Dr.A.Jasmine Christy Asst Prof-Pedagogy of Mathematics

Measurement & Algebra

Definition of Measurement A number that shows the size or amount of something. Usually the number is in reference to some standard measurement, such as a meter or kilogram.

History of Measurement ➢ Years ago people came up with standard length measures, but they didn't all agree on one system. ➢ The Imperial System (which uses yards, feet, inches, etc to measure length) was developed over hundreds of years in the UK ➢ French came up with the Metric System (origins in 1670, but developed in the 1790s), which soon spread through Europe, and then most of the world, even to England itself in 1965. ➢ The metric system was first proposed by the French astronomer and mathematician Gabriel Mouton in 1670 and was standardized in Republican France in the 1790s. ➢ USA developed their own version of the Imperial system (US Standard Units), but the Metric System is also used in the USA, particularly in Science.

Measurement

•How old are you? •How much do you weigh? •How tall are you? •How much water can be filled in your water bottle? •How hot is it today? To answer the above questions; what we need is to measure. •To find how old you are, you need to measure time. •To know how much you weigh, you must weigh yourself. •To know how tall you are, you need to measure your height (length) •To know how much water you can fill in your water bottle, you need to measure the capacity of your bottle. •To find out how hot it is today, you need to measure the temperature. So, what exactly is “measurement”? Measurement is a process of finding a number that shows the amount of something.

Time The ongoing sequence of events is time. We can measure time in seconds, minutes, hours, days, weeks, months, and years. A clock and a calendar help us to measure time

Measurement

Weight The amount of matter a thing consists of is called its weight. Measuring weight means to measure the heaviness of a thing. Weight can be measured in grams, kilograms, and pounds.

Capacity Capacity is a measure of how much quantity a thing can hold

Length The amount of something that is measured from one end to the other along the longest side is called its length.

Length is measured in centimeters, meters, kilometers, feet, and miles

Temperature The temperature of a thing is the measurement of how hot or cold it is Temperature is measured in Celsius, Fahrenheit, and Kelvin

Measurement

SI

Measurement

"English Units" or "US Customary Units" Liquids

Mass (Weight)

Length

Algebra What Is Algebra? ➢ The Number Theory, Geometry, and their analysis put together to make an extensive part of mathematics which is known as "Algebra". ➢ In other words, Algebra is a part of mathematics that deals with symbols and the rules to calculate those symbols.

History Of Algebra ➢ Muhammad ibn Musa al-Khwarizmi is known as the "Father of algebra". ➢ He was a Persian mathematician who wrote a book named Kitab Al Muhtasar fi Hisab Al Gabr Wa I Muqabala in the Arabic language, which was later translated into English as " The Compendious Book on Calculation by Completion and Balancing", from which the word ALGEBRA was derived. ➢ The book provides a systematic solution for linear and quadratic equations. ➢ According to Al-Khwarizmi, the word algebra is described as 'reduction' and 'balancing' of subtracted terms that is a transposition to other sides of the equation (cancellation of like terms).

A Puzzle What is the missing number?

In Algebra we don't use blank boxes, we use a letter (usually an x or y, but any letter is fine). So we write:

It is really that simple. The letter (in this case an x) just means "we don't know this yet", and is often called the unknown or the variable.

Why did we add 2 to both sides? To "keep the balance"...

Degree of a Polynomial

What is a Polynomial? ▪ Polynomial is made up of two terms, namely Poly (meaning “many”) and Nominal (meaning “terms.”). ▪ A polynomial is defined as an expression which is composed of variables, constants and exponents, that are combined using the mathematical operations such as addition, subtraction, multiplication and division (No division operation by a variable). ▪ Based on the numbers of terms present in the expression, it is classified as monomial, binomial, and trinomial

Remainder Theorem

Algebraic Identities

Least Common Multiple(LCM) is a method to find the smallest common multiple between any two or more numbers. Different Methods of LCM There are three important methods by which we can find the LCM of two or more numbers. They are: •Prime Factorization Method •Division Method

LCM of 24 and 15 = 2 × 2 × 2 × 3 × 5 = 120

GCD Find the GCD of the following polynomials. x4 + 3x3 −x −3, x3 +x2 −5x + 3 Solution : Let f(x) = x4 + 3x3 −x −3 g(x) = x3 +x2 −5x + 3 The degree of the polynomial f(x) is greater than g(x). Note : If we have any missing term, we have to replace that place by 0.

The remainder is 3(x2 + 2x - 3), which is not equal to 0. So, we have to divide x3 +x2 −5x + 3 by x2 + 2x - 3 by leaving the remainder.

We get 0 as remainder by dividing the given polynomial by x2 + 2x - 3. Hence the required GCD is x2 + 2x - 3.

CROSS MULTIPLICATION METHOD Let us consider the following system of linear equations. a1x + b1y + c1 = 0 a2x + b2y + c2 = 0

CROSS MULTIPLICATION METHOD Solve the following system of equations using cross multiplication method : 2x + 7y - 5 = 0 -3x + 8y = -11 Solution: First we have to change the given linear equations in the form a1x + b1y + c1 = 0, a2x + b2y + c2 = 0. 2x + 7y - 5 = 0 -3x + 8y + 11 = 0