# Mathematics Local News Update

### Did you know?

## Breaking news!!! New mathematical way has been discovred

## But what is the Pythagorean theorem about?

**Pythagorean theorem**, also known as

**Pythagoras' theorem**, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angled) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides

*a*,

*b*and

*c*, often called the "Pythagorean equation"

## 1st usage of the Pythagorean theorem Some repair men use the Pythagorean theorem in their repairs especially in repairing windows or other material | ## 2nd usage of the Pythagorean theorem Painters use ladders to paint on high buildings and often use the help of Pythagoras' theorem to complete their work. The painter needs to determine how tall a ladder needs to be in order to safely place the base away from the wall so it won't tip over. In this case the ladder itself will be the hypotenuse. Take for example a painter who has to paint a wall which is about 3 m high. The painter has to put the base of the ladder 2 m away from the wall to ensure it won't tip. What will be the length of the ladder required by the painter to complete his work? You can calculate it using Pythagoras' theorem | ## 3rd usage of the Pythagorean theorem The most ubiquitous "real-life" application of the Pythagorean Theorem is in the empirical observation that this describes how spatial distances in our world act (at least, at such ordinary scales as allow us to ignore relativistic curvature). Indeed, this is the whole reason the theorem is so celebrated in the first place (in the abstract context of inner product spaces, it's mere tautology. The wonderful thing is that the spatial world happens to be one of those inner product spaces). The Pythagorean Theorem tells us how to predict certain distance relationships from other distance relationships. If you see a pendulum swinging and you want to predict the coordinates of its tip over time, the Pythagorean Theorem is there. If you place a box on a ramp, and you want to know how quickly it will slide down, the Pythagorean Theorem is there. If you do any physics or engineering of any kind that involves the notions of "distance", "length", "angle", or "rotation"... well, the Pythagorean Theorem is there, and how much more real-life can you get? |

## 1st usage of the Pythagorean theorem

## 2nd usage of the Pythagorean theorem

## 3rd usage of the Pythagorean theorem

The Pythagorean Theorem tells us how to predict certain distance relationships from other distance relationships. If you see a pendulum swinging and you want to predict the coordinates of its tip over time, the Pythagorean Theorem is there. If you place a box on a ramp, and you want to know how quickly it will slide down, the Pythagorean Theorem is there. If you do any physics or engineering of any kind that involves the notions of "distance", "length", "angle", or "rotation"... well, the Pythagorean Theorem is there, and how much more real-life can you get?

## Radicals have been introduced to math!!!

*a*has a unique non-negative square root, called the

*principal square root*, which is denoted by √

*a*, where √ is called the

*radical sign*or

*radix*. The term whose root is being considered is known as the

*radicand*. The radicand is the number or expression underneath the radical sign,Every positive number

*a*has two square roots: √

*a*, which is positive, and −√

*a*, which is negative. Together, these two roots are denoted.

## The History of Radicals

The mathematical concept of square roots has been in existence for many thousands of years. A square root of a number can be defined as a number that when multiplied by itself produces the first number. For example, 2 is the square root of 4, 3 is the square root of 9. Exactly how the concept was discovered is not clearly known, but several different methods of exacting square roots were used by early mathematicians. Recently discovered Babylonian clay tablets from 1900 to 1600 b.c. contain the squares and cubes of the integers 1 to 30 in Babylonian base 60 Akkadian notation. Whole number roots were specifically stated, while irrational roots were expressed in surprisingly accurate approximations.

Egyptian papyrus dating from about 1700 b.c. have revealed that the Egyptians were knowledgeable of square roots, using a hieroglyphic symbol similar to ⌈ to denote square roots. By the Greek Classical Period (600 to 300 b.c.), square root operations were improved upon by the use of better arithmetic methods. However, because Greek mathematicians were unsettled by inharmonious phenomenon such as irrational numbers (e.g., the square root of 2 is 1.4142135...), they regarded geometry more highly than algebra and arithmetic for its elegance and harmony.

When Hindu mathematics became significant about a.d. 628, mathematicians accepted irrational numbers and used square root operations freely in their equations. They used the term *ka*, from the word *karana*, to denote a square root. Thus, *ka 9* would equal 3. Borrowing much from Hindu mathematics, Arabian mathematicians continued working with irrational number operations. It is the Middle Eastern mathematician al-Khwarizmi who developed our familiar term root to denote a solution to a problem. In the sixteenth century, the German mathematician Cristoff Rduolff was the first to use the square root symbol, , in his book *Coss*, written in 1525.

## 1st usage of radicalsIn the real world people are interested in finding out what values are "normal" and what values are outside of normal, those values that are in the tails of the distribution. Students can't control how tall they grow so you don't want to call the shortest and tallest kids in your class abnormal! Student height was just an easy example to look at and understand. But many factories use the normal distribution to make sure that the products that they are making are of good quality. The business people at the factory develop a normal distribution of the product and do not sell any items that measure in the tails of the distribution. There are lots of other uses for the normal distribution; factories are just one simple example. What does the normal distribution have to do with squares and square roots? Plenty! The equations for finding the tails of the normal distribution use squares and square roots! | ## 2nd usage of radicalsHere is one idea that showcases an important real-life application of square roots and at the same time lets students ponder where math is needed. This idea will work best after you have already taught the concept of square roots but have not yet touched on the Pythagorean theorem. **Draw a square on board or paper, and draw one diagonal into it**. Make the sides of the square to be, say, 5 units. Then make the picture to be a right triangle by wiping out the two sides of square. Then ask students**how to find the length of the longest side**of the triangle. The students probably can't find the length if they haven't studied the Pythagorean theorem yet — but that is part of the "game". Have you ever seen an advertisement where you couldn't tell what they were advertising? Then, in a few weeks the ad would change and reveal what it was all about. It makes you curious. So, let them think about it for a few minutes (don't tell them the answer at first). Hopefully it will pique their interest. Soon you will probably study the Pythagorean theorem anyway, since it often follows square roots in the curriculum.
- Then go on to the question:
**In what occupations or situations would you need to find the longest side of a right triangle**if you know the two other sides? This can get them involved! The answer is: in any kind of job that deals with triangles. For example, it is needful for carpenters, engineers, architects, construction workers, those who measure and mark land, artists, and designers. One time I observed construction people who were measuring and marking on the ground where a building would go. They had the sides marked, and they had a tape measure to measure the diagonals, and they asked ME what the measure should be, because they couldn't quite remember how to do it. This diagonal check is to ensure that the building is really going to be a rectangle and not a trapezoid or some other shape.
| ## 3rd usage of radicals Using Pythagorean's theorem, for instance. Let's say I have a backyard that is a rectangle, 30 yards by 40 yards, and I want to know how far the corners are from each other. Pythagorean's theorem states that a² + b² = c². So, if I square the two sides and add them, I get 30² + 40² or 900 + 1600, which equals 2500. By getting the square root of 2500, I can figure that the two corners are 50 yards apart. |

## 1st usage of radicals

In the real world people are interested in finding out what values are "normal" and what values are outside of normal, those values that are in the tails of the distribution. Students can't control how tall they grow so you don't want to call the shortest and tallest kids in your class abnormal! Student height was just an easy example to look at and understand.

But many factories use the normal distribution to make sure that the products that they are making are of good quality. The business people at the factory develop a normal distribution of the product and do not sell any items that measure in the tails of the distribution. There are lots of other uses for the normal distribution; factories are just one simple example.

What does the normal distribution have to do with squares and square roots? Plenty! The equations for finding the tails of the normal distribution use squares and square roots!

## 2nd usage of radicals

Here is one idea that showcases an important real-life application of square roots and at the same time lets students ponder where math is needed. This idea will work best after you have already taught the concept of square roots but have not yet touched on the Pythagorean theorem.

**Draw a square on board or paper, and draw one diagonal into it**. Make the sides of the square to be, say, 5 units. Then make the picture to be a right triangle by wiping out the two sides of square. Then ask students**how to find the length of the longest side**of the triangle.

The students probably can't find the length if they haven't studied the Pythagorean theorem yet — but that is part of the "game". Have you ever seen an advertisement where you couldn't tell what they were advertising? Then, in a few weeks the ad would change and reveal what it was all about. It makes you curious.

So, let them think about it for a few minutes (don't tell them the answer at first). Hopefully it will pique their interest. Soon you will probably study the Pythagorean theorem anyway, since it often follows square roots in the curriculum.

- Then go on to the question:
**In what occupations or situations would you need to find the longest side of a right triangle**if you know the two other sides? This can get them involved!

The answer is: in any kind of job that deals with triangles. For example, it is needful for carpenters, engineers, architects, construction workers, those who measure and mark land, artists, and designers.

One time I observed construction people who were measuring and marking on the ground where a building would go. They had the sides marked, and they had a tape measure to measure the diagonals, and they asked ME what the measure should be, because they couldn't quite remember how to do it. This diagonal check is to ensure that the building is really going to be a rectangle and not a trapezoid or some other shape.

## 3rd usage of radicals

## And for the last subject today, we have Quadrilaterals!!

**Polygon**comes from Greek.

**Poly-**means "many" and

**-gon**means "angle". Quadrilateral just means "four sides" (

*quad*means four,

*lateral*means side). There are five main types of quadrilaterals the rectangle, parallelogram, square, rhombus, and the trapezoid.

## Definition of every quadrilateral

**-Rectangle:**A rectangle is a four-sided shape where every angle is a right angle (90°).

Also **opposite sides** are parallel and of equal length.

**-Rhombus:**A rhombus is a four-sided shape where all sides have equal length.

Also opposite sides are parallel *and* opposite angles are equal. Another interesting thing is that the diagonals meet in the middle at a right angle. In other words they "bisect" (cut in half) each other at right angles. A rhombus is sometimes called a **rhomb** or a **diamond**.

**-Square:**A square has equal sides and every angle is a right angle (90°)

Also opposite sides are parallel.A square also fits the definition of a **rectangle** (all angles are 90°), and a **rhombus** (all sides are equal length).

**-Parallelogram:** A parallelogram has opposite sides parallel and equal in length. Also opposite angles are equal (angles "a" are the same, and angles "b" are the same).

NOTE: Squares, Rectangles and Rhombuses are all Parallelograms!

**-Trapezoid:**A trapezoid (called a trapezium in the UK) has a pair of opposite sides parallel.It is called an **Isosceles** trapezoid if the sides that aren't parallel are equal in length and both angles coming from a parallel side are equal, as shown. And a **trapezium** (UK: trapezoid) is a quadrilateral with NO parallel sides

## Who discovered them and when did they show up?

## 1st usage of quadrilaterals From ancient times, properties of quadrilaterals have been used especially in art, design and architecture. Diagonal of a rectangle divides it into two congruent triangles and the idea of congruency especially in triangles had been used by Egyptians to build The Great Pyramids of Giza!!!!! The idea of congruency of triangles initially from diagonal of quadrilaterals also helped Leonardo Da Vinci to paint the world famous 'Monalisa'!!!!! So, what other wonders do you want from quadrilaterals????!!!???? | ## 2nd usage of quadrilaterals The properties and rules governing the geometric shapes known as quadrilaterals are used to create floor plans for new buildings, or to create buildings or spaces through engineering and architecture. These distinctive polygon shapes are composed of a couple of triangles, and these two triangles can be arranged in different shapes, such as diamonds, arrows, and rectangles. Since these shapes are so common, quadrilaterals are also used in graphic art, sculpture, logos, packaging, computer programming and web design; in fact, there are few areas of daily life where there are no examples of quadrilaterals. | ## 3rd usage of quadrilaterals there are infinite quadrilaterals in real life! Anything with 4 sides, even if the sides are uneven, is a quadrilateral. Examples could be: table top, book, picture frame, door, baseball diamond, etc. There are a number of different types of quadrilaterals, some of which are harder to find in real life, such as a trapezoid. But, look around you – at buildings, at patterns on fabric, at jewelry – and you can find them! |

## 1st usage of quadrilaterals

## 2nd usage of quadrilaterals

## 3rd usage of quadrilaterals

There are a number of different types of quadrilaterals, some of which are harder to find in real life, such as a trapezoid. But, look around you – at buildings, at patterns on fabric, at jewelry – and you can find them!