**Number Properties:**

Number properties are one of the most important and tricky topic tested in the exams. So let’s learn what are the properties of numbers, and its classifications.

**1. Digits:**

Digits are basic elements of Math that is like a backbone of the whole math. There are only 10 digits exist as mentioned below:

0, 1, 2, 3, 4, 5, 6, 7, 8, and 9

If you say that digits are the base on which the whole math stands, you’re not wrong. Also when it is given that *a*, *b*, *c* and *d* are different digits, you should know what are the digits.

**2. Numbers:**

Digits combine to form numbers. Perhaps you’re thinking why you need to remember the definitions and concepts of these basic terminologies? Well, you will be given some technical information about variables in question; for instance, *a*, *b* and *c*, or *x*, *y* and *z* are numbers, integers, non-integers, positive integers, non-positive integers, negative integers, non-negative integers, even numbers or odd numbers. So if you know these, you’ll able to put the values from given definitions to answer the question. Therefore, before going to prepare for the above mentioned exams, you need to know what are these terminologies.

Numbers are of two basic types:

**I. Integers:**

Numbers that have 0 decimals are integers. For instance:

… –4.0, –3.0, –2.0, –1.0, 0, 1.0, 2.0, 3.0, 4.0, …

Or simply,

… –4, –3, –2, –1, 0, 1, 2, 3, 4, …

**II. Decimals (i.e non-integers):**

Numbers that have at least one non-zero digit decimal. For instance:

–4.9, –3.75, –1.9858, 0.5, 0.75, 2.5, 3.75, etc…

There are infinite decimals. These are also called non-integers, which we will discuss later on in detail in this free study plan for beginners.

Later on, in this lecture, you’ll learn how to approximate decimal into an integer or into another decimal.

**Integers and Decimals:**

Refer to the following table:

Combinations |
Addition or Subtraction (±) |
Multiplication (×) |
Division (÷) |
---|---|---|---|

Integer vs Integer |
Integer ± Integer = Must be an Integer | Integer × Integer = Must be an Integer | Integer ÷ Integer = Either Integer or Decimal |

Integer vs Decimal |
Integer ± Decimal = Must be a Decimal | Integer × Decimal = Either Integer or Decimal | Integer ÷ Decimal = Either Integer or Decimal |

Decimal vs Integer |
Decimal ± Integer = Must be a Decimal | Decimal × Integer = Either Integer or Decimal | Decimal ÷ Integer = Must be a Decimal |

Decimal vs Decimal |
Decimal ± Decimal = Either Integer or Decimal | Decimal × Decimal = Either Integer or Decimal | Decimal ÷ Decimal = Either Integer or Decimal |

From the table above, it’s clear that in addition or subtraction, when both numbers are different (i.e. one integer and other decimal), the result must be a Decimal; if both are integers, the result must be an integer, and if both are Decimal, the result may be an integer or a fraction {For instance, when 2.6 and 0.4 are added, the result is an integer; and when 2.2 and 0.4 are added, the result would be a decimal}.

In multiplication, only when both numbers are integers the result must be an integers, otherwise the result may be an integer or a decimal. For instance, when 2 is multiplied by 2.5, the result would be 5, which is an integer; but when 2 is multiplied by 2.2, the result would be 4.4, which is a decimal; Similarly, when both numbers are decimal i.e when ^{5}⁄_{2} (which equals to 2.5) is multiplied by ^{2}⁄_{5} (i.e. 0.4), the result would be 1, which is an integer; but when 2.5 is multiplied by itself, the result would be 6.25, which is a decimal}.

And, in division, only when a fraction is divided by an integer, the result must be a decimal, otherwise the result may be an integer or a decimal.

**Types of Integers:**

**I. Positive Integers:** 1, 2, 3, 4, 5, ….

**II. Negative Integers:** –1, –2, –3, –4, –5, ….

**III. Neutral Integer:** Only 0, which is neither negative nor positive.

**Sub-types:**

Non-Positive Integers: = All integers except positive = (Neutral + Negative) Integers: 0, –1, –2, –3, –4, –5, ….

Non-Negative Integers: = All integers except negative = Neutral Integer + Positive Integers: 0, 1, 2, 3, 4, 5, ….

**Classification of Integers:**

Integers are classified into two important types: *Even* and *Odd*. Remember that Even and Odd are always integers. Also Integers are always either *Even* or *Odd*. Former statement means when you see the word even, you should think about it as an integer rather than non-integers. Later statement means each integer is either an *Even integer* or an *Odd integer*.

**I. Even integers:**

Integers that are divisible by 2 are even integers. For instance:

… –8, –6, –4, –2, 0, 2, 4, 6, 8, …

**Important Note:** *Zero is an even integer, because zero is divisible by 2*.

**Important Note:** *Remember that, a is divisible by b, if a divided by b results to an integer*.

When 0 is divided by 2, it results to 0 (an integer). So 0 is divisible by 2. We’ll learn about this further later on while discussing Factors & Multiples.

**II. Odd Integers:**

Integers that are not divisible by 2 are odd integers. For instance,

… –7, –5, –3, –1, 1, 3, 5, 7 …

**Important Note:** *Remember that non-integers are not odd, because odd must be an integer that is not divisible by 2*.

**Important Note:** *Even or Odd integers can also be written as Even or Odd numbers, because all integers are numbers. Also, we know that fractions (i.e non-integers) are neither even nor odd, because both even and odd are integers. Therefore, you’ll sometimes see ‘even/odd number’ or simply ‘even/odd’, instead of even/odd integer. So don’t confuse from these terms*.

**Arithmetic operations with Odds and Evens:**

Refer to the following table:

Combinations |
Addition or Subtraction (±) |
Multiplication (×) |
Division (÷) |
---|---|---|---|

Even vs Even |
Even ± Even = Even | Even × Even = Even | Even ÷ Even = N/A |

Even vs Odd |
Even ± Odd = Odd | Even × Odd = Even | Even ÷ Odd = N/A |

Odd vs Odd |
Odd ± Odd = Even | Odd × Odd = Odd | Odd ÷ Odd = N/A |

Odd vs Odd |
Odd ± Even = Odd | Odd × Even = Even | Odd ÷ Even = N/A |

From the table above, it’s clear that in addition or subtraction, when both are same irrespective of even or odd, the result is even; Otherwise it’s odd. In multiplication, when at least one even exists, the result would always be even, otherwise it’s odd. And in division, as the result may be an integer or may be a fraction, so we can’t determine what’s the result; in short Not Applicable (N/A).

**Consecutive Integers:**

The series of integers that have common difference of 1 are consecutive integers. Let’s suppose ‘I’ is an integer,

*I*, (*I* + 1), (*I* + 2), (*I* + 3), (*I* + 4), …..

The above series is a set of consecutive integers.

Similarly,

1, 2, 3, 4, 5, …..

This series is consecutive positive integers.

And,

0, 1, 2, 3, 4, ……

This series is consecutive non-negative integers.

Please note the difference! When it says consecutive non-negative integers, 0 will be included because it’s an integer which is not negative.

**Application of this basic concept:**

If the average of 75 consecutive integer is 1023, what is the sum of first two of those integers?

**Solution:** Let’s assume the 75 consecutive integers are:

*I*, (*I*+1), (*I*+2), (*I*+3), ………. (*I*+74)

Remember that the last integer will not be *I*+75, because first integer is *I*+0, second integer is *I*+1, third integer is *I*+2 and so on…

From this trend, we find out the 75^{th} integer would be *I*+74

**Important Note:** *Remember that in any evenly spaced set of numbers, the average of all numbers is equals to average of (first number and last number). Evenly spaced set of numbers are those that have common difference in any two adjacent numbers*.

Therefore, as consecutive integers has common difference of 1 in any of its adjacent number, so you may say:

Average = ^{(First number + Last number)}⁄_{2}

⇒ = ^{{I + (I+74)}}⁄_{2} = ^{(2I + 74)}⁄_{2}

By taking 2 common from the numerator,

⇒ = ^{2(I + 37)}⁄_{2}

As the 2, which has taken out as common, is multiplying with (*I* + 2), so we can cancel it out with the 2 in denominator. You’ll learn this in divisibility rules, if few of you not understood why this happen.

⇒ = *I* + 37

Now, it is given that average of five consecutive integers = 1023

And from our set of supposed consecutive integers, average = *I* + 37

⇒ *I* + 37 = 1023

⇒ *I* = 1023 – 37

⇒ *I* = 986

Now, we are asked to find the sum of first two of those integers. i.e

*I* + *I* + 1 = 2*I* + 1

By putting value of integer *I*, we’ll get:

Sum of first two integers = 2(986) + 1 = 1972 + 1 = **1973** **Answer**

**Consecutive Even Integers:**

Let’s suppose *n* is an integer, the series of consecutive even integer would be:

2*n*, (2*n*+2), (2*n*+4), (2*n*+6), (2*n*+8), ……

You’ll learn the application of this concept in advance level in medium to hard level question scenarios of *Arithmetic* topic of quantitative lecture. You’ll learn short-cut technique there to solve such complex things, which requires problem solving skill. 🙂

**Consecutive Odd Integers:**

Its very simple if you know Even + Odd = Odd

Let’s suppose *n* is an integer, the series of consecutive odd integer would be:

(2*n*+1), (2*n*+3), (2*n*+5), (2*n*+7), (2*n*+9), ……

This has obtained by adding consecutive odd integers to an even integer *‘2n’*.

As 1, 3, 5, 7, 9, … are consecutive odd integers, so the above series is a general form of consecutive odd integers that will be used to solve complex questions related to series and sequences and averages etc. We’ll learn this in detail in advance level.

At this level, you just need to remember this general form of series.

Many books and instructors misguide students by informing wrong general form of odd integers series as:

(*n*+1), (*n*+3), (*n*+5), (*n*+7), (*n*+9), ……

If you put n any odd number, it will result to an even integer. But as I said n must be an integer, whether even or odd. You’ll only able to solve complex questions from this topic, if you remember correct general form of odd series as we’ve discussed above. Similar mistake these books and instructors commit and misguide students by informing wrong general form of even integers series as below:

*n*, (*n*+2), (*n*+4), (*n*+6), (*n*+8), ……

So, never allow yourself to be misguided at anywhere. 🙂

**Prime Numbers:**

Numbers that have exactly two **different** factors. For instance:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, …

Many people define prime number wrongly by saying, an integer that is only divisible by itself. If we assume this version as correct, then 1 is also divisible by itself, but 1 is not prime.

In above series of prime numbers, each number has two different factors. In other words, each integer in above series is divisible by two different integers, i.e 1 and itself. Therefore, from this definition, 1 is not a prime number. There are infinite prime numbers.

Note that, I’m most of the time saying the above mentioned series of prime as prime **numbers**, rather than prime integers. As I said earlier, we can say any integer or series of integers to be numbers, but cannot say any number to be an integer. So do not confuse. 🙂

**Important Note:** *Factors: When a is divisible by b, then b is factor of a. So when the ratio of two numbers results to an integer, the number on denominator is factor of the number on numerator*.

**Important Note:** *There’s only one even prime number (i.e 2), which is the least prime number as well*.

Prime numbers are the backbone of whole *Integer theory* topic, just as digits are the backbone of whole *math* section.

**Whole Numbers:**

Non-negative integers are termed as *‘Whole Numbers’*. i.e

0, 1, 2, 3, 4, 5, …

**Number-Line:**

In a number-line, numbers are always increases from left to right. For instance, consider the number-line below:

Now, suppose we need to find the distance between 2 and 5 in a number-line. Let’s consider a number-line below:

Mathematically, the distance between 2 and 5 is:

5 – 2 = 3

Similarly,

The distance between 5 and –3 (blue region + green region) can be calculated as follows,

Mathematically, the distance between 5 & –3 is:

5 – (–3) = 5 + 3 = 8

The distance between any two numbers is always larger number – smaller number.

So far, we’ve learned number-line from one basic scenario. Let’s try to play with the common sense of your brain through this topic. This will help you improving some problem solving skill and improve your thinking power in broader way. 🙂

**Basic sample question:**

Let’s consider the following number-line:

Which of the following statement MUST be true? Indicate all such statements.

i) *y* > *x* + *b*

ii) *y* + *b* > *x* + *a*

iii) *a* × *b* > *y*

**Solution:**

First of all, pay attention to the words *‘MUST be true’* in question. Secondly, note that there’s no information given about the space between marks on the number-line. So we are not sure whether all the marks are equally spaced. So in that scenario, remember that the marks may be evenly spaced or not evenly spaced. Therefore, you need to be careful!

Now, let’s start with first statement:

i) *y* > *x* + *b*

As you can see clearly, *y* > *x*

Furthermore, *b* is a negative number. How much is its value, is not a matter of significance. You have enough information that *b* is negative. So, when a negative value is added to a positive value *x*, the result will be less than *x*.

Now, as *y* is greater than *x*, therefore, *y* MUST always be greater than *x* + *b*.

**Therefore, statement (i) MUST be true.**

Now, let’s analyze second statement:

ii) *y* + *b* > *x* + *a*

Again, here you may quickly see that

*y* > *x* ———— (eq. 1)

and,

*b* > *a* ———— (eq. 2)

Now, from equation 1 and 2, we can conclude that *y* + *b* > *x* + *a*

**Therefore, statement (ii) MUST be true as well.**

Now, let’s analyze third statement:

iii) *a* × *b* > *y*

Here, both *a* and *b* are negative. But the product of two negatives is always positive. Therefore, the product of *a* and *b* could be greater than *y*. But this product may also be less than *y* or even it may be equals to y. Therefore, we do not have sufficient information whether the product of *a* and *b* is greater than *y*.

Therefore, **Therefore, statement (iii) is NOT a MUST be true statement.**

Also, in advance study plan, we’ll learn number-line question scenarios that have came in past exams. There you’ll learn in much detail that will clear your every concept in this topic. Furthermore, practice questions in advance study plan and expert level study plan will make you expert in this topic. 🙂

**Fractions and decimals:**

All those numbers that have decimals are fractions, and decimals are those digits that are placed after dot / point (.), e.g.,

Don’t confuse ^{8}⁄_{4} as decimal or fraction, it can further be simplified as 2, i.e.,

^{8}⁄_{4} = 2, which is a non-fraction.

**Parts of Fractions:**

There are basically two parts of fractions. One is known as ‘nominator’, and other is known as ‘denominator’. The value that placed upper side of the fraction bar is ‘nominator’, and the value below the bar of the fraction is named as ‘denominator’. i.e

In above fraction, value above fractional bar is ‘1’ (numerator), and the value below the fraction bar is ‘2’ (denominator).

**Different places of digits in a number:**

For instance, consider a number below:

4321.765

(In words: *Four thousand three hundred twenty one point seven six five*)

In this number:

Unit digit = 1

Ten**s** digit = 2

Hundred**s** digit = 3

Thousand**s** digit = 4

Ten**th** digit = 7

Hundred**th** digit = 6

Thousand**th** digit = 5

Note that there’s only one unit digit exists in any number.

Also in an integer, the unit digit is the first digit from the right side of the integer. For instance

245

This is an integer, which can be written as

245.00

Here unit digit is 5.

We’ll discuss Fractions & decimals further later on in detail, after you complete *FREE Basic Math Concept-01*.

**Approximation of Non-Integer to Integer or to another non-integer:**

Let’s approximate 424.89 to nearest *0.1*, i.e approximate the number to nearest tenth digit.

424.89 ≈ 424.9

The reason is simple, the above number can be written as:

424.890 = 424.800 + 0.090

If we remove 0.090, it’ll make bigger change than if we add 0.010 to the above number in order to get a value nearest tenth digit. Because removing 0.090 will reduce the number by larger amount than adding 0.010 to the number. Therefore, according to the rule, we need to find nearest; so we must add 0.010 to the number.

⇒ 424.89 **≈ 424.9** **Answer**

Let’s approximate 424.83 to nearest *0.1*, i.e approximate the number to nearest tenth digit.

424.83 ≈ 424.8

Again the reason is same as discussed above.

424.83 = 424.80 + 0.03

Either we need to remove 0.03 or add 0.07. But 0.07 will make bigger change and will not give answer to nearest, therefore we’ll remove 0.03 to get answer nearest to tenth digit.

⇒ 424.83 **≈ 424.8** **Answer**

But what if the digit is exactly 0.05? Let’s discuss this scenario:

Let’s approximate 424.85 to nearest *0.1*, i.e approximate the number to nearest tenth digit.

424.85 ≈ 424.9

Here, many people will ask we either can remove 0.05 or add 0.05, thereby making same change. Then why we’ll add 0.05 rather than subtract?

The answer is that it’s a rule in which all test governing bodies (i.e ETS, GMAC, LSAC, and NTS etc) believe in.

Therefore, you should remember that if the rounding digit (i.e hundredth digit in above examples) is less than 5, the resultant digit (i.e tenth digit in above examples) will remain the same. If the rounding digit is 5 or greater than 5, the the resultant digit will increase by 1 unit.

To make further clear your understanding, let’s approximate 424.849 to nearest tenth digit.

424.849 ≈ 424.8

Because 0.049 has hundredth digit 4, which is less than 5; therefore the resultant digit(i.e tenth digit) will remain 8.

**Arithmetic & PEMDAS Rule:**

This rule is very famous and applied in every arithmetic calculations. Violation from this may result to wrong answer. Many people commit mistake in this rule while solving an arithmetic expression or equation.

According to PEMDAS rule:

**1.** Every calculation must begin from the smallest parenthesis *“( )”* and then gradually move to the larger parenthesis *“{ }”* and finally largest parenthesis *“[ ]”*.

**2.** Exponential form of expressions must solved first before other arithmetic operations (i.e ×, ÷, +, and –).

**3.** Then do either multiplication or division as convenient.

**4.** Finally do either addition or subtraction as convenient.

**Multiplication & Division:**

**Very important Points:**

In some instances where multiplications and divisions are adjacent to each other, always start solving from left side and gradually move towards right side.

For instance,

4 ÷ 2 × 2 = ?

Incorrect ways: 4 ÷ 2 × 2 = 4 ÷ 4 = 1

As I said, whenever you see multiplication and division adjacent to each other, start solving from left to right

Therefore, Correct way is:

4 ÷ 2 × 2 = (4 ÷ 2) × 2 = (2) × 2 = 4 (First doing division, and then multiplication)

**Addition & Subtraction:**

**How to add big integers without calculator?**

**Addition of decimals:**

In case when the decimal add to give result 10 or more, then use the same method of addition as discussed above, i.e only write the unit digit after addition and take 1 carry to the next column for addition. Do this unless you’ll reach to the final column. And then write down the complete result in addition of final column of the addition of digits. Column represents each calculation that we made in addition of big integers. i.e first column is 4 + 5, second column is 9 + 4, third column is (1)+5+9 and so on. Did you noticed that in last column (1)+4+5 resulted to 10. So we’ll no more wright only unit digit here, and write down complete result in this last column addition. This similar process we’ll follow in case of addition of digits.

**Addition of Integer and decimal combined:**

**Basic rules of addition:**

Let’s consider *a*, *b* & *c* are numbers.

*a + b = b + a*

*a + (b + c) = (a + b) + c*

If *a = b*, then, *a + c = b + c*. In other words, if we add both sides of an equation with a number(i.e. adding ‘c’ on both sides), Left hand side would remain equal to Right hand sight of the equation. So we can add any number on both sides of equation.

**Subtraction of big integers:**

**Important Note:** *In subtraction operation, when the upper digit is less than lower digit(as in above figure, upper digit 4 which is less than lower digit 9), we need to take 1 carry from next left upper digit (as in above figure, next left upper digit is 9, which is left of upper digit 4).*.

**Basic rules of Subtraction:**

Let’s consider *a*, *b* & *c* are numbers.

*a – b = a + (–b)*

*a – b = a + (–b)*

Therefore, *a – b* ≠ *b – a*

And,

*a – (b – c) = a – b + c* {–ve sign will multiply inside parenthesis and will change signs of *b* & *c*

Therefore, *a – (b – c)* ≠ *(a – b) – c*

But

*(a – b) – c = a – b – c*

As if we open the parenthesis, nothing will change.

Also,

If *a = b*, then, *a – c = b – c*. In other words, if we subtract both sides of an equation with a number, Left hand side would remain equal to Right hand sight of the equation. So we can subtract any equation on both sides by same number.

**Addition & Subtractions combined:**

At this point we need to learn how to deal with addition & subtraction at a time in an expression. For this purpose, let’s solve:

4 + 3 – 5 = ?

First, we need to solve addition and then, come to subtraction. i.e.,

4 + 3 – 5 = (4 + 3) – 5 = 7 – 5

= 2

Similarly,

Let’s solve:

– 44 + 22 + 84 + 61 – 88 – 20 = ?

Again, place positive numbers altogether,

– 44 + 22 + 84 + 61 – 88 – 20 = 22 + 84 + 61 – 44 – 88 – 20

By Taking –ve common, = (22 + 84 + 61) – (44 + 88 + 20)

After simplifying the above, we’ll get; = 167 – 152

Now, by subtracting 152 from 167, we’ll get = 15

**Multiplication of Negatives vs addition of negatives:**

Few students who are new to arithmetic have asked me difference between multiplying two or more negative numbers and adding two or more negative numbers. In this final topic of this *FREE Basic Math Concept-01*, we’ll learn this difference and solution thereafter.

Before learning this, you should know that if the two numbers are adjacent to each other with parenthesis, this means that the two numbers are multiplying with each others. i.e:

(–3)(–4) = (–3) × (–4)

And remember:

(–3)(–4) ≠ –3 – 4

Now, you are ready to attempt a mini Quiz. Click on *‘Mini Quiz’* button below:

This is a really good refresher. Things are summarized in a simple, step by step way. Grammatically errors are very common, so one should be careful, and the author should try to correct the mistakes. Keep up the good work!

Hi Tayyab,

Thanks for the comment and suggestion. I have not made proof read so far, but will soon do it for grammatical correction and further improvement in content.

If the sum of 75 integers is 1023 and average of these integers is I+37, then shouldn’t be

75 (I+37) = 1023

whereas in the above discussion it is mentioned that

I+37 = 37

Please advise.

Hi Ahmed,

As it is given in question that average of 75 integers is 1023. And in terms of I, the average is I + 37.

So simply equate these i.e. I + 37 = 1023, because both are average of those 75 integers and thus equal.

Sorry for typo

It is mentioned as

I+37 = 1023

Author!

You done a great job, really informative. Keep it up.

is this for gat preparation ? I’m confused coz top of page Gmat preparation is mentioned

Hi Tahira,

It’s for GMAT, GRE, and GAT.

As this free beginners study plan is free, so its same for all. Later on advance level study plan is different because GMAT and GRE are hard level exams, and covers same topics but in hard level scenarios.