Mathematical Sentence

In Algebra, Mathematical sentence is a term which is also known as an expression. An expression is usually defined as a sentence that has a number, an operation and a letter in it.

When a mathematical sentence is not in an algebraic form, it will just have two numbers and an operation. In other words, an expression is the mathematical analogue of an English noun. It is a correct arrangement of mathematical symbols which is used to represent a mathematical object of interest.

Example for Mathematical Sentence:

In general, a mathematical sentence is a formula that is right or wrong, true or false. If a mathematical sentence has an equal sign, it is referred as an equation.

Let us consider the following simple examples to show what a Mathematical Sentence means:

3 + 2 = 5. We know this to be true, it is a mathematical sentence.

3 + 4 = 5

we know this to be false, however, since we know definitely that it is false, But still it is a mathematical sentence.

5x + 8 = 13, for all values of x

This statement is neither true nor false: For some values of x, the statement is true and for some other values, it is false. Hence as the statement is neither true nor false, this is not a mathematical sentence.

Mathematical Sentence as an Open Sentence:

A mathematical sentence that contains one or more variables is referred as an open sentence

Some examples for an open sentence are listed below:

2a = 5+ b, 4x = b + 2a

Solved example for an open sentence:

To show ‘3(2b) = 3' an open sentence?

Solution:

Step 1: In the equation given, the number of variable is One

Step 2: Open Sentence is a mathematical sentence with one or more variables

Step 3: So, '3(2b) = 3' is an open sentence.

Subtracting Fractions

Subtraction is one of the basic operations in math. It is a process of finding the difference of two numbers. In case of whole numbers the process is simple but in case of fractions, the process may involve a number of steps.

How to Subtract Fractions
As mentioned earlier, subtraction of fractions require a number of steps. Of course, in the simplest cases, where the denominators of the fractions are same, the process is just like subtracting whole numbers. All that need to be done is just subtract the numerators over the given common denominator.

But in case of fractions with different denominators, first task is to find the equivalents of the given fractions with same denominator.  Such a equivalent common denominator is also called as Lowest Common Divisor. It is same as the lowest common multiple of the denominator.

How do you Subtract Fractions
Let us concentrate on the case of subtractions of fractions with different denominators, as the case of subtraction of fractions with same denominators is as simple as subtraction of whole numbers.
The easiest method for students to understand is the method of equivalent fractions. The first step here is, determine the lowest common multiple of the given denominators.

Then find ‘equivalent fractions’ of the given ones with the lowest common multiple as the denominator. That is rewrite the given fractions as if their denominators are changed to the lowest common multiple found. For example, 1/2 can be rewritten in the equivalent form as 2/4, 4/8,8/16 etc. depending on the need.
The subtraction of given fractions is same as the subtraction of their equivalent fractions and now the process is simple because the denominators are made same.

Subtract Fractions
Let us discuss a specific example to illustrate how to subtract fractions with different denominators. Let is consider that 2/5 is subtracted from 3/7.

The denominators are 5 and 7 and the lowest common multiple is 35. The equivalent fraction of 2/5 with denominator is 14/35 and that of 3/7 with the same denominator is 15/35.

Now subtraction of 2/5 from 3/7 is same as subtraction of 14/35 from 15/35 which is (15 – 14)/35 = 1/35. Thus the answer is 1/35.

Subtracting Fractions with Whole Numbers
For subtracting fractions with whole numbers, first subtract the fraction parts and then the whole number parts and combine them For example, 5(1/2) – 3 (1/4) is done by (1/2) – (1/4) 1/4 and 5 – 3 = 2, which gives the final answer as 2(1/4).

However trouble arises when the fraction part of the first mixed number is less than that of the second mixed number. In such cases, convert the first fraction part to an improper fraction by borrowing 1 from the whole number. The rest of the procedure is same as explained but remember the whole number part of the first one is reduced by 1!

How to Multiply Two Digit Numbers

If we have practiced and hopefully memorized the multiplication tables and Multiplication Rules, then we can solve any multiplication problem. We just have to understand the system to how to do it. We know that multiplication is just repeated addition but if we have large numbers, we cannot add them repeatedly to get the solution. To multiply one digit number and two digit number with a one digit number is an easy process but multiplying two double digits numbers uses a different process that needs to be followed.

How to Multiply Two Digit Numbers - We will learn here how Double Digit Multiplication works. Let us start with an example of the multiplication problem of two double digits numbers. Let us do 16 times 19; we can break down the Two Digit Multiplication into series of steps which are given as follows: -

• In 2 Digit Multiplication, firstly we take the numbers present in ones place and multiply them together; in this case we multiply 9 times 6 which equal 54.
• With the product of the digits at ones that is 54, we only write the 4 down while multiplication and 5 is carried forward on the tenth place just like when we add two numbers.
• Then we multiply 9 times 1 which is 9 and then add 5, and which is 14 so the solution to 9 times 16 is 144. Thus 16 times 9 is 144.
• Then we take the digit on the tens place on one number that is one in the number 19 and multiply it with 16, which gives us 16.
Now we have two solutions, one is 144 and other is 16, to find the final answer we add a zero to the second solution and then add the solutions together. That means 16 will become 160 and then we add it to 144 which equals 304. So the solution of 16 times 19 is 304.

 Only by using these breaking down method we have solved this big problem.  To multiply two digit numbers we can either break the one double digit number into two digits. For example, in the previous example, we can break 19 into 1 and 9. 9 is at ones place so we can write it as 9 but 1 is at tens place so we can use its face value which is 10. Then we can multiply 9 by 16 which give us 144 and then we multiply 10 by 16 which equals 160. We can add both the solutions to get the solution 304.

Definition of an Acute Angle

The different types of angle have different types of names. Where the angle is less than 90 radiant is called as acute angle. The acute angle is smaller angle compared with other type of angle such as right angle, obtuse angle, straight angle, reflex angle, and full angle.  The angle which is larger is called as reflex angle.

Definition of a Acute Angle
Definition acute angle, an angle which is less then 90 radiant is called as acute angle. Compared to the other types of angle, an acute angle is very smaller that is in between 0-90 radiant. Also that acute triangle are those where all the interior angle are acute.

Definition of an Acute Angle
An angle of less then 90˚ is called as acute angle and it has three angles. An angle which is exactly 90˚ is a right angle but it is not an acute angle, since it is less than 90˚ or between 0˚-90˚.

Acute Angle Definition Math
An angle which measures less than 90˚ or in between 0˚-90˚ is called as acute angle. In other words, acute angle is a positive angle that measures less than 90˚. Simply it is defined as, an angle of less than 90˚.

Geometry Acute Angle
In geometry, an angle is line segments that intersect in the same plane or it is the amount of bend between two lines. The angles are formed by two intersecting rays at the point of intersection or sharing of same end point. The angles are used to measure the turns in between the two arms. The unit of an angle is degree or radiant. In order to measure an angle from a circle that is a two dimensional plane, if the angle measures greater than 0˚ and less than 90˚ is called as acute angle. The acute angle is referred by the alphabets like ABC or DEF.

In above figure angle A is acute which measures greater than 0˚ and less than 90˚. It is formed by the intersection of two rays like AB and AC, which are called as sides of angle or arms. Where the angle is formed is called as vertex. In figure, the angle 1 which is, marked by square sign is called acute angle. Acute angle is smaller than right angle which is exactly equal to 90˚. Similarly the angle 2 is complementary of an angle 1. Complementary angle of an acute angle is also acute as it measures greater than 0˚ and less than 90˚.

Exterior Angles in Polygon and Triangle

Exterior Angle
Consider a shape such as square ABCD. Extend the horizontal bottom of the square at the point D up to F. Now we have a vertical line BD and a horizontal line DF joining at the common vertex D. The angle BDF formed between the original side BD of the mathematical shape i.e. square and the extended side DF is termed to be the exterior angle. The angle formed on the inner side of BD is the interior angle BDC.  The sum of the interior angle and the exterior angle formed using the side BD is 180 degrees. This forms a horizontal straight line CF measuring 180 degrees.

Polygon
A mathematical shape which has straight sides and flat shape is termed to be polygon.

Exterior Angles Polygon
Exterior Angles Polygon definition states that: With one of the angles of the polygon, a linear pair is formed by an angle which is termed as an exterior angle of the polygon.

At every vertex of a polygon, two exterior angles can be formed at the maximum.  Each exterior angle of a polygon is formed in between a side of the polygon and the line extended from its adjacent side.

Finding Exterior Angles of Polygons
We can find the exterior angles formula for polygons. The important point to be noted here is that at each vertex of the polygon, two equal exterior angles can be drawn but the formula for finding the exterior angles of polygon uses only one exterior angle per vertex.

Polygon Exterior Angles Formula:
The polygon exterior angles formula states that the sum of Polygon Exterior Angles is 360 degrees, irrespective of the type of the polygon. In other words, we can say that the sum of all the exterior angles of a polygon is equal to one full revolution.

Sum of one exterior angle of all vertex of any polygon = 360 degrees.

In case of a regular polygon, the formula to find any exterior angle is obtained by dividing the sum of the exterior angles i.e. 360 by the number of angles, say “n”.

The value of an angle of a regular polygon = 360/n

Here are few examples of Hexagon:

Example 1:
Hexagon has 6 sides. Therefore every exterior angle in hexagon = 360/6 = 60 degrees.

Example 2:
The value of an angle of a regular polygon is 30 degrees. How many sides are there in the polygon?
360/n = 30 which can be rewritten as 360 = 30n.
Thus, n is given by dividing 360 by 30 which results in 12.

Exterior Angles Triangle
In case of triangle, exterior angle lies in between one side of the given triangle and the extension of the other side of the same triangle. Exterior Angles of Triangle can be obtained by adding the measures of the two of the non-adjacent interior angles.