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Real Numbers

Exclusively for CAT/CSAT/CLAT and Other Compititions

Agenda

Indices and Surds

Number System – Understanding of Real Numbers

HCF and LCM

Base System of Numbers

Unit’s Digit

Algebra of Numbers

Prime Numbers

NumSys

Number System - Understanding of Real Numbers 

(Excerpts from Course Material)

All numbers which are not square root of -1 (minus one), i.e. they are not √(-1) or are not multiple thereof are REAL NUMBERS.

 

The square root of -1 (minus one), i.e. √(-1) is represented by ‘i’ and is called imaginary number.

All number of the form of a+ib where a and b are real numbers and i is imaginary number are called COMPLEX NUMBERS. Complex numbers are not real numbers.

 

By Number System, we mean system of REAL NUMBERS.

Rules:

  1. Natural numbers (1, 2, 3, 4,  …) are at the lowest rung of the system.

  2. Next, we have set of whole numbers (0, 1, 2, 3, 4,  …) and, obviously, every natural number is a whole number.

  3. Next, we have set of integers (……….., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …) and, obviously, every whole number is an integer.

  4. Next, we have set of rational numbers (like ) and, obviously, every integer is a rational number as 2 can be written as   or   and so on.

  5. The numbers which are not rational numbers are irrational number. And of course the numbers which are not rational are NOT integers, whole numbers or natural numbers also.

So, in view of (5) above, the world of real numbers is composed of two types of numbers, i.e. rational number and irrational number. Hence, a real number is either a rational number or irrational number.

 

You must take not of the following:

 

Every rational number is not an integer (for example,  is a not an integer).

Every integer is not a whole number (‘-1, -2, -3, …… are not whole numbers).

Every whole number is not a natural number (‘0’ is not a natural number).

Every real number can be put in the form of a decimal, for example, 4 can be written as 4.0, 3/2 can be written as 1.5, 10/6 = 1.66666666666666…..

 

This suggests that every real number can be expressed in a decimal form.

 

We shall now understand it this way:

The Non-terminating decimals are of two types as you see in the table given above.

 

  1. Non-Terminating – Repeating Decimal (or non-terminating recurring decimal): These numbers are strictly rational numbers. Such numbers are those numbers where a set of decimals is repeating itself, or repeating an innumerable number of times. The example, other than the given above, may be: 20.11111111, 0.32323232……, 14.234234234234234…………, -6.203203203203203……. etc. We denote such repetitions by putting a bar on the set of digits being repeated, as the numbers given above can be written as 20.1, 0.32, 14.234, and -6.203 respectively. All such numbers are strictly “Rational” as these can be put in the form of p/q with q ≠ 0.We shall see it happening a little later. We always place a bar on the set of numbers being repeated, for example, if X = 6.203203203203203……., it is, X = . Or, X = 16.56012012012012……., it is, X = 1 .

  2. Non-Terminating Non-Repeating Decimal (or non-terminating non-recurring decimal): These numbers are strictly irrational numbers. Such numbers with decimals are the ones where there is no rule of repetition. The examples, other than that given above, are: 20.100165210116980124875……………, 0.37523923022363020439827……, 14.2130340025436492030………… etc. All such numbers are strictly “Irrational” as these numbers CAN NEVER be put in the form of p/q with q ≠ 0. Values of e (exponential function with x=1, i.e. Euler’s number), (Pi), are examples of irrational numbers. Though, the expressed values of e and  i.e. 2.71828 and   or 3.14 respectively seem to be rational ones, they are irrational numbers. The rational values are only approximations to their irrational numbers values. Further, as stated above, all square-root values of all such numbers which are not perfect square are irrational numbers.

AlgNumbers

Algebra of Numbers (Excerpts from Course Material)

Properties of Algebra of Real Numbers

  • Commutative property for addition - If we have two real numbers m and n, then m + n = n + m.

  • Commutative property for multiplication - If we have two real numbers m and n, then m*n = n*m.

  • Associative property for addition - If we have three real numbers m, n and r, then m + (n + r) = (m + n) + r.

  • Associative property for multiplication - If we have three real numbers m, n and r, then m * (n * r) = (m * n) * r.

  • Distributive property- If we have real numbers m, n and r, then: m * (n + r) = m*n + m*r and (m + n) * r = m*r + n*r

  • Identity element for addition: ‘0’ is the identity element for every real number.

  • Inverse element for addition: For every real number m, there exists a real number (-m) such that m + (- m) = 0 (the identity element), this number (-m) is called inverse element of m under addition.

  • Identity element for multiplication: ‘1’ is the identity element for every real number except 0.

  • Inverse element for addition: For every real number m except 0, there exists a real number (1/m) such that m * (1/m) = 1 (the identity element), this number (1/m) is called inverse element of m under multiplication.

Rules for Divisibility

 

  • Divisibility by 1: Every number (including zero) is divisible by 1 and results in number itself.

  • Divisibility by 2: A number is divisible by 2 if its last digit (at unit place) is divisible by 2.

  • Divisibility by 3: A number is divisible by 3 if the sum of all its digits is a multiple of 3. Further remember that any digit when written continuously in number of multiples of 3, it is divisible by 3. For example, 777 (7 written 3 times), 888888 (8 written 3 times), 111111111 (1 written 9 times) are divisible by 3. The reason is very simple. The sum of the digits SHALL be divisible by 3.

  • Divisibility by 4: A number is divisible by 4 if the number formed by last two digits is divisible by 4 or is a multiple of 4.

  • Divisibility by 5: A number is divisible by 5 if its last digit is either 0 or 5. Remember that the sum of five successive whole numbers is always divisible by 5, as a multiple of 5 shall appear in these 5 numbers.

  • Divisibility by 6: A number is divisible by 6 if it is divisible both by 2 and 3.The product of 3 consecutive natural numbers is divisible by 6.

  • Divisibility by 7: A number is divisible by 7 if we remove the last digit of the number, and subtract its double from the remaining number, and repeat the process unless the number is reduced to one digit number. If this last digit is either 0 or 7, the number is divisible by 7.

First example: 123654

Step I: 12365 – 8 = 12357

Step II: 1235 – 14 = 1221

Step III: 122 – 2 = 120

Step IV: 12 – 0 = 12

Step V: 12 - 4 = 8, which is a single digit number and is neither 0 nor 7. Hence, the given number (123654) is not divisible by 7.

 

Second example: 123655

Step I: 12365 – 10 = 12355

Step II: 1235 – 10 = 1225

Step III: 122 – 10 = 112

Step IV: 11 – 4 = 7, which is a single digit number and is 7. Hence, the given number (123655) is divisible by 7.

​

Divisibility by 2^n: To examine whether a number (howsoever large) is divisible is by 2^n, is to see the divisibility of last n digits of this number by 2^n. Hence, a number is divisible by 2^n if last ‘n’ digits of the number are divisible by 2^n, otherwise it gives the remainder. For example, the number 3456789876543212345678987648 is divisible by 32, i.e. 2^5, as last 5 digits of the number, i.e. 87648 is divisible by 32.

Base System of Numbers (Excerpts from Course Material)

BaseSystem

3.2 What is base system for numbers?

 

Suppose we have a number 6789. This number is six thousand seven hundred eighty nine. That is, 6000 + 700 + 80 + 9. We may write it as: 6x103 + 7x102 + 8x101 + 9x100. This type of representation of numbers is called ‘Decimal System” or “Decimal Base System” and various indices, starting from 0 are called place values. The index 0 (on 10) is called unit value, index 1 is called tens value, index 2 is called hundreds place, index 3 is called thousands place and so on. This system has 10 primary numbers which make all other numbers by various permutations and combinations obeying the rules of indices as given above. These 10 numbers are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

 

Various Base Systems

 

  1. Base 2, called binary system. Every number of this system will use only two digits, i.e. 0 and 1 and every number shall be a permutation combination of these two digits only. For example, some number ‘n’ in this base 2 system can be written as 001011010. This system is widely used in computer science.

  2. Base 3. Every number of this system will use only three digits, i.e. 0, 1 and 2.

  3. Similarly for all base systems upto decimal base (base 10), the basic principle is that given a base ‘n’, all the numbers under that base (‘n’) shall use digits starting from 0 to (n-1), where .

  4. Base 11. This system has eleven numbers (0 to A). The numbers are: 0 = 0, 1 = 1, 2 = 2, 3 = 3, 4 = 4, 5 = 5, 6 = 6, 7 = 7, 8 = 8, 9 = 9, A = 10.

  5. Base 12. The duodecimal (base 12) or dozenal numbering system. This system has twelve numbers (0 to B). The numbers are: 0 = 0, 1 = 1, 2 = 2, 3 = 3, 4 = 4, 5 = 5, 6 = 6, 7 = 7, 8 = 8, 9 = 9, A = 10, B = 11.

  6. Base 13. This system has thirteen numbers (0 to C). The numbers are: 0 = 0, 1 = 1, 2 = 2, 3 = 3, 4 = 4, 5 = 5, 6 = 6, 7 = 7, 8 = 8, 9 = 9, A = 10, B = 11, C = 12.

  7. Base 14. This system has fourteen numbers (0 to D). The numbers are: 0 = 0, 1 = 1, 2 = 2, 3 = 3, 4 = 4, 5 = 5, 6 = 6, 7 = 7, 8 = 8, 9 = 9, A = 10, B = 11, C = 12, D = 13.

  8. Base 15. This system has fifteen numbers (0 to E). The numbers are: 0 = 0, 1 = 1, 2 = 2, 3 = 3, 4 = 4, 5 = 5, 6 = 6, 7 = 7, 8 = 8, 9 = 9, A = 10, B = 11, C = 12, D = 13, E = 14.

  9. Base 16. This system has sixteen numbers (0 to E). The numbers are: 0 = 0, 1 = 1, 2 = 2, 3 = 3, 4 = 4, 5 = 5, 6 = 6, 7 = 7, 8 = 8, 9 = 9, A = 10, B = 11, C = 12, D = 13, E = 14, F = 16.

Fermat's Little Theorem

3.31    Conversion from Decimal System to any other base system

 

This procedure gives you an idea as to how to convert a number GIVEN in Decimal system into one of another system

 

You must watch closely the steps taken to solve the conversion problem, because it is the only easiest way to learn the method of conversion.

But this is a lengthy process, especially for larger numbers.

Now the simple procedure for converting a number of decimal base to a number of another base, say n, is given below. Given a number with base ten, we shall begin with dividing the given number with n and collect so obtained remainders one by one till we don’t get zero as quotient. The last remainder is most significant digit and the first remainder is least significant digit. Remember that lower the base, higher the number.

3.33    Conversion of a number of any base system to other system of base: The procedure is simple. First convert to base 10, then to the desired one.

When, finally you get C = 1.00, means that the value of A in the next row shall be 0.00, hence no further calculation is possible. Now we have got the number with base 10 equivalent to given number. Take the values of I one by one starting from the last row indicated with an arrow sign and place the decimal sign in the beginning. Hence the number with base 10 equivalent to the given number in base 2 is: 0.101

Indices

Indices and Surds (Excerpts from Course Material)

4.1      Indices or Powers (or exponent)

Let there be a real number, say r, such that, r is being multiplied by r, say n times.

That is: r x r x r x r x r x r x r x r……. n times.

We can write this huge multiplication process in brevity as r^n and call it as r raised to power n. Here r is called base and n is called power or exponent or index (indices in plural).

Hence in (¾)^4, ¾ is the base and 4 is the exponent. It is equal to calculating ¾ x ¾ x ¾ x ¾  = 243/256.

​

4.2      Rules of Indices

  1. If any number except 0, say a (≠0), is raised to the power 0, the result is always 1. That is, a^0 = 1.

  2. 1 is raised to any power (any real number), the answer is always 1. Means, 1^n = 1, where n is any real number (1x1x1x1x……. = 1).

  3. The reciprocal of the number has same power with sign changed, i.e. a^n = 1/ a^-n. For example, 5^-4 = 1/(5^4) and 15^(3/4) = (1/15)-3/4

  4. When the base of two numbers is same, the powers of the two numbers, when two numbers are multiplied, are added, i.e. a^n x a^m = a^(n+m). For example, 17^12 x 17^5 = 17^17.

  5. When the base of two numbers is same, the powers of the two numbers when two numbers are divided are subtracted, i.e. a^m ÷ a^n = a^(m-n). For example, 17^12 ÷ 17^5 = 17^(12 – 5) = 17^7.

  6. In case, base of two equal exponents are same, their indices are equal. That is if a ^m = a^n, then m=n, in as much as a^m/a^n = 1 è a^(m-n) = 1 = a, that is, m-n = 0, that is, m=n.

  7. When a power is raised on a power, it results in multiplication of two powers, i.e. (a^n)^m = a((nm). For example, (17^12)^5 = 17^60.

  8. Let a^m = b. Then a = b^(1/m), or . And a is called mth root of b. For example, 4 = 64^(1/3), and 4 is the cube-root of third root of 64.

  9. Let a^m = b^n. Then a = b^(n/m), and b = a^(m/n)

​4.5      How to find out square-root?

There are two techniques and both are very simple, Factor Method and Division Method. I shall suggest that for larger numbers you should use division method.

Factor Method (you learn about factors a little later); you make factors and assemble equal factors in pairs, take one factor from each pair and multiply all. The unpaired factors remain inside the square-root. But many a times this method is not suitable for large numbers.

Division Method: This will be clearer to you when you see it doing.

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