Category: Maths Notes PPT

Before proving [sqrt{2}] as irrational, first, let us understand what an Irrational number.

What is an Irrational Number?

For any integers, an irrational number is a number that can not be represented as a fraction, and irrational numbers have decimal expansions that do not terminate.

The best example of an irrational number is Pi (𝝅) which is has a non-terminating number 3.14159265359.

Here we have to prove the irrationality of [sqrt{2}]. This proof is a classic example of Proof by Contradiction. 

In proof by contradiction, at the start of the proof, the opposite is believed to be valid. The assumption is shown not to be valid after rational reasoning at every stage. This is also known as indirect proof and proof by assuming the opposite.

Euclid developed this proof by contradiction and applied for [sqrt{2}] to prove as an irrational number

Euclid Square Root 2 Irrational Proof

According to proof by contradiction given by Euclid, the first step of the proof, we will assume the opposite is true. In the same way here will we assume that [sqrt{2}] is equal to some rational number a/b.

[sqrt{2}] = [frac{a}{b}]……………………(1)

Now, we will square on both sides of equation (1),

([sqrt{2}])[^{2}] = ([frac{a}{b}])[^{2}]…………………(2)

We will simplify and rewrite the equation (2) as 

2b2 = a2……………………….(3)

If we observe, here the value of a2 will be positive because b2 is multiplied by an even number ‘2’. Since a2 is positive, we can conclude that a is also positive. Since ‘a’ is positive, we can write a=2c where c is any whole number. Since ‘a’ is even number 2 multiplied by any whole number will satisfy the definition of even number.

Now let substitute a=2c in equation (3),

2b2 = (2c)2

2b2 = 4c2………………..(4)

Now divide by 2 into both sides of equation (4), we get

b2 = 2c2……………………(5)

Here b2 is multiplied by 2 and c2 which satisfies the definition of even number. Therefore, b2 is also an even number which concludes that ‘b’ is an even number. 

So, we have proved ‘a’ and ‘b’ even numbers.

In the next step, let us assume that b=2d in the same of assuming a=2c which satisfies the even number definition.

Now let us substitute a=2c and b=2d in equation (1) where we have assumed 

[sqrt{2}] = [frac{a}{b}]

[sqrt{2}] = [frac{2c}{2d}]

[sqrt{2}] = [frac{c}{d}] ………………(6)

Now we have obtained c/d which is in simpler form compared to p/q. Also from equations (1) and (6) 

a/b = c/d ……………….(7)

Here we can further simplify c/d into say e/f by carrying out the same process. Again e/f will be put through the same process and we obtain g/h is simpler. 

Rational number definition states that “a number cannot be simplified indefinitely it has to terminate at some point. 

So, the basic assumption that [sqrt{2}] is a rational number will fail here. So the answer contradicts the basic assumption that [sqrt{2}] as a rational number is unreasonable.

So, we can conclude that the contradiction has been reached that [sqrt{2}] is not a rational number. 

Hence we have proved that [sqrt{2}] is irrational.

Exponents and Powers

Exponents and Powers are mathematical operations used to represent and represent large sums of numbers or minimal numbers in a simplified manner. Students know the basics of how to calculate an expression 5 x 5, but, this expression can be simplified shortly and crisply using a concept called the exponents. An expression that represents the repeated multiplication of the same factor is known as the power.

For example, the sum 3 x 3 x 3 x 3 is an equation, but a simple way to write it is 34, where 4 is the exponent power and 3 is the base number. The full operational expression 34 is said to be the power. 

Power is an expression that presents the repeated multiplication of the same factor or number. The exponent value is based on the repeated number of times that the base is multiplied to itself.

Exponent Formula

From the expression a2, ‘a’ is known as the base number, and the two is referred to as the exponent. The exponent number represents the number of times the base number is to be multiplied. 

a[^{m}] . a[^{n}] = a[^{m+n}]

(a[^{m}])[^{n}] = a[^{mn}] 

For example, in a2, the base number ‘a’ will be multiplied twice, that is, a x a and similarly, a3 = a × a × a. Here are a few of the important exponent form-

• a0 =1

• a1= a

• √a=a1/2

• n√a=a1/n

• a−n  = 1/an

• an = 1/a−n

• aman=am+n

• am/an=am−n

• (am)p=amp

• (amcn)p=ampcnp

• (am/cn)p=amp/cnp

Laws of Exponents

There are three basic exponent laws that every student needs to comprehend. Each rule of exponents help students solve different types of mathematical equations and teach them basic concepts like addition, subtraction, multiplying, and even division exponents.

Law #1: a[^{n}] x a[^{m}] = a[^{n+m}] 

Law #2: [frac{a^{m}}{a^{n}}] = a[^{m-n}]

Law #3: (a[^{n}])[^{m}] = a[^{n times m}] 

The laws of exponents are demonstrated based on the powers each expression carries.

1. Division Law

The division law is applicable when two exponents have the same base numbers but different powers. The expression is divided, thus, resulting in the base number raised to the difference between the two power numbers.

An example of dividing exponents: am ÷ an  = am / an = am – n

2. Multiplication Law

The Multiplication law is applicable when the product of two exponents comprise the same base numbers but comprise different powers. The result of the multiplying exponents equals the base raised to the sum of the two integers or powers.

An example of multiplying exponents: am × an = am + n

3. Negative Exponent Law

The Negative Exponent Law is applicable when any base numbers comprise a negative power. This expression results in the reciprocal but with the positive integer or positive to the base number. 

An example of negative exponent law: a – m  = 1/am

Exponent Rules

The exponent laws follow the exponent rules. There are four basic exponent rules to follow-

Let’s consider the following- suppose the integer values are ‘a’ and ‘b’ and the power values are ‘m’ and ‘n’, then the rules of exponent and power are as follow-

1. (ax)y = a(xy)

The expression results as- ‘a’ raised to the power of ‘x’ presented to the power ‘y’ outputs to ‘a’ submitted to the power of the product numbers of ‘x’ and ‘y’.

The example of this expression is (52)3 = 52 x 3

2. ax/bx = (a/b)x

The expression results as- the division of the ‘a’ raised to the power of ‘x’ and ‘b’ presented to the power of ‘x’ outputs to the division of ‘a’ and ‘b’, the whole raised to the power ‘x’.

The example of this expression is  52/62 = (5/6)2

3. a0 = 1

As per this exponent rule, if the power of any integer is zero, the output of the expression leads to one or unity. 

The example of this expression is 50 = 1

4. ax × bx =(ab)x

The expression results as- the product of the expression ‘a’ raised to the power ‘x’ and ‘b’ presented to the power ‘x’ outputs to the product of ‘a’ and ‘b’ the whole raised to the power ‘x’.

Example is 52 × 62 =(5 x 6)2

The two important concepts of mathematics are factors and multiples. These concepts are typically studied together as they are related to each other. The number that divides the other number exactly is called a factor. When you want to obtain a specific number you multiply a number with another number, this is called multiples of that number.

Factors Definition 

The number that divides the other number exactly leaves the remainder as zero, it is called the factors of numbers. This can alternatively be explained as when the dividend is exactly divisible by the divisor leaving a zero remainder, then the divisor is the factor of the dividend. The common factor of every number is the number itself and the number 1. In the given image, we can observe that the number 20 is the multiple of numbers 4 and 5. Alternatively, the numbers 4 and 5 are the factors of the number 20.

When any natural number has only two factors (which are the number itself and the number 1), then such a number is termed as a prime number. The examples of factors representing prime numbers can be numbers 2, 5, 7, and so on.

What is Multiple?

A multiple can be defined as a product of the given number and any other natural number. To understand and observe the multiples, study the multiplication table.

Some of the basic observations for the multiples of numbers can be mentioned below:

• Multiples of the number 2 will always be even numbers and end with 8, 6, 4, 2, and 0.

• The multiples of the number 5 will always end with either 0 or 5.

If we represent the numbers in alphabets then where A and B are two numbers, we can conclude that:

• If number A divides number B, then A is the factor of B

• If number B is divisible by number A, then B is the multiple of A

Difference Between Factors and Multiples

To explain what is a factor and what is multiple, it is essential to understand the differences between both concepts. The differences between factors and multiples are tabulated below:

How to Find Factors of a Number?

To find the factors of a number we have to find all the finite numbers that divide the given number in a way that no remainder is left after the operation of division. For instance, if we take an example of the number 28, then all the numbers that exactly divide the number 28 are 28, 14, 7, 4, 2, and 1. So all these numbers are the factors of number 28.

It should be noted however that every number has two essential factors namely the number itself and the number 1.

How to Find Multiples of a Number? 

Pick a number for which you want to find a multiple and then multiply it by continuing whole numbers. For example, let’s say we wish to find the multiples of 4.

Start by: 

4 x 1 = 4

4 x 2 = 8

4 x 3 = 12, and continue multiplying other whole numbers. 

Now, determine if the particular number, which in this case is 4, can be divided without a remainder. A multiple of any number should be able to be divided by the initial number. For example, 8 is a multiple of 4, and 4 x 2 = 8. Therefore, 8 divided by 4 will give us 2 as the answer. In this example, both 4 and 2 are considered to be factors of 8 and there is no remainder left either. 

What Does the Least Common Multiple Mean? 

When a number tends to be a multiple of 2 or more numbers, then it is referred to as a common multiple of the said numbers. 

Let’s take an example:

We know that 4 x 2 = 8. So, 8 becomes a multiple of both 4 and 2, which is why it is considered to be a common multiple of both those numbers. 

The smallest, other than zero, which happens to be the multiple of 2 or more numbers, is considered to be their Least Common Multiple or their LCM. In simpler words, the smallest number that tends to be divisible by all the given is the least common multiple of two or more numbers. 

For example, let’s take two numbers 6 and 8. 

Now, the multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, and so on. And the multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, and so on. From this, we can infer that the common multiples of 6 and 8 are 24, 48, etc… And out of these, the least common multiple of 8 and 6 is 24, therefore it is the LCM of 6 and 8. 

Methods to Find LCM of Two Numbers

Two primary methods can be used to find the least common multiple of two or more numbers namely, the LCM method or the Common Division Method. If you want to find LCM by using the prime factorisation method, then each number is to be expanded as a product of its prime factors. Then, using the highest power in the prime factorisation of each number, you find the product of all the different prime factors. And if you want to find LCM with the help of the common division method, then the numbers that are divisible by the number chosen in the previous step are to be divided after which you write down the quotients just below them. Then, the numbers that are not divisible, are carried forward. 

Properties of Factors and Multiples

To understand the concept of factors and multiples you should go through the types of factors and multiple examples. Also, certain distinctive properties make the concept clear and concise. Some of the major properties are mentioned below:

• Every number has a common factor: the number 1.

• Every number has a multiple: the number 0.

• The concept of multiples and factors applies to only whole numbers.

• Every number has a minimum of two factors namely the number 1 and the number itself. Number 1 is the smallest factor and the number itself is the largest factor.

• Each number has one multiple that is the number itself.

• For each number, the number of factors is finite and the number of multiples is infinite.

• The number is termed a prime number if the number has only two factors namely the number itself and the number 1.

Applications of Factors and Multiples 

There are various applications and uses of factors and multiples. Multiples are used in various ways in the transactions carried out in our everyday lives as well as in complicated scientific and mathematical calculations that are fundamental to physics and computer science. Both the concepts of factors and multiples are very important to understand and comprehend.

They tend to set the foundation of various mathematical aspects and operations like division, measurement, patterns, etc. And if one understands the concept of factors and multiples and the relationship between them, then it would become much easier for them to navigate the overall relationship of numbers in real life. 

Conclusion

The concept of multiples and factors are the backbone of many operations in mathematics. It is very essential to be clear with these concepts so that the other operations become simple and easy.

Finding the factors of a small number is an elementary concept in Mathematics.  For finding a number of factors of larger numbers, we have a factor theorem and some other techniques. In this article, we are going to find the factors of 23 and what makes 23 a prime number. Consider a number N. Then N’s factors will be those numbers that will divide N leaving the remainder zero. Similarly, factors of 23 will be those numbers which will divide 23 leaving the remainder zero. All the numbers have 1 and the number itself as the two factors. As 1 × N = N.

When we divide N by 1 or by N we will always get remainder zero. We can say there are at least two factors of a given number. Fraction numbers or decimal numbers are not considered as factors.

Method of Prime Factorization

In this method, we will continuously divide the given number by its prime factors. Let’s say we are given number 16. Then by prime factorization method, we get

16 = 2 × 2 × 2 × 2.

Negative Factors of 23

To find negative factors, simply assign the -ve sign behind the numbers.

From the above intuition, we get factors of 23 are 1 and 23. So it’s negative factors come out to be -1 and -23.

The Number of Factors of 23

The number of factors of a given number let’s say 8 ( by factor theorem)?

Step 1 — from prime factorization method we can get 8 = 2 × 2 × 2

Step 2 — write the prime factors in exponential al form — 2^3

Step 3 — leaving the base we consider the number in the exponent.

3 +1 = 4 hence 8 has 4 factors.

In case of our number 23 = 1 × 23

It is a prime number hence the factor theorem has a different approach to prime numbers.

The number of positive factors of a prime number is always 2.  The factor theorem has this special case for prime numbers.

Facts About Factors

1. 1 is always a factor of every number

2. The number itself is the largest factor of that number

            e.g –  The largest factor of 23 is 23.

3. Every factor of a number is the exact divisor of that number.

4. For every number, there are only finite factors for that. 

What can we conclude from these factors?

We can conclude that 23 is a prime number as it has only two factors: the number 1 and the number itself. There are many other ways to prove a number is prime by using a primality test but it is beyond the scope of this article. We got how the number 23 has only two factors and hence it is a prime number.

In Math, factorisation is the breaking up of a number into smaller numbers. The lower numbers when multiplied together give you the original number. The process of splitting a number into its factors and divisors is known as factorisation. So factors of 56 will be some prime numbers that when multiplied, give the result, as 56. This is known as prime factorisation of 56. Factors of a number are those that are evenly divisible into another number. It is of immense importance in elementary arithmetic. It can be used in many ways like determining the least common multiple or highest common factor. Typically, we factorize a number till we reach a point where it can no longer be divisible. This is generally known as prime factorization.

Definition 

When two numbers are multiplied together, and the result is a given number, then the two numbers are called factors of that particular number. For example, 5 x 3 equals 15; hence the numbers 5 and 3 are the factors of 15. The numbers that are multiplied together are called factor pairs.

What are the Factors of 56 in Pairs?

Every number is divisible by itself and one. For example, 56 /1 = 56. Since multiplication is the other side of the division, we can say that 56 x 1 = 56, so factors of 56 are 1 and 56. 1 and 56 are also known as factor pairs. Similarly, there are other factors and factor pairs of the number 56, such as

7 x 8 = 56 or 56/7 = 8 and 56/8 = 7

2 x 28 = 56 or 56/2= 28

4 x 14 = 56 or 56/4=14 and 56/14 = 4.

Therefore, the factor pairs of number 56 are 1 & 56, 7 & 8, 2 & 28, and 4 & 14.

Distinct Factors

You can consider 1, 2, 4, 7, 8, 14, 28, and 56 to be distinct factors of 56. Factors and distinct factors are the same. Factors are positive and are evenly divisible.

However, factors can be negative also. For example, it is -56, then the factors of – 56 are

-1 x 56 or 1 x -56 = -56

-2 x 28 or 2 x -28 = -56

-7 x 8 or 7 x -8 = -56

-4 x -14 or 4 x -14 = -56

In simple words, negative factors are factors with a (-) negative sign. Similarly,

-1 x -56 = 56

-2 x -28 = 56

-7 x -8 = 56

-4 x -14 = 56

Calculation of the Factors 

Only whole numbers and integers can be converted into factors. For 56 to be an integer, you should be able to write 56 without a fraction or decimal parts, e.g. 56/3, 23/56 or 56.01. For 56 to be an integer, it has to be the whole number that is not a fraction. Thus, 56 is an integer as well as a whole number. When you divide 56 by its factors, the remainder is always a zero. For example, 

56/1 = 56, results in zero and is divided by 1

56/2 = 28, results in zero and is divided by 2

56/4 = 14, results in zero and is divided by 4  

56/7 = 8, results in zero and is divided by 8.

56/56 = 1, results in zero and is divided by 1

56/ 28 = 2, results in zero and is divided by 28   

56/8 = 7, results in zero and is divided by 8

56/14 = 4, results in zero and is divided by 14.

How to Write the Factors of 56

Let us start by writing the number 56 first. Now, we have enough knowledge about the factor and the factor pairs of 56, so let’s begin any pair. For example, 7 x 56

= 7 x 8 = 56

= 7 x 2 x 2 x 2 = 56  

Now, 7 is a prime number and it cannot break down further like the number 8. 7 can be written only as 7 x 1. So, we write the factors of 56 as = 7 x 2 x2 x 2 x 1.

Let us take another factor pair, 2 and 28

= 2 x 28 = 56  

= 2 x 14 x 2 = 56

= 2 x 7 x 2 x 2 = 56

= 2 x 2 x 2 x 7 = 56

= 2 x 2 x 2 x 7 = 56

= 2 x 2 x 2 x 7 x 1 = 56

Therefore, factors of 56 can be written as 7 x2 x 2 x 2 x 1. You will get the same answer with any other factor pair that you use. The conclusion is that the factors of number 56 are always written as 7 x2 x 2 x 2 x 1.

Prime Factorization of 56 

Firstly, know that 56 is a composite number. You already know that 56 is a whole number as well as an integer (not a fraction and not a decimal). The multiplication can take 7 x 8 or 2 x 28, so 56 is a composite number. A composite number is made up of prime factors also. As you know, prime numbers are 1, 2, 3, 5, 7, 11, 13, 17, and 31 as they have no other factors apart from 1.

To begin with, divide the number 56 with 2, one of its prime factors.

56/2 = 28

28/2 = 14

14/2 = 7

7/2 = 3.5, a fraction. So, the next option is

7/7 = 1

It puts a stop to further divisions. Hence, prime factors of 56 are written as 2 x 2 x 2 x 7 or 23 x 2 where 2 and 7 are the prime numbers. It is the prime factorization of 56.

Fun Facts about Factors 

1. Factors are whole numbers, integers, and never decimals or fractions.

2. All the even numbers always have two as a factor. 

3. All numbers that end with five have five as a factor.

4. All numbers ending with a zero have 2, 5 and 10 as factors.

Factors of 56

When two numbers are multiplied together, the result is 56, and the factors of 56 are the numbers that generate that result. The full numbers, positive or negative, but not a fraction or decimal number, are factor pairs of the number 56. Consider the factor pair of 56, which is written (1, 56) and (-1, -56). As a result, both positive and negative factor pairs of 11 can be considered. The factorization method will be used to identify the factors of an integer, 56. Using the prime factorization approach and several solved cases, we will discover the factors of 56, pair factors, and prime factors of 56 in this article.

1,2,4,7,8,14,28,56

 

What are the 56 Factors?

The factors of 56 are the numbers that divide 56 perfectly. In other terms, the factors of 56 are the integers multiplied in pairs to produce 56. The number 56 has several factors other than 1 and 56 since it is an even composite number. As a result, the 56 factors are 1, 2, 4, 7, 8, 14, 28, and 56.

1, 2, 4, 7, 8, 14, 28, and 56 are the factors of 56.

Prime Factorization of 56: 2×2×2×7 or 23 × 7.

Pair Factors of 56

Multiply the two numbers in a pair to produce the original number of 56; examples of such numbers are as follows:

As a result, the positive pair factors of 56 are (1, 56), (2, 28), (4, 14), and (5, 14), respectively (7, 8). (-1, -56), (-2, -28), (-4, -14), and (-1, -56) are the negative pair factors of 56, respectively (-7, -8).

56 Factors Using the Division Method

The factors of 56 can be calculated using the division method by dividing 56 by different consecutive integers. If the integers divide 56 perfectly, they are the 56 factors. Let’s begin by dividing 56 by one, and then move on to the other integer values.

(Factor = 1 and remainder = 0) 56/1 = 56

(Factor = 2 and remainder = 0) 56/2 = 28

(Factor = 4 and remainder = 0) 56/4 = 14

(Factor = 7 and remainder = 0) 56/7 = 8

(Factor = 8 and remainder = 0) 56/8 = 7

(Factor = 14 and remainder = 0) 56/14 = 4

(Factor = 28 and remainder = 0) 56/28 = 2

(Factor = 56, remainder = 0) 56/56 = 1

As a result, the 56 factors are 1, 2, 4, 7, 8, 14, 28, and 56.

Factors of a number are the product of such numbers which completely divide the given number. Factors of a given number can be either positive or negative numbers. By multiplying the factors of a number we get the original number. For example 1, 2, 3, 6 are the factors of 6. On multiplying two or more numbers we get 6. Hence we have 2 x 3 = 6 or 1 x 6 = 6. On this page, we will study the factors of 98 definitions, what are the factors of 98, what is the prime factorization of 98, factor tree of 98, and examples. Factor pairs of the number 98 are the pairs of the whole numbers which could be either positive or negative but not a fraction or decimal number. Factorisation is the common method to find the factors of 98. Let us study the prime factorization of 98.

What are the Factors of a Number?

In elementary school pupils start the subject of mathematics by learning how to count numbers. Various props and activities are performed to practice counting. After the students get comfortable with the digits and can identify their values it is important to understand the values of each number and the quantity they represent. Students learn addition, subtraction, multiplication and division and solve questions based on these basic operations. It helps them to understand and calculate the total amount for any quantity they are provided within their day to day activities. To make it easier there are tables of multiplication that the students practice and by heart. They learn to divide the higher digit numbers by recalling the table they have practised. In the subsequent classes, the problems get advanced such as identifying the multiples of a number and finding out the numbers which can divide a particular number with the remainder zero. These numbers that can perfectly divide a number are known as the factors of that number.

Let’s take the example of the number 98 to understand what the factors of a number are.

98 is an even number so it is obvious that the number is divisible by 2. It is not divisible by 3, 4, 5 or 6 so these are not the factors of 98. If we divide the number by 7 then we get the dividend of 14. So both these numbers are the factors of 98. It is again not divisible by 8, 9, 10, 11, 12, or 13. Thus we have found that the 6 factors of 98 are 1, 2, 7, 14, 49, and 98.

What are the Factors of 98 ?

According to the definition of factors of 98 , we know that factors of 98 are all the positive or negative integers which divide the number 98 completely. So let us simply divide the number 98 by every number which completely divides 98 in ascending order till  not divides completely

98 [div] 1 = 98

98 [div] 2 = 49

98 [div] 3 = not divides completely

98 [div] 4 = not divides completely

98 [div] 5 = not divides completely

98 [div] 6 = not divides completely

98 [div] 7 = 14

98 [div] 14 = 7

98 [div] 49 = 2

98 [div] 98 = 1

All Factors of 98 can be Listed as Follows

Hence 98 have a total of 6 positive factors and 6 negative factors. All Factor Pairs of 98

All Factor Pairs of 98 are combinations of two factors that when multiplied together give. According to the prime factor definition, we know that the prime factor of a number is the product of all the factors that are prime( a number that divides by itself and only one). Hence we can list the prime factors from the list of factors of Or the other way to find the prime factorization of 98 is by prime factorization or by factor tree.