Rational numbers & Properties.

 

Rational numbers

Rational numbers are represented in the form of a/b where a and b are integers and  b ≠ 0. It is also a type of real number. Any fraction with non-zero denominators is a rational number. Hence, we can say that ‘0’ is also a rational number, as we can represent it in many forms such as 0/1, 0/2, 0/3, etc. But, 1/0, 2/0, 3/0, etc. are not rational, since they give us infinite values.

How to identify rational numbers?

The set of rational numerals:

1.     Include positive, negative numbers, and zero

2.     Can be expressed as a fraction

Examples of  Rational Numbers: 

a	b	a/b
10	2	10/2 =5/1
1	1000	1/1000
10	50	10/50 = 1/5

Types  of  Rational Numbers

A number is rational if we can write it as a fraction, where both denominator and numerator are integers and denominator is a non-zero number.

The below diagram helps us to understand more about the number sets.

• Real numbers (R) include all the rational numbers (Q).
• Real numbers include the integers (Z).
• Integers involve natural numbers(N).
• Every whole number is a rational number because every whole number can be expressed as a fraction.

 

 

Standard Form of Rational Numbers

The standard form of a rational number can be defined if it’s no common factors aside from one between the dividend and divisor and therefore the divisor is positive.

For example, 12/36 is a rational number. But it can be simplified as 1/3; common factors between the divisor and dividend is only one. So we can say that rational number     ⅓ is in standard form.

Positive and Negative Rational Numbers

Positive Rational Numbers	Negative Rational Numbers
If both the numerator and denominator are of the same signs.	If numerator and denominator are of opposite signs.
All are greater than 0	All are less than 0
Example: 12/17, 9/11 and 3/5 are positive rational numbers	Example: -2/17, 9/-11 and -1/5 are negative rational numbers

Arithmetic Operations on Rational Numbers

Arithmetic operations are the basic operations we perform on integers. Let us discuss here how we can perform these operations on rational numbers, say p/q and s/t.

Addition: When we add p/q and s/t, we need to make the denominator the same. Hence, we get (pt+qs)/qt.

Example: 1/2 + 3/4 = (2+3)/4 = 5/4

Subtraction: Similarly, if we subtract p/q and s/t, then also, we need to make the denominator same, first, and then do the subtraction.

Example: 1/2 – 3/4 = (2-3)/4 = -1/4

Multiplication: In case of multiplication, while multiplying two rational numbers, the numerator and denominators of the rational numbers are multiplied, respectively. If p/q is multiplied by s/t, then we get (p×s)/(q×t).

Example: 1/2 × 3/4 = (1×3)/(2×4) = 3/8

Division: If p/q is divided by s/t, then it is represented as:
(p/q)÷(s/t) = pt/ qs

Example: 1/2 ÷ 3/4 = (1×4)/(2×3) = 4/6 = 2/3

Properties of Rational Numbers

·        The results are always a rational number if we multiply, add or subtract any two rational numbers.

·        A rational number remains the same if we divide or multiply both numerator and denominator with the same factor.

·        If we add zero to a rational number then we will get the same number itself.

Rational Numbers and  Irrational Numbers

There is a difference between rational and Irrational Numbers. A fraction with non-zero denominators is called a rational number. The number ½ is a rational number because it is read as integer 1 divided by the integer 2. All the numbers that are not rational are called irrational. Rational numbers   can be either positive, negative or zero. While specifying a negative rational number, the negative sign is either in front or with the numerator of the number, which is the standard mathematical notation. For example, we denote negative of 5/2 as -5/2.

An irrational number cannot be written as a simple fraction but can be represented with a decimal. It has endless non-repeating digits after the decimal point. Some of the common irrational numbers are:

Pi (π) = 3.142857…

Euler’s Number (e) = 2.7182818284590452…….

√2 = 1.414213…

Solved Examples

Example  1:

Identify each of the following as irrational or rational: ¾ ,  90/12007,  12 and √5.

Solution:

Since a rational number is the one that can be expressed as a ratio. This indicates that it can be expressed as a fraction wherein both denominator and numerator are whole numbers.

• ¾ is a rational number as it can be expressed as a fraction. 3/4 = 0.75
• Fraction 90/12007 is rational.
• 12, also be written as 12/1. Again a rational number.
• Value of  √5 = 2.2360679775…….. It is a non-terminating value and hence cannot be written as a fraction. It is an irrational number.

Example 2:  

Identify whether mixed fraction, 1 ¹/₂ is a rational number.

Solution: 

The Simplest form of 1¹/₂  is 3/2

Numerator = 3, which is an integer

Denominator = 2, is an integer and not equal to zero.

So, yes, 3/2 is a rational number.

Example 3:

Determine whether the given numbers are rational or irrational.

(a) 1.75  (b) 0.01   (c) 0.5  (d) 0.09   (e) √3

Solution:

a ,b, c and d are rational numbers and e is irrational numbers

The given numbers are in decimal format. To find whether the given number is decimal or not, we have to convert it into the fraction form (i.e., a/b)

If the denominator of the fraction is not equal to zero, then the number is rational, or else, it is irrational.

 

 

 

 

 

Real life context for teaching rational numbers.

1.     Mohan hikes for 2.5 miles and stops for lunch. Then he hikes for 1.5 more miles. How many miles did he hike altogether?

 

 

 

 

 

Solution : 

Step 1 : 

Use positive numbers to represent the distance Mohan  hiked. 

Step 2 : 

Find 2.5 + 1.5.

Let us use the real number line to add 2.5 and 1.5.

Step 3 : 

Start at 2.5.

2.     The temperature on an outdoor thermometer on Monday was 5.5 °C. The temperature on Thursday was 7.25 degrees less than the temperature on Monday. What was the temperature on Thursday ?

Answer : 

Step 1 : 

Find 5.5 - 7.25.

Step 2 :

Start at 5.5.

Step 3 :

Move |7.25| = 7.25 units to the left, because we are subtracting a positive number

 

3.     The science teacher is filling her new fish aquarium. The aquarium holds 40 gallons. If she fills the aquarium 4/5 of the way full, how many gallons will she need?

Solution :

If 4/5 of the aquarium is filled, then 1/5 th of the aquarium to be filled to make it full. Because 5 - 4  =  1.

To find number of gallons, we have to multiply 40 by 1/5. 

Step 1 : Multiply 40 by 1/5

40 x 1/5     

Step 2 : Simplify

8 x 1/1

Step 3 : Multiply 

8 x 1  =  8    

4.     Cooper's bird feeder holds 9/10 of a cup of birdseed. Cooper is filling the bird feeder with a scoop that holds 3/10 of a cup. How many scoops of birdseed will Cooper put into the feeder?

Solution : 

Step 1 : 

To get answer for the above question, divide the total amount of birdseed by the size of each scoop.

That is, we have to find the value of (9/10) / (3/10).

Step 2 : 

Determine the sign of the quotient.

The quotient will be positive, because the signs of both numerator (9/10) and denominator (3/10) are same. 

Step 3 : 

Write the complex fraction as division :

(9/10) / (3/10)  =  (9/10) ÷ (3/10)

Step 4 : 

Rewrite the above division as multiplication by taking the reciprocal of the second fraction. 

(9/10) ÷ (3/10)  =  (9/10) x (10/3)

Step 5 : 

Simplify

(9/10) x (10/3)  =  (3/1) x (1/1)= 3

So, Cooper will put 3 scoops of birdseed into the feeder.

Note:  A gallon : It is a  unit of liquid or dry capacity equal to eight pints or 4. 55 litres.

 

 

 

Exploring properties associated with numbers including their geometric representations

 

Exploring properties associated with numbers including their geometric representations

Arithmetic  is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them—addition, subtraction, multiplication, division, exponentiation and extraction of roots.  Algebra is described as an area in mathematics that uses variables, in the forms of letters and symbols, to act as numbers or quantities in equations and formulas.  Geometry, on the other hand, is described as an area in mathematics that studies points, lines, varied-dimensional objects and shapes, surfaces, and solids.

Relationships Between Geometry, Arithmetic  and Algebra

As we said, algebra has to do with equations and formulas, and geometry has to do with objects and shapes, so how can these two things be related? Well, as one example, you are probably familiar with the fact that an equation can be graphed. For instance, the equation

y = x + 3 is the graph of a set of points that satisfy the equation, and it turns out to be a straight line.

 

We can take an equation, which is an algebraic concept, and graph it, making it a geometric concept.

In general, the equation of a line has the form y = mx + b, where m is equal to the slope of the line (where the slope is the change in y-values divided by the change in x-values from one point to the next on the line) and b is equal to the y-intercept of the line. We see that the variables within the equation (both algebraic concepts) can actually be used to refer to geometric concepts (slope and y-intercept) of the line.

 

Consider the shapes of a circle, rectangle, and square. These shapes fall under the category of a geometric concept. However, what about the areas of these shapes? To find the areas of each of these shapes, we use algebraic formulas and use the numbers to find the final value.

Pythagorean Theorem:

 It's a theorem that states that if a right triangle has legs of lengths a and b, and a hypotenuse of length c, then a² + b² = c².

The theorem itself shows a relationship between geometry and algebra by relating the lengths of the sides of a right triangle (a geometric concept) to an equation (an algebraic concept).

Relationships  between Geometry & Arithmetic:  

Geometry, an important branch of Mathematics, has a place in education for the development  of critical thinking and problem solving, furthermore, that geometrical shapes are parts of our  lives as they appear almost everywhere, geometry is utilized in science and art as well. Using the concept of  Geometry and their shapes we can teach Arithmetic in a  better way. Suppose we are going to teach addition and subtraction, we may use the line concept or shape of any objects. When we are going to teach Integers we may apply draw the line & explain it. If we are going to teach fraction we may use circle shapes. We divide circle in different part and give the concept of different types of fraction ,  If we are going to teach Arithmetic progression we may apply the rectangles shapes like stairs,  etc.

 

 

Concept of   Integers:

 

 

 

 

          

 

   Concept of   Tn terms of Arithmetic progression: