Category: Maths Notes PPT

The term ‘Mean’ is used constantly in the field of Statistics and is one of the basic methods used to obtain a result. It is also known as arithmetic mean or the average of a given set of data. It also measures the central tendency of data. The definition of a mean for a given set of data is the average calculated for a given set of numbers or data. This is referred to as the total of all the values of data provided divided by the number of data values in total for any given set of data.

The mathematical symbol or notation for the mean is ‘x-bar’. This symbol appears on scientific calculators and in mathematical and statistical notations.

The ‘mean’ or ‘arithmetic mean is the most commonly used form of average. To calculate the mean, you need a set of related numbers (or data set). At least two numbers are needed in order to calculate the mean.

The formula denoting the mean of a given set of data is as follows:

Mean = Sum of Observations/Total number of observations

The other two statistical methods used are median and mode to obtain a result for a given set of data. The median is defined as the value present in the middle of a given set of data and the mode is the frequency with which a particular number occurs in a given set of data.

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In order to find the mean of 4, 5, 6, 3, and 7, first, we have to add the numbers and then divide the sum by the number of items. 

4 + 5 + 6 + 3 + 7 = 25 i.e. the sum of the numbers is 25.

Mean =   =   =   = 5

So, the mean of the data set 4, 5, 6, 3, and 7 is 5.

How to Find Mean?

The mean value for a given set of data is calculated in a two-step process:

1. The values given in the data set are added up together.

2. The total of the values obtained is then divided by the number of values given.

Mean Formula

The measure of central tendencies is used to describe data clusters around a central value. The mean definition indicates a varied formula used to calculate the mean depending on the data provided. The general formula to calculate the mean is as follows:

[Mean = frac{text{Sum of Given Data}}{text{Total Number of Data}}]

When using the Sigma (∑) notation, the mean formula is:

[frac{sum_{i=1}^{n} X_{i}}{N}]

Here,

N = it is the Total number provided in a given data set.

∑ Xi = Total sum of all the data values.

Mean of Negative Numbers

We have seen examples of finding the mean of positive numbers till now. But what if the numbers in the observation list include negative numbers. Let us understand with an instance,

Example: Find the mean of 9, 6, -3, 2, -7, 1.

Add all the numbers first:

Total: 9+6+(-3)+2+(-7)+1 = 9+6-3+2-7+1 = 8

Now divide the total from 6, to get the mean.

Mean = 8/6 = 1.33

Mean Formula with Example

Find the mean for the given set of random data,

3, 5, 9, 17, 19

• The given set of data contains the numbers 3, 5, 9, 17, 19

• The total number of numerals given is 5

• Sum of the given numbers in the data set = 3 + 5 + 9 + 17 + 19 = 53

Therefore, Mean = Sum of given data/Total number of data

=[ frac{53}{5}] =10.6

Hence, the mean for the given data is 10.6.

Different Types of Mean

A. Arithmetic Mean

The arithmetic mean is one of the foremost methods used to obtain the central tendency of a set of data. It encompasses all the values provided by the data set. It is referred to as the ratio of the total sum of given observations to the total number of observations. The arithmetic mean can be positive, negative, or zero. There are two types of Arithmetic Mean,

The formula to calculate Arithmetic mean is as follows:

[X = frac{sum_{i=1}^{n} x_{i}p_{i}}{N}]

The arithmetic mean is easy to calculate and is rigidly defined.

B. Geometric Mean

The second type of Mean is the Geometric Mean (GM). It is defined as the average value signifying the set of numbers of central tendencies by calculating the product of their values. Multiplication of the numbers provided and take out the nth root of the multiplied numbers.

Here, n is the total number of values.

Taking an example of two numbers in a given set of data as 4 and 2, the geometric mean is equal to. [sqrt{(4+2)} = sqrt{6} = 2.5]

The difference between the arithmetic mean and the geometric mean is the method. In the arithmetic mean, we add the numbers whereas in the geometric mean we calculate the product of the numbers.

[text{Geometric Mean = } sqrt[n]{prod_{i=1}^{n} x_{i}}]

C. Harmonic Mean

This is one of the methods of central tendency used in Statistics. It is the reciprocal of the arithmetic mean for a given set of data. The Harmonic Mean is based on all values from the data set and it is defined rigidly. It also provides the weightage of the mean in terms of large or small values depending on the data set. This is applied in time and average analysis.

To calculate the harmonic mean for a given set of data, where  x1, x2, x3,…, xn are the individual items up to n terms, then,

[text{Harmonic Mean = } frac{n}{[(frac{1}{x_{1}}) + (frac{1}{x_{2}}) + (frac{1}{x_{3}}) + . . . + (frac{1}{x_{n}})]}]

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Properties of Mean

• The sum of the deviations taken from the arithmetic mean is zero.
  If the mean of n observations x1, x2, x3….,xn is x then (x1-x)+(x2-x)+(x3-x)…+(xn-x)=0. In short, ∑ (x-x)=0

• If each observation is increased by p, the mean of new observations is also increased by p.
  If the mean of n observations x1, x2, x3….,xn is x then the mean of (x1+p), (x2+p), (x3+p),….,(xn+p) is (x+p).

• If each observation is decreased by p, the mean of new observations is also decreased by p.
  If the mean of n observations x1, x2, x3….,xnis x then the mean of (x1-p), (x2-p), (x3-p),….,(xn-p) is (x-p).

• If each observation is multiplied by p (where p≠0), the mean of new observations is also multiplied by p.
  If the mean of n observations x1, x2, x3….,xn is x then the mean of px1, px2,px3,pxn is px.

• If each observation is divided by p (where p≠0), the mean of new observations is also divided by p.
  If the mean of n observations x1, x2, x3….,xn is x then the mean of

[ frac{X_{1}}{p} ,frac{X_{2}}{p} ,frac{X_{3}}{p} ,…frac{X_{n}}{p} is frac{bar{X}}{p}]

Why is the Average Called the Mean?

To find the mean, add all the data points and divide it by the total number of data points. In the case of Mathematics, we have been always taught that the average is the middle point of all the given numbers. The central value which is called the average in mathematics is called the mean in statistics.

Important points:

• The mean is the mathematical average of a set of two or more numbers.

• The arithmetic mean and the geometric mean are two types of mean that can be calculated.

• Summing the numbers in a set and dividing by the total number gives you the arithmetic mean.

• The geometric mean is more complicated and involves the multiplication of the numbers taking the nth root.

• The mean helps to assess the performance of an investment or company over a period of time, and many other uses.

Metric Speed

How do you find how fast an object is moving? Measured as distance travelled per unit of time, metric speed is the speed in meters per second (m/s). Thus, the SI derived unit for speed is meter per second. That said, a metric speed is described as the rate at which an object is moving (covering a specific distance). It is referred to as a scalar quantity as it only describes the magnitude and not direction. Do not confuse speed with velocity.

Formula of Speed

Speed = Distance traveled / Time taken

Introduction To Velocity

Velocity is described as the rate of change of an object’s position in reference to a frame of time. Velocity falls under the category of a vector quantity as it defines both the magnitude as well as direction. The international system of unit (SI) derived unit for velocity is also meter per second (m/s) alike speed.

Formula of Velocity

Velocity= Change in position / Change in time

Introduction To Acceleration

Acceleration too falls under the classification of a vector quantity as it is described as the rate of change of velocity with reference to change in time.

The SI derived unit for acceleration is meter per square seconds (m/s2).

 

What is Speed in Meters Per Second (m/s)?

Speed in meters per second (m/s)

If an object is travelling at a speed of 1 m/s, it moves 1 meter every second.

1 m/s is like a very normal walking speed.

One hour of gentle walking at 1 m/s moves you about 3.6 km.

 

What Is Speed In Kilometers Per Hour (km/h)?

If an object is travelling at a metric speed of 2 km/h, it moves 2 kilometres every hour.

It is quite a slow walking speed. A kilometre per hour (km/h) is often used to express speed for a car.

Example: Highway speed of a car is around 150 km/h

One hour at this speed moves you 150 km.

Metric Speed Conversion

In this section, you will learn how to perform metric speed conversion using various formulae for the conversion. These formulae for the conversion consider values between different unit representations for speed/velocity.

Usually, the technique used to arrive at the formula is dependent upon the individual units inserted in the numerator and the denominator.

Unit Conversion For Meter/Second To kilometer/Hour (m/s to km/h)

M/s = ÷1,000km / ÷ 3,600 hour

M/s → × 3.6 → km/h

Unit Conversion For Kilometer/Hour To Meter/Second (km/h to m/s)

Km/hr = × 1000km / × 3600 hour

km/hr → × 0.28 → m/s

Unit Conversion For Kilometer/Second To Miles/Hour (km/s to mi/h)

Km/s = × 0.62137 mi / × 3600 hour

Km/s → × 2236.9 → mi/hr

Unit Conversion For Feet/Second To Meter/Second (ft/s to m/s)

Ft/s = × .3048 m/ × 1 s

Ft/s → × .3048 → m/s

 

Unit Conversion For Miles/Hour To Meter/Second (mi/h to m/s)

Mi/hr = × 1609.34 m / × 3600 hour

Mi/hr → × 0.447→ m/s

Unit Conversion For Meter/Second To Feet/Second (m/s to ft/s)

m/s = × 3.28084 ft / × 1 s

m/s → × 3.28→ ft/s

Unit Conversion For Kilometer/Second To Meter/Second (km/s to m/s)

km/s = ÷1000m / × 1 s

km/s → × 1000→ m/s

Unit Conversion For Feet/Minute To Meter/Second (ft/min to m/s)

ft/min = ×.3048m / × 60s

ft/min → ×.00508→ m/s

Unit Conversion For Miles/Hour To Feet/Second (mi/h to ft/s)

mi/hr = × 5280ft / × 3600s

mi/hr → ×1.47→ ft/s

Unit Conversion For Centimeter/Second To Meter/Second (cm/s to m/s)

cm/s = ÷100m / × 1s

cm/s → ÷100m → m/s

Unit Conversion For Rotations/Minute To Meter/Second

Rotation/min = × 2 Π× 2 r m / × 60 s

rpm → × 2 (Π× r/30) → m/s

Here,

r = radius

2 × π × r = Linear Velocity

Unit Conversion For Radians/Second To Meter/Second (rad/s to m/s)

Radian/sec = × r m / × 1 s

rad/s → × r → m/s

Unit Conversion For Meter/Second To Mach (m/s to Mach)

Mach refers to the ratio of the speed of a moving object through a fluid to the speed of sound via the same medium. because it is a ratio, it does not contain any dimension. The speed of the sound does not remain constant. It differs depending upon the temperature and atmospheric pressure.

m/s → × .0029104→ Mach

Solved Examples

Example: Convert a speed of a moving object 70 meters per second to kilometer per hour

Solution: For metric speed conversion of m/s to km/hr, we need to multiply it by 3.6

Thus, 70 m/s = 70 × 3.6 = 252 km/hr

Example: Convert a speed of of a car 20 feet per second (ft/s) to meters per second (m/s)

Solution:

Converting value from ft/s to m/s, we would require multiplying it by 0.3048

Thus, 20 ft/s = 20 × 0.3048 = 6.096 m/s

Conclusion: Metric speed helped you learn about the Speed,Time, Velocity and Acceleration,  their definitions, units and the metric conversion, rules/formulae between different units. Thus, this may serve as a quick guide for any of the aforementioned concepts. 

A monomial in Maths is a type of polynomial that has only one single term. For example, 4p + 5p + 9p is a monomial because when we add the like terms it will obtain the result as 19p. Furthermore, 4x, 21x²y, 9xy, etc are monomials because each of these expressions includes only one single term. As we know, polynomials are the algebraic expressions or equations which include variables and coefficients and have one or more than one term.Each term of the polynomial is a monomial. A polynomial includes the operations of addition, subtraction, multiplication and non-negative integer exponents of variables. In this article, we will study polynomial, monomial definition, monomial problems monomial examples, degree of monomial etc.

Polynomial

Polynomials are expressions that include exponents that are added, subtracted, multiplied, or divided. There are different types of polynomial such as monomial, binomial, trinomial, and zero polynomial. A monomial is a type of polynomial with one single term. A binomial is a polynomial with a maximum of two unlike terms. A trinomial is a polynomial with a maximum of three unlike terms.

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Monomial Definition

A monomial is a special kind of polynomial, which is an algebraic-expression having only one single term which is non-zero. It includes only one single term which simplifies the operation of addition, subtraction or multiplication. It includes only a single variable or coefficient or product of variable or coefficient with exponents as a whole number, which denotes only one single term unlike binomial or trinomial which includes two or three terms. Monomials in Maths do not have a variable in the denominator.

For example, 3x is a monomial, as it represents only one single term. Similarly, 23, 3x², 7xy, etc.are examples of monomials but 13 + x, 3xy, 4xy -1 are not considered as monomials because  they don’t satisfy conditions.

Monomial is a product of powers with non-negative integers. For example, if there is a one- variable p, then it will have a power either 1 or power of pⁿ with n as any positive integer and the product of the multiple variables such asPQR, then the monomial will be represented in the form of paqbrc.  Here a, b and are non-negative integers. Monomial in terms of the coefficient is defined as the term with a non-zero coefficient. 

Binomial

A binomial is an algebraic expression or a polynomial which has a maximum two, unlike terms. For example, 2x + 5x² is a binomial as it has two unlike terms that is 2x and 5x².

Trinomial

A trinomial is an algebraic expression or a polynomial that  has a maximum of, three unlike terms. For example, 2x + 5x² + 7x³ is a trinomial as it has three unlike terms that is 2x ,5x² and  7x³.

Finding the Degree of a Monomial

The degree of a monomial is the addition of the exponent of all the included variables which together forms a monomial. For example, pqr³ have 4 degrees 1,1,and 3. Therefore, the degree of pqr³ is 1 + 1 + 3 = 5.

Monomials Examples

• x – Here, the variable is one i.e x  and degree is also one.

• 6x² – Here, the coefficient is 6 and the degree is 2

• x³y – Here, x and y are two variables and the degree is 4(3+1).

• -6xy – Here, x and y are two variables and a coefficient is – 6.

Monomial Problem

1. Identify which one of the given below is a monomial.

1. 4xy

2. 4y + z

3. 3x² + 3y

4. x + y + z²

Solution: 4xy is a monomial whereas 4y + z , 3x² + 3y are binomials and x + y + z² is a trinomial.

Solved Examples –

1. (x³y) (x²y³) 

 Solution: (x³x²) (yy³)  

= x3 + 2 y1 + 3

= x5y4   

2. Solve the monomial expression 16q + 7q – 2q-(-4q)

Solution:  23q – 2q + 4q

= 21q + 4q

= 26q

3. (4x² + 3x -14 )+ (x³ – x² + 7x + 1)

Solution: (4x² + 3x -14 )+ (x³ – x² + 7x + 1)

Grouping the like terms, we get

x³ + (4x² – x²) + ( 3x+ 7x) + (-14 +1)

Simplifying the above expression we get,

= x³ + 3x² + 10x – 13

Quiz Time

1. In the expression -3x⁴, the coefficient is

1. x

2. 4

3. -3

2. Which of the following has a similar value as ( t t t t t t) (zzzz)

1. (t²z)⁴

2. (6t) (4z)

3. (t³z²)²

Do you know an easy trick to solve multiplication story sums? Let us assume there are 5 students in Class 4 in each section- A, B, C, and D. If you have to find the total number of students in all four sections, you can just do 5 times 4, i.e., 5 * 4 = 20 students in total.

However, another way to find the total number of students is, just add 5 + 5 + 5 + 5, you get 20 again. Awesome isn’t it? Well, here, we notice that multiply sums can be done by performing the addition of numbers. Multiplication is one of the interesting concepts of maths. Here, you will learn interesting tricks on solving multiplication problems. Let’s go through Saumya’s story and understand how multiplication works.

Story of Water Supply in ABC Society

During summers, water supply is a major problem in ABC society. The water supply starts at 8: 00 a.m. and ends at 9: 00 p.m., which means 13 hrs supply, the same is continued each day. Now, if I say how many hours did the water supply was there in a week? How can you determine the total number of hours in a week?

Well, a simple trick is to write 13 * 7, you get 91 hours. However, there is another approach to to make easy multiplication sums, which is as follows:

• Till First day = 13hrs of water supply

• Till Second day = 13 + 13 = 26 hrs

• Till Third day = 13 + 13 + 13 = 26 + 13 = 39 hrs

• Till Fourth day = 13 + 13 + 13 + 13 = 39 + 13 = 52 hrs

• Till Fifth day = Water supply till fourth day + 13 = 52 + 13 = 65 hrs

• Till Sixth day = Water supply till fifth day + 13 = 65 + 13 = 78 hrs

• The Seventh day = Water supply till sixth day + 13 = 91 hrs

So, this is how multiply sums can simplify the long additions.

Now, let us have a look at one of the interesting multiplication sums with a story.

Story 2 on Lily’s Flower Shop

Let us go to Lily’s flower shop to understand multiplication word problems. 

A few days ago, my friends went to purchase a flower bouquet to gift Neha on her birthday. On the way, they found 13,000 roses packed in a basket. Now, they wanted to see how many flowers are there in 65 such baskets? Well, for that, they needed some approach, but what is that approach? Voila! That is multiplication. Now, let us understand it.

What is 13, 000 …(a)

What is 65?….(b)

Firstly, we will write the statement for solving all multiplication questions. So, answers to the above questions (a) and (b) will help us in framing the statements for the solutions.

Statements:

No. of roses in a basket =  13, 000

No. of baskets = 65

Now, let us see how to solve multiplication in this case.

13, 000 roses—— are there in one basket………?  

Number of roses in 65 such baskets

Here, we multiply 13, 000 with 65 to get the number of roses needed to pack similar 65 baskets:

   1 3 0 0 0

x           6 5

_________

8 4 5 0 0 0

_________

Thus, there are 8,45,000 roses in total in 65 such baskets.

Math Sums Multiplication: Story-telling Method

Now, let us have a look at one of the interesting multiplication word problems.

Question: 

Yesterday, Monika went to purchase one competitive book from a whole-seller book shop, and that cost around 67 rupees. Before purchasing from this shop, she went to many shops to get to know the price. Finally, she got it at the lowest price.

The next day, she goes to school and tells her classmates that she got the book at the lowest price.

On Saturday, students from the same and the neighbouring schools went on to purchase the book from the same shop and around 101 books were sold on the same day at the same price, as the girl bought it for. Overall, how much did the shopkeeper earn after selling 102 books till Saturday? 

Well, these types of questions are easy multiplication sums. Now, let us solve this sum.

Solution:

The cost of 1 book  =  67 rupees

Total number of books sold till Saturday = 102

The cost of 102 books  = 102 * 67 rupees

Therefore, in solving this problem, we find that the shopkeeper earned 6834 rupees. 

Now, let us look at some multiplication puzzles.

Interesting Puzzle on Multiplication Sums

We need to put numbers from 1 to 9 to complete the multiplication puzzle without repeating any numbers. 

Firstly, let us look at the bottom values:

Column 1: Divisors of 96  =  1, 3, 4, 6, 8, 12, 16, 24, 96 

Column 2: Divisors of 180 = 1, 2, 3, 4, 5, 6, 9, 12, 15, 18, 20, 45, 90, 180

Column 3: Divisors of 21 = 1, 3, 7, 21

We find three blocks corresponding to the following.

Row 1:  54 

Row 2: 120 

Row 3: 56 

Here, the number we fill in each column, we get the answer below, which is 96, 180, and 21 respectively.

Step 1: Since 21 is 7 * 3 and we don’t find a multiple of 7 in rows 1 and 2, i.e., 54 and 120, we mark ‘7’ in row 3.

However, ‘3’ is a divisor of both 54 and 120. For this, we will apply our logic in step 2.

Step 2: In Row 3, we see that 7 * 8 gives 56. Divisors of 8 are 1, 2, 4, and 8. Also, we have put such a number in the middle square of all rows such that the product is 180. Let’s say, we put ‘2’in the first square of Row 3, ‘4’in the middle, and 7 was already there, so we have:

So, we have 2 * 4 * 7 = 56. Now, let us look at the above two rows.

Step 3: For filling the middle column, we have to obtain 180 as our result, which is shown below:

Step 4: Now, in row 1, we put the following values, to obtain the row-wise result as 54.

Since no number must be repeated, we have to look for another arrangement to make it to 54, which is as follows.

Step 5: Now, we are left the middle row to get our result as 120. For this, we are left with only two numbers, i.e., 8 and 3, so let us do that as well.

Hence, our multiplication puzzle is complete. 

So, this was all about the sums on multiplication for Class 4 students. Having command over this concept will help you solve a variety of questions in the simplest tricks and in lesser time.

Definition of Whole Number and Natural Number:

Numbers are the symbols used to represent the amount of things in Mathematical calculations. There are several types of numbers used in the Number system of Mathematics. Numbers play a very vital role in almost all the Mathematical calculations. There are several types of numbers which include natural numbers, whole numbers, prime numbers, odd numbers, even numbers, composite numbers, rational numbers, irrational numbers, decimal numbers, fractions, real numbers, imaginary numbers, etc.

What are Natural Numbers?

The numbers which are used to count are called the natural numbers. Generally counting starts with 1 and increments. Counting can never be a negative number. So natural numbers are positive numbers starting from 1. Natural numbers are denoted as ‘N”. The natural numbers are 1, 2, 3…….

Numbers which can be regarded as natural numbers include:

The following types of numbers do not have properties of whole number and natural number.

• Negative integers

• Fractions and decimals

• Irrationational numbers

• Imaginary numbers

What are Whole Numbers?

All natural numbers along with zero are categorized as whole numbers. Whole numbers are denoted by the symbol ‘W’. Zero is not categorized as a natural number because zero is not used in counting. Both natural numbers and whole numbers are rational numbers because they can be expressed in the form of a fraction in which denominators are not equal to zero. 

Properties of Whole Number and Natural Number:

Property 1: Closure Property:

Natural numbers are closed under addition and multiplication. However, they are not closed under subtraction and division. 

• When two natural numbers are added, the result obtained as a sum is also a natural number.

• Product of 2 natural numbers is a natural number.

• Difference between two natural numbers need not be a natural number. 

Ex: 2 – 5 = -3 which is not a natural number.

Ex: If 5 is divided by 2, the answer is 2.5 which is not a natural number.

Property 2: Commutative Property

Natural numbers satisfy commutative law for addition and multiplication. However, they are not commutative for subtraction and division.

i.e. If m and n are two natural numbers, then

• m + n = n + m

• m x n = n x m

• m – n ≠ n – m

• m ÷ n ≠ n ÷ m

Property 3: Associative Property

Addition and multiplication of natural numbers are associative. But, subtraction and division of natural numbers are not associative. If E, F and G are 3 natural numbers, then

• (E + F) + G = E + (F + G)

• (E x F) x G = E x (F x G)

• (E – F) – G ≠ E – (F – G)

• (E ÷ F) ÷ G ≠ E ÷ (F ÷ G)

Property 4: Distributive property

Natural numbers obey distribution of multiplication over addition. If k, l and m are two natural numbers, then

k (l + m) = (k x l) + (k x m)

Property 5: Identity Property

The additive identity of a number is that number which when added to a number gives the same number as the sum. Additive identity of any natural number is 0. i.e. If zero is added to any natural number, the sum is the natural number itself.

If ‘a’ is any natural number, then a + 0 = a

Multiplicative identity of a number is that number which when multiplied by the given number gives the given number as the product. Multiplicative identity of all natural numbers is 1. i.e. when any natural number is multiplied by 1, the product is the number itself.

If ‘a’ is a natural number, then a x 1 = a

Property 6: Inverse Property

Additive inverse of any natural number is that number which when added to a number yields the sum of additive identity. Additive inverse of any natural number is the negative of that number. 

If ‘a’ is a natural number, then its additive inverse is -a.

Multiplicative inverse of a natural number is that number which when multiplied by the given number yields the product equal to multiplicative identity. Multiplicative inverse of any natural number is the reciprocal of the number.

If ‘a’ is a natural number, then its multiplicative inverse is 1/a.

All the above-mentioned statements are true for both the properties of the whole number and natural number. 

Fun Facts:

• All the natural numbers are whole numbers. But all whole numbers are natural numbers except ‘0’. i.e. all the numbers except zero are natural numbers and whole numbers examples.

• All the natural numbers and whole numbers examples are real numbers. However, all real numbers are not natural numbers and whole numbers examples. 

• The product of any number multiplied by 0 is 0. The quotient of 0 divided by any number is 0 and that of any number divided by 0 is undefined.

What is a Number System?

In the simplest terms, a number system can be described as a system representing numbers. There are different types of number systems in mathematics, including binary number systems and decimal number systems.

In this article, readers will be able to understand the concept of the number system. They will also be able to learn the method of solving number system conversion questions. There is a number conversion table for a better understanding of these concepts.

As mentioned above, a number system can be explained as a system of writing that expresses numbers. It can also be defined as the mathematical notation for representing various numbers of a given set. This is done by using digits or other symbols in a very consistent manner. Students should use this knowledge to work on number system conversion with solutions.

It should be noted that number system conversion online also provides a very unique representation of every single number. It also represented the arithmetic and algebraic structures of the figures. It also allows an individual to operate different arithmetic operations like subtraction, addition, and division.

If a student has to solve questions related to converting any base to decimal, then he or she should know that the value of any digit in a number can also be determined by:

Types of Number Systems

There are many different types of number systems. In this section, readers will learn about those different types of number systems. This discussion is carried out below.

The decimal number system is characterized by having a base of 10. This is because in this number system ten digits are used from 0 to 9. Further, the positions successive to the left of the decimal point represent units, tens, hundreds, thousands, and other numbers. This system can be described in terms of decimal numbers.

Every position in a decimal number system is used for highlighting a particular power of the base (1). For example, the decimal number 1457 consists of the digit 7 in the units place, 5 in the tens place, 4 in the hundreds place, and 1 in the thousands place. This value can also be written as:

(1 x 103) + (4 x 102) + (5 x 101) + (7 x 100)

(1 x 1000) + (4 x 100) + (5 x 10) + (7 x 1)

1000 + 400 + 50 + 7

= 1457

The binary number system or the base 2 number system has only two binary digits, including 0 and 1. It should be noted that the usual base 2 is a radix of 2. The digits that are described under the binary number system are simply known as binary numbers, which are combinations of 0 and 1. For example, 101001110 is a binary number. It is possible to convert any number system into a binary system. This is also true vice versa.

In the base 8 number system or the octal number system, the numbers from 0 to 7 are used for representing numbers. Students must remember that octal numbers are usually used in computer applications. Also, converting an octal number to decimal is similar to the conversion of the decimal number system. To understand this in a clearer manner, let’s look at the example that is given below.

Convert 2158 into decimal.

2158 = 2 x 82 + 1 x 81 + 5 x 80

= 2 x 64 + 1 x 8 + 5 x 1

= 128 + 8 + 5

= 14110

As the name indicates, the hexadecimal number system is represented or written with a base of 16. In this number system, the numbers are usually represented just like a decimal system, which is from 0 to 9. After that, the numbers are represented by using alphabets from A to F.

To summarize all of this, students should go through a number system chart. Luckily, we have compiled a similar chart and we are attaching that chart below.

Number System Conversion

As it was mentioned before, any number that is represented in a number system type can be converted into a number of other systems. There are several methods that can be used for solving these types of questions. Apart from that, students can also refer to a number system conversion table.

Using a number system conversion table can be very useful. Keeping this in mind, we have mentioned a similar table below. Students should go through this table to solve questions related to number system conversions.

Now, let’s look at the other methods of converting numbers into other systems. This is done on the basis of the base of the numbers. For example, if an individual wants to change a decimal number with a base 10 to a binary number with base 2, then we must start with the basic representation of the number system base conversion. The general form of any base number is:

(Number)_b = d_{n – 1}, d_{n – 2}, –, d₁ d₀. D_-1 d_-2 —- d_-m

In this expression, d_{n -1} d_{n -2} — d₁ d₀ represents the values of the integer part, and d_-1 d_-2 — d_m represents the fractional part of the equation.

Also, d_n-1 is also known as the Most Significant Bit (MSB) and dm is the Least Significant Bit (LSB). Let’s move on to specific cases of number system conversion. These specific cases are discussed in a list below.

For converting a decimal number to some other base number, we have to divide the decimal number after converting the value of the new base. This might seem complex but is quite easy. Let’s suppose that if we have to convert decimal to binary number system, then we need to divide the decimal number by 2. For example, let’s try to convert (25)₁₀ to binary numbers.

To get a better handle on the answer, we will look at the answer in the form of a table. This table is mentioned below.

Hence, from this table, we can conclude that

(25)₁₀ = (11001)₂

If you want to convert decimal to octal number, then we need to start by dividing the given original number by 8. This should be done in a manner so that the base 10 changes to a base 8. For example, if we have to convert (128)₁₀ to an octal number, then, we can arrive at the answer with the help of the table that is mentioned below.

Hence, it can be said that the equivalent octal number is (200)₈.

When it comes to the conversion of decimal to hexadecimal conversion, we need to divide the given decimal number by the number 16. For example, if we have to convert (128)₁₀ to a hexadecimal number, then, the answer can be arrived at with the help of the table that is mentioned below.

Hence, the equivalent hexadecimal number is (80)₁₆.

Did You Know?

• The Hindu-Arabic numeral system is often called the decimal number system.

• There are almost 200 different number systems like mix radic, non-integer bases, negative bases, etc.

• There is only one even prime number.

• Humans have been using the number system since 35,000 BCE.