Buildings and dice designs were based on geometrical shapes. Calculating the areas of various shapes and volumes of cylinders is some of the other geometrical activities that took place in this period.
- The popular Plimpton 322 clay tablet bears evidence of the knowledge of the principle of the right-angled triangle much before the Pythagoras theorem came to light. However, this fact has been disputed as many suggest that the tablets were used for academic representations and do not interpret anything else.
Mathematics in Egypt [2000 – 6000 BCE]:
Settled along the Nile valley, early Egyptians started to note the lunar phases, agricultural and religious reasons. Measurements were based on body parts, and a decimal numeric system was in place depending on our ten fingers. They had no concept of place value but used a stroke for units, a heel-bone symbol for 10s, a coil of rope for 100s and a lotus plant for 1000s.
Famous Texts Available:
- The most prominent Egyptian mathematical text available from 2000-1800 BCE is the Rhind Papyrus. It contained several formulae for geometry, division, multiplication, knowledge about prime and composite numbers and solutions to linear equations, etc.
- Moscow Papyrus is another text from the same period that contained word problems, making mathematics more entertaining.
- Evidence of solving a second-order algebraic equation is available in the Berlin Papyrus.
Egyptian Pyramids and Rules of Triangles:
- Egyptian pyramids depict the marvellous wit of Egyptian mathematicians. The pyramids suggest that the mathematicians knew the perfect formula for the volume of a pyramid as these structures observed the golden ratio of 1:1.618.
- They knew the rules of a triangle, and Egyptian builders yielded perfect right-angled triangles.
Greek Mathematics [4th – 7th Century BC]:
The Greek empire spread enormously and conquered many societies. They were wise to adopt useful and mathematical elements from those societies, including the Babylonians and the Egyptians. Slowly they made innovations on their own and brought about a revolution in the world of Mathematics. A fully developed Greek numeral system (Attic or Herodianic numerals similar to the Roman system) was used.
The Great Greek Mathematicians:
- Thales, a Greek mathematician and one of the Seven Sages of Ancient Greece, laid down the foundation guidelines for the abstract invention of geometry. He gave the world Thales’s Theorem and the Intercept Theorem.
- Another legendary mathematician who is the dawn of Greek mathematics is Pythagoras, believed to have coined the terms “philosophy” and “mathematics”. He has given the Pythagoras Theorem, which is undoubtedly one of the best mathematical theorems ever known.
- Democritus was the first to note that a cone has 1/3rd the volume of a cylinder with the same height and base.
- Other renowned mathematicians include Plato, Aristotle, Archimedes, Euclid and Pappus of Alexandria, who provided theorems with a logical approach.
Influence on Geometry:
- Greek geometry has seen three major geometrical problems (the squaring of the circle, the doubling of the cube, and the trisection of an angle), which have influenced the future of geometry and resulted in many fruitful developments.
- The solutions to these problems were compiled in the 19th Century. Greeks were pioneers in introducing the concept of infinity with Zeno’s Dichotomy Paradoxes.
Mathematics in India [5th – 12th Century]:
The history of Mathematics in India can be traced to very early stages. The time from the 5th to 12th Century is said to be the Golden Era of Indian Mathematics. Several mathematical developments took place in India simultaneously with the West, which led to a few claims of plagiarism by a few prominent European mathematicians.
Arithmetic System:
- The Vedic period (before 1000 BCE) has mantras including powers of 10 from 100 and bears evidence of the application of arithmetic solutions like addition, subtraction, division, multiplication, cubes, squares, fractions and roots.
- Indians perfected the decimal place value number system like the Chinese, which is one of the most valued intellectual innovations to date.
Famous Texts Available:
- A CE Sanskrit text as old as the 4th Century shows Buddha mentioning different number systems and estimating the number of atoms in the universe.
- Sulba Sutras, also known as Sulva Sutras, is an 8th Century BCE text that has laid down several simplified statements of the Pythagoras theorem. It is believed that Pythagoras got his inspiration from this text. It also contains linear and quadratic geometric solutions and an accurate result for the square of 2.
- Ancient Buddhist literature and Jain mathematicians of the 3rd or 2nd Century recognised the different types of infinities.
The Greatest innovation of all times – The invention of Zero:
- The invention of Zero has been instrumental in the history of Mathematics and is solely attributed to the Indian mathematician, Brahmagupta of the 7th Century.
- He laid down the principles for using zero and negative numbers. In addition, he has done pioneering work in quadratic equations and the concepts of algebra.
- It was Bhaskara II of the 12th Century who corrected the wrong result of division by zero. He explained that dividing 1 by zero will result in infinite pieces.
Significant Contribution to Trigonometry:
- The theory of trigonometry introduced by the Greeks had huge progress in India during the Golden Age of Indian Mathematics. The various functions of sine, cosine and tangent were used to measure lands, navigate through the seas, and calculate the distance between the Moon and the Earth and between the Earth and the Sun.
- The first use of trigonometry is contained in the text named “Surya Siddhanta” by an anonymous author that dates back to 400 CE.
- The great Indian mathematician Aryabhata of the 6th Century CE has provided proper definitions and complete tables of all trigonometry functions. He specified and used the value of π to give an approximate circumference of the Earth.
Mathematics in Medieval Europe and Renaissance:
During the 4th to 12th Century, when great mathematical innovations were taking place in several parts of the world like China, India and Islamic regions, very little or no progress took place in Europe in the field of intellectual sciences. The focus was majorly on spiritual, literary and philosophical subjects. Their mathematical knowledge was based on the findings and theories of Greek masters like Nicomachus and Euclid. Greek and Roman-based abaci and Roman numerals were used for trade and calculations.
Rise of Mathematics in Europe:
- It was after the 12th Century when Mathematics played a more practical part in the lives of European people. This shift from the academic realm to real-life uses was brought about by the rapid progress of trade and commerce with the East and the West.
- The introduction of the printing press in the mid-15th Century was another catalyst in the process of imparting knowledge.
- Business people were educated on computational methods for trade and commercial needs through various books on arithmetic.
The Great European Mathematicians:
- Leonardo of Pisa, popularly known as Fibonacci, was Europe’s 1st great middle-aged mathematician who is best known for introducing a Fibonacci sequence of numbers. The spread of the Hindu-Arabic numeral system across Europe during the 13th Century is attributed to him. This led to the extinction of the Roman numeral system and opened the doors for great innovations in the field of Mathematics in Europe.
- Frenchman Nicole Oresme of the 14th Century is known for introducing time-speed-distance graphs. He also worked on the system of rectangular coordinates, fractional exponents and the infinite series.
- The great German scholar of the 15th Century, Regiomontanus, has established math importance in the area of trigonometry. He separated it from astronomy and made trigonometry an independent part of Mathematics. His first significant book on trigonometry, “De Triangulis”, exhibits basic trigonometry knowledge and is widely used for high school and college studies.
Mathematics During Renaissance:
During the Renaissance, Italian artists and merchants played an influential role in the development of Mathematics. Both algebra and accounting developed together though there is no direct relationship between them. Arithmetic and algebra were essential in complex bartering operations and compound interest calculations.
The Great Mathematicians:
- During the 14th Century, Piero Della Francesca wrote many books on linear perspective and solid geometry.
- Another great mathematician of this time was Luca Pacioli, whose book Summa de Arithmetica introduced the plus and minus signs for the 1st time. It is also the first book printed in Italy with algebra.
- Scipione del Ferro and Niccolò Fontana Tartaglia of the 16th Century provided solutions for cubic equations.
- Ars Magna, a 15th Century book by Gerolam Cardano, had solutions for the quartic equations.
- The real number system is influenced by Simon Stevin’s “De Thiede”, which showed the 1st systematic dealing of decimal notation.
Mathematics and the Scientific Revolution:
Mathematics in the 17th Century:
The 17th Century, also called the Age of Reason, saw many mathematical and scientific ideas explode across Europe.
The Great Mathematicians and their Innovations:
- The most significant innovation during this period was the introduction of the logarithm by John Napier, which helped Kepler and Newton to perform complex calculations for their innovations in the field of Physics. Logarithms helped the advances in science, astronomy and mathematics by making complex calculations relatively easy.
- René Descartes is known for developing analytic geometry and Cartesian coordinates, which enabled the plotting of orbits of the planets on a graph. He also laid the foundation for calculus.
- Pierre de Fermat and Blaise Pascal are two other great French mathematicians known for establishing math importance and the concept of probability and expected values.
- By laying down the laws of Physics, single-handedly, Sir Issac Newton is often regarded as one of the greatest mathematicians in the history of mathematics. In addition, Newton and Gottfried Leibniz developed two operations called differentiation and integration, which are used widely in different fields of study, highlighting the importance of math.
Mathematics in the 18th Century:
The Great Mathematicians and their Innovations:
- The Bernoulli of Basel and Leonhard Euler are the two prominent mathematicians who dominated the field during the 18th Century. They worked on calculus, probability, number theory, geometry, trigonometry and algebra.
- Christian Goldbach has proposed the Goldbach Conjecture and proved many number theory theorems, such as the Goldbach-Euler Theorem on perfect powers.
- Abraham de Moivre is known for de Moivre’s formula, which links trigonometry and complex numbers. In addition, his work on Newton’s famous binomial theorem, analytic geometry and probability theory is of great value in the history of mathematics.
- Joseph Louis Lagrange is credited with the four-square theorem and is also popular for Lagrange’s Theorem or Lagrange’s Mean Value Theorem.
- Pierre-Simon Laplace, also known as “the French Newton”, is famous for his work on “Celestial Mechanics”.
- Adrien-Marie Legendre’s contributions to abstract algebra, statistics, number theory and mathematical analysis deserve special attention.
Mathematics in the 19th Century:
The Great Mathematicians and their Innovations:
- Joseph Fourier studied infinite sums where the terms are trigonometry functions. He introduced the Fourier Series, which became a powerful tool in applied and pure mathematics.
- Jean-Robert Argand represented complex numbers on geometric diagrams using trigonometry and vectors, which are popularly known as the Argand Diagrams.
- Évariste Galois advocated that polynomial equations with any degree greater than 4 had no general algebraic solution. He worked in areas such as group theory, rings, vector spaces, algebraic geometry and non-commutative algebra.
- Carl Friedrich Gauss, also named the “Prince of Mathematics”, is one of the greatest in the history of mathematics who worked in various fields of study.
Mathematics in the 20th Century:
Mathematics moved towards generalization and abstraction during this period, and it became a profession in teaching and other industries with a specialization in fields of study.
The Great Mathematicians and their Innovations:
- G.H. Hardy and Srinivasa Ramanujan are note-worthy mathematicians of this time who tried solving the Riemann hypothesis.
- The “Principia Mathematica”, a joint work by Bertrand Russell and A.N. Whitehead, greatly influenced mathematics and philosophy.
- The other great mathematicians who highlighted math importance include Johann Gustav Hermes, David Hilbert, Kurt Gödel, Alan Turing, John von Neumann, Claude Shannon, Andrey Kolmogorov, André Weil, Paul Erdös, Paul Cohen, Julia Robinson and Yuri Matiyasevich.
Mathematics in the 21st Century:
Many online versions and print versions of mathematical journals were made available recently. In addition, open-access publishing is gaining momentum. With computers becoming more powerful and significant, the subject is growing, and the application of mathematics to bioinformatics is increasing at an alarming pace.
Math importance in
JEE Main is one of India’s highly competitive exams for engineering aspirants. Preparing for such exams can be challenging, and proper knowledge of all subjects can greatly help score good marks in JEE Main.
Question Paper Pattern:
The questions are from 3 major subjects Physics, Chemistry and Mathematics. The question paper pattern has different sections of 100 marks each, with each section having 25 questions.
