Algebraic expressions

Introduction to algebraic expressions:

Algebraic expressions are formed from variables and constants. We simply use the combination of operations of addition, subtraction, multiplication and division on the variables and constants to form expressions. Algebraic expressions does not contain equal sign.

Some examples of algebraic expressions:

2x + 2y , 2xy , x² - y² , 3m + n - 5.

There are one or more terms in the algebraic expressions. For example, 3x² + 4x + 2 is algebraic expression. It has three terms.

Let us discuss about some example problems on Algebraic expressions.

An algebraic expression is a number, a variable or a combination of the two connected by some mathematical operation like addition, subtraction, multiplication, division, exponents, and/or roots.

For eg.: 4x, 3p, 2a+b, 6z/5

Introduction about Algebraic expressions:

Expressions are a central concept in algebra. A variable can take various values. Its value is not fixed. On the other hand, a constant has a fixed value. We combine variables and constants to make algebraic expressions. For this, we use the operations of addition, subtraction, multiplication and division. We have already come across simple algebraic expressions like x + 3, y – 5, 4x + 5, 10y – 5 and so on. The above expressions were obtained by combining variables with constants. We can also obtain expressions by combining variables with themselves or with other variables.In this article we will see some definitions, facts about algebraic expressions

Terms about Algebraic Expressions

Properties about algebraic Expressions:

Like terms: When the terms are same variables then that terms are called as like terms. Otherwise it is called as unlike terms

2xy and 5xy because the factors of 2xy are 2 and x and y. and factors of 5xy are 5 and x and y both factors are same. So these are like terms.

Unlike terms: The terms 2xy and –3x , have different algebraic factors. They are unlike terms. Similarly, the terms, 2xy and 4 are unlike terms. Also, the terms –3x and -4 are unlike term.

Coefficient of the Term: The value in front of the variable is called as the coefficient of the term. The above example the coefficient of x is -3.

Facts about Algebraic Expressions

Monomial: The number of terms in an expression is one then the expression is called as monomial. For example 5x

Binomial:The number of terms in an expression is two then the expression is called as binomial. For example,

x + y, m – 5

Trinomial: The number of terms in an expression is three then the expression is called as trinomial. for example, the expressions x + y + 7, a b + a +b

Polynomial:An expression with one or more terms is called a polynomial. Thus a monomial, a binomial and a trinomial are all polynomials. For Example: 2x³+5x²+15x+22.

Finding the value of an expression:

This is nothing but substituting the values to the variables in the expression. That gives the solution to the expression. For example the area of a square is length², where l is the length of a side of the square. If l = 8 cm., the area is 8² cm²or 64 cm²; if the side is 12 cm, the area is 12² cm² or 144 cm² and so on.

When an expression is written using words or phrases, it is a verbal expression. It is in a statement form. The symbols and statements used have the same meaning as they have in arithmetic.

Signs and symbols:

‘=’ means ‘is equal to’

‘≠’ means ‘is not equal to’

‘<’ means ‘is less than’

‘>’ means ‘is greater than’

‘≤’ means ‘is less than or equal to’

‘≥’ means ‘is greater than or equal to’

The following are the list of words used and their meaning:

1. Addition : sum, plus, add to, more than, increased by, total

2. Subtraction: difference of, minus, subtracted from, less than , decreased by, less

3. Multiplication: product, times, multiply, twice, of

4. Division: quotient , divide, into, ratio

Variable expressions to algebraic expressions:

In Algebra an expression is a term which consists of both symbols and numbers. There are two types in that. They are Algebraic and Verbal expressions. Algebraic expression consists of signs and symbols. The signs involve signs of several operations like addition, subtraction, multiplication and division. The symbol involves all Arabic numerals and literal numbers. Ex: 3ax, 4yz…Verbal expression is the expression which consists of verbal phrases like sum, product, less than, etc…..

Verbal Expressions into Algebraic Expressions Explanation:

Some of the important operations that are used for this verbal expression into algebraic expression are,

• Increased, Plus, more, total, add, sum +

• Difference, Minus, less, decreased, subtract, -
• Product, Times, multiply, double, twice, product *

In words they can be expressed as

Sum of a and b a + b

Difference of a and b a – b

In addition with these basic operations some other operations like less than and greater than...

• Less than <
• Greater than >
• Less than or equal <=
• Greater than or equal >=

Types of Algebraic Expressions

Monomial

A monomial is an algebraic expression formed by a single term.

Binomial

A binomial is an algebraic expression formed by two terms.

Trinomial

A trinomial is an algebraic expression formed by three terms.

Polynomial

A polynomial is an algebraic expression consists of more than one term.

In math, An algebraic expression is an expression containing symbols, variables and constants together. In math, Verbal expression is a sentence forming from an algebraic expression. Some of the verbal phrases for arithmetic operation is given below. Using these, we can frame the verbal expression for given an algebraic expression.

Let us see brief about verbal expression in math.

Verbal Expression in Math:

The verbal expression for an algebraic expression a + b may write as following,

a plus b , a added to b, a is increased by b, the sum of a and b, b is added to a, b more than a.

The verbal expression for an algebraic expression a - b may write as following,

a minus b , a is decreased by b, b subtracted from a, b less than a, a diminished by b, a reduced by b, the difference between a and b.

The verbal expression for an algebraic expression a x b may write as following,

a times b , the product of a and b, b is multiplied by a.

The verbal expression for an algebraic expression a ÷ b may write as following,

The quotient of a and b, a is divided by b.

Let us learn about how to translate algebraic expression into verbal expression.

Introduction to verbal expression in math:

In math, An algebraic expression is an expression containing symbols, variables and constants together. In math, Verbal expression is a sentence forming from an algebraic expression. Some of the verbal phrases for arithmetic operation is given below. Using these, we can frame the verbal expression for given an algebraic expression.

Let us see brief about verbal expression in math.

Verbal Expression in Math:

The verbal expression for an algebraic expression a + b may write as following,

a plus b , a added to b, a is increased by b, the sum of a and b, b is added to a, b more than a.

The verbal expression for an algebraic expression a - b may write as following,

a minus b , a is decreased by b, b subtracted from a, b less than a, a diminished by b, a reduced by b, the difference between a and b.

The verbal expression for an algebraic expression a x b may write as following,

a times b , the product of a and b, b is multiplied by a.

The verbal expression for an algebraic expression a ÷ b may write as following,

The quotient of a and b, a is divided by b.

Let us learn about how to translate algebraic expression into verbal expression.

Example Problems of Verbal Expression in Math:

Problem 1:

Translate this algebraic expression into verbal expression: 3 + b

Solution:

We can write it as,

b more than 3, 3 plus b, 3 added to b, b is increased by 3.

Problem 2:

Translate this algebraic expression into verbal expression: 3 - b

Solution:

We can write it as,

b less than 3, 3 minus b, b subtracted from 3, b is decreased by 3.

Comparing Decimals

Let Us Learn About Comparing Decimals

Decimal number is a number. It has two parts. That is whole number part and a fractional part. The “.” is called the decimal point.

Ordering decimals:

Suppose if we have a two or more decimal, we need to arrange the decimals using tenths, hundreds and thousands place.

Comparing decimals:

Suppose if we have a two decimal numbers we can compare them. Normally one number is greater than, less than or equal to another number. Here we are going to see some examples for how to compare and order decimal values.

Mathematical Concepts in Comparing and Ordering Decimals

• To compare decimals start at the left and compare digits in the same places, find first the place where the decimals differs,
• Equal decimals can be written by adding zeros or deleting zeros from the right of the last non zero digit.
• Adding zeros at the end of the decimal does not change it value. On the other hand, adding zeros before the first decimals digit changes its value.
• To compare and order decimals having different number of digits, put zeros at the end of the shorter decimals to have the same number of digits, then compare and order the numbers as if the decimal points.

Comparing and Ordering Decimals:

• Compare and order decimals through hundred thousandths.
• Use the symbols >, <, or = correctly in comparing decimals.
• List decimal in the order least to greatest.

Problem in comparing and ordering decimals

When you compare decimals, think about place value you can add or remove zeros at the end of the decimal without changing its meaning.

0.2 = 0.20 =0.200 =0.2000

0.54 = 0.540 = 0.5400 = 0.54000

3.5 = 3.50 = 3.500 = 3.5000

The ordering decimal is 3.5, 0.2,0.54.

Rule : To compare two decimals, first convert the given decimals into like decimals and then compare the whole number parts. The decimal with greater whole number part is greater. If the whole number parts are equal, then compare the tenths digits. The decimal with the bigger digit at tenths place is greater. If the tenths digits are equal, then compare the hundredth digits and so on.

i)In 2.5 and 3.5

Here we can notice that both are like decimals number and the tenths place is same so, comparing the whole number parts of two decimals, we have 3 > 2

Hence, 3.5 > 2.5

ii) In 15.12 and 15.39

Here we can notice that both are like decimals num

ber so, comparing the whole number parts of two decimals, we have 15=15

Now comparing the digits at tenths place, we have 1 <>

Hence, 15.12 <>

iii)In 5.147 and 5.14

Here we need to convert the unlike decimal to like decimal i.e., 5.14 = 5.140, so compare 5.147 and 5.140

Comparing the whole number parts of two decimals, we have 5 = 5

Now, comparing the digits at tenths place, we have 1 = 1

Next, comparing the digits at hundredths place, we have 4 = 4

Finally, comparing the digits at thousandths place, we have 7 > 0

Hence, 5.147 > 5.14.

iv)In 3.159 and 3.15

Here we need to convert the unlike decimal to like decimal i.e., 3.15 = 3.150, so compare 3.159 and 3.150

Comparing the whole number parts of two decimals, we have 3 = 3

Now, comparing the digits at tenths place, we have 1 =1

Next, comparing the digits at hundredths place, we have 5 = 5

Finally, comparing the digits at thousandths place, we have 9 > 0

Hence, 3.159 > 3.150.

5)Arrange 12.4, 135.25, 14.648, 115.35 in ascending order and descending order.

Converting the given decimals into like decimals, we have,

12.400, 135.250, 14.648, 115.350.

Here we notice that in all the numbers the whole number part is different so we need to arrange the whole number from smaller to bigger.

Thus arranging in ascending order, 12.400 <>

Hence, the given decimals in ascending order are :

12.400 <>

Thus arranging in descending order we need to arrange the whole number from bigger to smaller,

135.250 > 115.350 > 14.648 > 12.400 since 135 > 115>14>12.

Hence, the given decimals in descending order are :

135.250 > 115.350 > 14.648 > 12.400 (descendingorder).

Comparing Decimals:

This step explains the kids that how to compare the decimal valuesWhen 0's are add to the right of the decimal number, the number is not changed.

To compare decimals, we compare the values,

Step 1: Start at the left side; Solve the first place where the numbers are different

Step 2: Compare the digits that are different.

If there are two decimal numbers we can compare them. One number is either greater than, less than or equal to the other number.

A decimal number is just a fractional number. Comparing 0.7 and 0.07 is clearer if we compared 7/10 to 7/100. The fraction 7/10 is equivalent to 70/100 which is clearly larger than 7/100.

Therefore, when decimals are compared start with tenths

place and then hundredths place, etc. If one decimal has a higher number in the tenths place then it is larger than a decimal with fewer tenths. If the tenths are equal compare the hundredths, then the thousandths etc. until one decimal is larger or there are no more places to compare. If each decimal place value is the same then the decimals are equal.

Place Value

To understand decimal numbers you must first know about

Place Value.

When we write numbers, the position (or "place") of each number is important.

In the number 327:

• the "7" is in the Units position, meaning just 7 (or 7 "1"s),
• the "2" is in the Tens position meaning 2 tens (or twenty),
• and the "3" is in the Hundreds position, meanin g 3 hundreds.

As we move left, each position is 10 times bigger!
From Units, to Tens, to Hundreds

... and ...

As we move right, each position is 10 times smaller

From Hundreds, to Tens, to Units

And that is a Decimal Number!

Decimal Point

The decimal point is the most important part of a Decimal Number. It is exactly to the right of the Units position. Without it, we would be lost ... and not know what each position meant.

Now we can continue with smaller and smaller values, from tenths, to hundredths, and so on, like in this

example:

Large and Small

So, our Decimal System lets us write numbers as large or as small as we want, using the decimal point. Numbers can be placed to the left or right of a decimal point, to indicate values greater than one or less than one.

17.591

The number to the left of the decimal point is a whole number

As we move further left, every number place gets 10 times bigger.

The first digit on the right means tenths (1/10).

As we move further right, every number place gets 10 times smaller (one tenth as big).

Types of Cylinders

Let Us Learn About Types of Cylinders

There are two types of cylinders.

Right Circular Cylinder: When the base of a right cylinder is a circle, it is called a right circular cylinder.

Oblique Cylinder: When the centers of the bases of a cylinder are not aligned directly one above the other, it is called an oblique cylinder.

Examples:

Right Cylinder: A right cylinder is a cylinder in which the centers of the bases are aligned dire ctly one above the other.

In the figure, r = radius of the base and h= height of the cylinder

A cylinder is mainly referred to a Right circular cylinder , if specifically not mentioned of t he type.

Weight of the cylinder

Let Us Learn About Weight of the cylinder

Introduction to Weight of the cylinder:

Cylinder is the three dimensional figure. Weight of the cylinder is same as the volume of the cylinder. The formula to find the weight of the cylinder is pi * radius² * height. Height is the total height of the cylinder and radius is the radius of the circular face which is at the bottom and top of the cylinder.

Some basic concepts of a Cylinder:

A cylinder is a basic geometric shapes which is curvilinear.

It has two congruent and parallel bases and is a three-dimensional geometric figure that. Its bases are circles rather than polygons.

The cylinder is similar to a prism, it has two faces, zero vertices, and zero edges

The line formed by the centers of the bases of a cylinder, which coincides with the Altitude of the cylinder.

Any solid which is enclosed within the surfaces and two planes perpendicular to the axis is called a cylinder.

In the figure, h = the height and r = the radius of the circular base.

Altitude:

Altitude is the height of the cylinder measured between the two bases.

Altitude Cylinder

Let Us Learn About Altitude Cylinder

Introduction to altitude cylinder:

A cylinder is one of the curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since deep antiquity. Its cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder

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Introduction to Weight of the cylinder:

Cylinder is the three dimensional figure. Weight of the cylinder is same as the volume of the cylinder. The formula to find the weight of the cylinder is pi * radius² * height. Height is the total height of the cylinder and radius is the radius of the circular face which is at the bottom and top of the cylinder.

circumference of a cylinder

Let Us Learn About circumference of a cylinder

Introduction for circumference of a cylinder:

A cylinder is a 3-D geometry with two circular surfaces and one curved surface. Let us know how the surface area of a cylinder or circumference is determined. Cylinder has height and radius. The cylinder has two bases, the base has radius r.

Formula for finding circumference of a cylinder.

Circumference of cylinder = 2 x π x r(r + h)

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Introduction to Cylinder volume:

A Cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder.

The surface area and the volume of a cylinder have been known since deep antiquity. In this article of cylinder volume, finding volume of the cylinder is explained.

Formula for Finding Volume of Cylinder:

The diagrammatic representation of a cylinder is shown below:

Formula for finding volume of a cylinder:

Volume of a cylinder = r² h

where, r ----> radius

h----->height

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