Real numbers

Real Number Definition– As the name says “Real”, so these numbers are actually numbers that really exists. Any number that we think of is considered as a real num, be it positive or negative, fraction or decimal. Real num. are numbers those includes both rational and irrational number. A real-number has to have a value. If there is no value to any number then we can call that number as an imaginary number. All integers like -75, 89, and 84 etc. are considered as real num. All fractions like 3/5, 7/2, -9/7 are considered as real no. too.

Decimals along with repeating decimals are also considered as real num.
These can be any positive or negative number. We can plot All Real Numbers on the number line too.  Therefore we can order these numbers and that we cannot do in case of imaginary numbers. The name of imaginary numbers itself says that they are imaginary so we can just imagine them; they do not have a specified value. These numbers can be plotted the same way we plot the integers that is smaller numbers on left and larger numbers on right. So greater the number, more it will be towards right side of number line.
So we can call real nos. as all those numbers which are present on number line are termed as such.
Some Real Numbers Examples are pi, 34/7, 5.676767, -1034, 45.87 etc.
Some examples of imaginary numbers are square root of -34 or square root of -2, as there is a negative under the root, so the value of this number cannot be found.

Similarly value of infinity cannot be determined too .
Hence these numbers are not considered as real and are considered as an imaginary numbers. So these numbers are all integers, fractions decimals and repeating decimals numbers. We can add, subtract, multiply and divide real nos. just like another numbers. We can perform the operations on real-numbers same way as we do them on other numbers.

Now the question arises that Is 0 a Real Number– Zero is considered as an integer and all integers are real-numbers. Therefore zero is considered to be as a real-number.
 All Real Numbers Symbol is R which is used by many mathematicians. The symbol R is used to represent the set of real nos. All such numbers can be seen on the number line but we cannot find imaginary numbers on that.

Construct angles tutoring

In geometry, we have many constructions of different geometrical shapes, angles,angle bisectors etc, these constructions are really challenging and mind blowing,now let see and the steps of  few constructions of angles.

Constructions of angles can be done by two methods,one by using ruler and compass and another by using protractor. Now we see and understand the steps of angles constructions by using both the methods.

Construct Angle 60' by Two Methods

Construct angle 60' by a ruler and compass.

Construction method I: Draw a convenient line segment AB,keep the compass at A ,draw a semi circle.the semi circle should intersect the line segment AB,then mark the intersect the point as C, then again keep the compass at C,without changing the measurement of the compass draw an arc , which cut the semicircle at the point D.Now join A and D and extend the line.Hence we get angle 60'.

Construction method II:  

In this method we draw a line segment AB and keep the protractor at A ,mark 60' from right to left .Mark the new point as ,C Join AC.Now we get angle 60' using protractor.

Construct Angle 90' by Two Methods

Construct angle 60' by a ruler and compass.

Construction method I:

Draw a line segment AB,draw a semi circle at A,which intersect C,then keep the compass at C draw an arc which intersect the semi circle at D,again with the same measurement keep the compass  at D ,draw one more arc ,which intersect the semi circle at E,do not change the measurement of compass, keep the compass at D and E, cut two more arc's which intersect each other exactly at F, join A and F.The line AF exactly perpendicular to the line segment AB.Hence angle 90' constructed.

Construction method II :

Now in this method, draw a line segment AB keep the protractor at A and mark 90', name as C , join A and C , we get line AC is perpendicular to the line AB. Hence  angle 90' constructed using protractor.

Exercises on Construct Angles

1) Construct angle 120' using ruler and compass.

2) Construct angle 120' using protractor.

3) Construct angles 15' , 30', 45' , 75', 105', 135' using ruler and compass.

4) Construct angles 15' , 30', 45' , 75', 105', 135' using protractor.

Learning to Simplify Mixed Numerals

We have studied the number line. We know about the natural numbers, whole numbers, integers, real numbers and also imaginary numbers. When these numbers are combined we get interesting combinations. We need to do a study on fractions. Fractions can be numbers in the form of P/Q. P and Q are integers. These fractions now are of three types namely proper, improper and mixed fractions. Mixed fractions are bit complicated compared to the other two. So we need to study them in detail. First we need to learn the process of simplifying mixed fractions as a first step in this direction. It is a relatively simple process.  To simplify mixed fractions we convert them to improper fractions. Improper fractions are easy to handle and can be used for the purpose of calculations. After converting the mixed fraction into an improper one, we can further simplified if need be. For this we need to convert mixed fraction to decimal and solve this problem.

After getting the improper fraction, we take the numerator of the fraction as the dividend and the denominator as the divisor. We divide the dividend with the divisor till we get the remainder as zero. When the digits in the dividend get over we continue the process of division by adding zeros to it and a decimal point to the quotient. Sometimes we don’t get the remainder as zero. In that case the decimal part might contain digits that are repeating. It can be ended there. So the question arises how to solve mixed fractions after we have converted them into decimals. Solving can involve addition of two mixed fractions or it can also be subtraction of two mixed fractions. Both the processes can be relatively simple if the denominators in both the cases are the same. But if the denominators are different the process is bit different than when they are same. Now we will learn subtracting mixed fractions with different denominators to get the final answer. When the denominators are not same first convert into similar ones and then do the process of addition or subtraction. This can be done by taking the LCM of the denominators. After taking the LCM the denominators become common and the numerators can be added or subtracted to get the answer. This helped us to know what is a mixed fraction better. It contains a whole number part and a proper fraction.

Line Plot Examples

One of the important educational tools is graphs. Through graphs we learn how to organize and interpret information. Graphs help not only to analyze data in math but also help convey information in business. Many types of graphs exist, but we use line plots to show frequency of data.

A line plot shows frequency of data along a number line with “x” mark or any other marks to show frequency. It is best to use a line plot while comparing fewer than 25 frequency data. It is a quickest and simplest way to organize data.

Steps to Follow to Plot the Examples on Line Plot

Here are few steps to show how to sketch a line plot;

Step 1 : Gather the given  information which is called data for which a line plot has to be drawn. Look for those data sets that need to show frequency.

Step 2 :Group the data items that are the same and then create and label a chart to help you organize the list from the data.

Step 3 : Determine an approximate scale to draw the line plot. If the scale consists of numbers, then break it into even parts.

Step 4 : Draw a horizontal line and label it according to the scale chosen. This looks  similar to a number line.

Step 5 : Put a mark, say X that corresponds to the number on the scale according to the data that is organized. This is the line plot.

For example,

We use the above steps to give the line plot examples

Sketch the line plot for the following given data of heights of students of a class in centimeters.

120,110,100,120,105,110,105,100,110,105.

Solution : Let the labels be the heights of the students (in centimeters) in a class. If two students are of 120cms , you'd place two X's above 120. If three students are of 110cms , you'd place two X's above 110. If three students were 105cms , you'd put three X's above 105. If two students are of 100cms , you'd place two X's above 100. The total would be ten students, thus ten X's altogether.

Some more Line Plot Examples

Use all the steps to draw the line plot examples:

(1) Sketch the line plot to show the test scores of 17 students that are given below:

10,  30,  10,  10,  20,  30,  50,  40,  45,  50,  10,  30,  35,  30,  20,  40,  50

 (2) Represent the data in a line plot. The following data is obtained from a survey made in an area for number of pets in each household.

4, 0, 1, 3, 2, 6, 1, 4, 2, 8, 10, 2, 1, 3, 0, 13 ,4, 2, 1, 14, 0, 3, 0, 2, 1, 0, 3, 1

Multiplying mixed fractions

Mixed fractions are just the mixture or a combination of two things of whole number and a fraction. Mixed fractions are specially used when we have to show how much a whole thing we have and how much of part of something we have. The form for mixed fraction looks like this, a and b/c, where ‘a’ is our whole number and b over c is b/c is our fraction.

Multiplying Mixed Number Fractions - For Multiplication of Mixed Fractions, we first convert the mixed fractions into improper fractions and the multiply them and then convert the solution back into mixed fraction. How do I Multiply Mixed Fractions – let us understand by taking an example, if we have to multiply 1 times 3/4 and 5 times 7/9 then we can convert 1 times 3/4 to improper fraction by multiplying 4 with 1 and adding 3 to it. We will get 7, which will be our numerator for the improper fraction. The denominator will remain same that is 4. So our improper fraction will be 7/4. Similarly we can convert 5 times 7/9 as (9*5) +7/9 which will be 52/9.

Hence we got two fractions as 7/4 and 52/9. We can multiply both the numerators and the denominators. We will get 7*52/4*9 which will be 364/36. We can reduce this fraction by dividing both the numerator and the denominator by 4 which makes the fraction as 91/9.

Now we can convert it into a mixed fraction by dividing. So 91/9 can be written as 10 times 1/9. Multiplying Mixed Fraction always gives a mixed fraction as the solution. The same we witnessed in the above example. We multiplied two mixed fractions and we got a mixed fraction as an answer. So to multiply mixed fractions we need to follow the following steps: -
1. Convert the mixed fractions that we have to multiply into improper fractions.
2. Multiply the improper fractions by multiplying the numerators and the denominators separately that is if we have two fractions, the numerator of both the fractions should be multiplied together and the denominator of both the fractions together. For example if we have 11/2 times 8/5. Then we can multiply 11 with 8 which will be 88 and 2 with 5 which will be 10. Therefore 11/2 times 8/5 gives 88/10
3. Reduce the fraction as much as we can.
4. Convert the answer back into the mixed fraction form.