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Number Theory

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CAT 2025 Lesson : Number Theory - Types of Numbers

2. Types of Numbers

All numbers can be divided into real and imaginary numbers. Real numbers are those which can be plotted on a number line.

Imaginary numbers are those which do not exist on a number line. These can be written as a real number multiplied by an imaginary unit

i

, where

i

= \sqrt{-1}

. A complex number is a number that can be expressed in the form

a + bi

, where

a

and

b

are real numbers and

i

is the imaginary unit. Complex and Imaginary numbers are not in the curriculum of most management entrance exams including CAT. Therefore, we will not be discussing this further.

2.1 Real Numbers

Real numbers are categorised as rational numbers and irrational numbers.

2.1.1 Rational numbers

Numbers that can be expressed in the form

\dfrac{p}{q}

, where

p

and

q

are integers and

q \ne 0

, are called rational numbers. The following are examples of rational numbers.

5 = \dfrac{5}{1}

;

-1.25 = \dfrac{-125}{100} = \dfrac{-5}{4}

;

2.66666..... = 2.\overline{6} = \dfrac{8}{3}

-1.428571428571... = -1.\overline{428571} = \dfrac{-10}{7}

Rational numbers can be numbers with an infinite set of digits after the decimal point, as long as these digits are recurring. An

\overline{\text{overline}}

is used to represent the recurring set of digit(s).

The following example shows how to convert a number with recurring digits after the decimal to the

\dfrac{p}{q}

form.

Example 2

Express

0.575757

... as a fraction.

Solution

Let

x = 0.\overline{57} \longrightarrow (1)

100x = 57.\overline{57} \longrightarrow (2)

(2) - (1)

⇒

100x - x = 57.\overline{57} - 0.\overline{57}

⇒

99x = 57

⇒

x = \dfrac{57}{99}

Answer:

\dfrac{57}{99}

Example 3

Let D be a recurring decimal of the form, D =

0.a_1a_2a_1a_2a_1a_2 .......

, where digits

a_1

and

a_2

lie between

0

and

9

. Further, at most one of them is zero. Then which of the following numbers necessarily produces an integer, when multiplied by D?
[CAT 2000]

(1)

18

(2)

108

(3)

198

(4)

288

Solution

As exactly two digits are recurring,

\mathrm{D} = 0.\overline{a_{1}a_{2}} \longrightarrow (1)

100\mathrm{D} = a_{1}a_{2}.\overline{a_{1}a_{2}} \longrightarrow (2)

(2) - (1

) ⇒

99\mathrm{D} = a_{1}a_{2}

⇒

\mathrm{D} = \dfrac{a_{1}a_{2}}{99}

198

is the only option which perfectly divides

99

\therefore 198 \times \mathrm{D} = 2 a_{1}a_{2}

is always an integer.

Answer: (3)

198

If every three digits are recurring, then we multiply by

10^3

and subtract. Therefore, if every

n

digits are recurring, then we multiply by

10^n

2.1.2 Irrational numbers

Numbers that cannot be expressed in the form

\dfrac{p}{q}

are irrational numbers. The digits after the decimal place in these numbers will not be recurring. The following are examples of irrational numbers

\pi = 3.14159265359...

;

\sqrt{2} = 1.4142135624...

;

3^\frac{1}{3} = 1.4422495703...

Note that

\dfrac{22}{7} = 3.\overline{142857}

is a close approximation of

\pi

. Therefore,

\pi = \dfrac{22}{7}

is used to approximate and simplify calculations in geometry. However,

\pi

is an irrational number.

2.2 Other Terminologies

Following are some of the different types of real numbers.

Integers: All numbers that have

0

after the decimal point. Examples are

5

and

-59

.

Decimals: Numbers expressed with a denominator of

1

. These are typically expressed with a decimal point. The digit placed to the immediate left of the decimal point is the units digit and that to the immediate right is the tenths digit. Integers are decimals as well. Examples are

4.04

-5.25

1.5555

...

Fractions: Numbers expressed in the form

\dfrac{p}{q}

, where

p

and

q

are integers and

q \ne 0

. Examples are

\dfrac{5}{1}

\dfrac{-15}{4}

\dfrac{465}{41}

.

Natural numbers: This is the set of all positive integers. The sequence of natural numbers is

1

2

3

4

, ... There is an infinite number of natural numbers.

Whole numbers: This is the set of all non–negative integers. They include the number

0

and all the natural numbers. The sequence of whole numbers is

0

1

2

3

, ...

Even numbers: These are integral multiples of

2

. These can be expressed in the form

2n

, where

n

is an integer. These numbers end with a units digit of either

0

2

4

6

8

. Examples are

-36

0

12

and

894734

.

Odd numbers: These are integers that leave a remainder of

1

when divided by

2

. These can be expressed in the form

2n + 1

, where

n

is an integer. These numbers end with a units digit of either

1, 3, 5, 7

9

. Examples are

-4593

1

19

and

864207

.

Arithmetic Properties of Odd and Even numbers:

Addition	Multiplication
Even + Even $=$ Even	Even $\times$ Even $=$ Even
Even + Odd $=$ Odd	Even $\times$ Odd $=$ Even
Odd + Odd $=$ Even	Odd $\times$ Odd $=$ Odd

Example 4

x = 3 - 6 + 9 - 12 + ... - 90

, then which of the following is true?

(1)

x

is even but not divisible by

4

(2)

x

is divisible by

4

(3)

x

is odd
(4) None of the above

Solution

The terms are the first

30

multiples of

3

with alternating signs

(+

and

-)

.

∴ There are a total of

30

terms.

Starting from the first term, if we group every

2

terms as a pair, we get

15

terms

x = -3-3-3...._{\text{15 terms}}

x = -3 \times 15 = -45

Answer: (3)

x

is odd

Example 5

Where

x

and

y

are integers, if

x + y

is odd and

xy

is even, then which of the following is true?

(1)

x + xy

is even (2)

x^2 + y^2

is odd (3)

(x + y)^2

is even (4)

5xy

is odd

Solution

x + y

is odd and

xy

is even, one of

\bm{x }

\bm{y}

is odd, while the other is even.

Option 1: We do not know if

x

is even or not.

\therefore

Inconclusive

Option 2:

a^2

would remain odd if

a

is odd and remain even if

a

is even. ∴

x^2 + y^2

is odd, as it is the sum of 1 odd and 1 even number. True

Option 3: As

x + y

is odd,

(x + y)^2

will also be odd. False

Option 4: As

xy

is even,

5xy

is also even. False

Answer: (2)

x^2 + y^2

is odd

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