These are the foundation of the Number System. For RRB NTPC, Group D, ALP and similar exams, you should know the definitions, properties, comparisons, and question patterns.
Natural numbers are the numbers we use for counting.
N = {1,2,3,4,5,6,...}
So:
Which are natural numbers?
3, 17, 100, 999
All are natural numbers.
But:
-3, 0, 1/2
are not natural numbers.
Whole numbers consist of 0 and all natural numbers.
W = {0,1,2,3,4,5,...}
Therefore:
W=N∪{0}
| Number | Natural? | Whole? |
|---|---|---|
| 0 | ❌ | ✅ |
| 1 | ✅ | ✅ |
| 5 | ✅ | ✅ |
| 100 | ✅ | ✅ |
| -5 | ❌ | ❌ |
| 1/2 | ❌ | ❌ |
Every natural number is a whole number.
But every whole number is not a natural number because:
0 is whole but not natural
Integers include:
Therefore:
Z={...,-5,-4,-3,-2,-1,0,1,2,3,4,5,...}
Negative integers
-1,-2,-3,-4,...
Zero
0
Positive integers
1,2,3,4,...
This is extremely important.
N⊆ W⊆ Z
Think of them as boxes:
Natural Numbers
→ 1, 2, 3, 4...
⬇
Whole Numbers
→ 0, 1, 2, 3, 4...
⬇
Integers
→ ..., -3, -2, -1, 0, 1, 2, 3...
Therefore:
Every natural number is a whole number and every whole number is an integer.
But the reverse is not true.
For example:
Integers can be represented on a number line:
... -3 -2 -1 0 1 2 3...
Moving right means numbers become larger.
Moving left means numbers become smaller.
Which is greater?
-3 or -7?
On the number line, -3 lies to the right of -7.
Therefore:
-3>-7
This is a very common mistake.
Any integer greater than zero:
1,2,3,4,...
Any integer less than zero:
-1,-2,-3,-4,...
0 is neither positive nor negative
This is important for exams.
The number immediately after a number.
Successor of n=n+1
Example:
Successor of 8:
8+1=9
The number immediately before a number.
Predecessor of n=n-1
Example:
Predecessor of 8:
8-1=7
Successor of -5:
-5+1=-4
Predecessor of -5:
-5-1=-6
999+1=1000
1000-1=999
-10+1=-9
-10-1=-11
Numbers occurring one after another are called consecutive numbers.
Example:
4,5,6,7,8
These are consecutive integers.
If the first number is n, then three consecutive integers are:
n, n+1, n+2
Four consecutive integers:
n, n+1, n+2, n+3
Example:
2,4,6,8
Difference between consecutive even numbers = 2.
Therefore, four consecutive even numbers can be written as:
n,n+2,n+4,n+6
Example:
1,3,5,7
Difference = 2.
Four consecutive odd numbers:
n,n+2,n+4,n+6
where n is odd.
frac{n(n+1)2}
Example:
Find:
1+2+3+...+10
=frac{10(11)}2
=55
n^2
Example:
1+3+5+7+9
There are 5 odd numbers.
5^2=25
n(n+1)
Example:
2+4+6+8+10
There are 5 terms.
5(6)=30
This is an important RRB question type.
100
Use:
Last-First+1
50-25+1=26
Therefore:
26
50-0+1=51
Therefore:
51
Use:
Last-First+1
10-(-10)+1
=10+10+1
21
15-(-20)+1
=15+20+1
36
Positive integers are:
1,2,3,...,10
Therefore:
10
Negative integers:
-10,-9,...,-1
Therefore:
10
Be careful about the word between.
Numbers:
6,7,8,9
Answer:
4
Formula:
10-5-1=4
-4,-3,-2,-1,0,1,2,3,4
Answer:
9
Formula:
5-(-5)-1=9
This is an important exam trap.
Both 10 and 20 are included.
20-10+1=11
Answer:
11
10 and 20 are excluded.
20-10-1=9
Answer:
9
Same signs → add and keep the sign.
5+3=8
-5+(-3)=-8
Different signs → subtract and keep the sign of the number with greater absolute value.
-8+5=-3
8+(-5)=3
Remember:
a-b=a+(-b)
Example:
5-8
=5+(-8)
=-3
Example:
-5-(-8)
Change subtraction of negative into addition:
-5+8=3
Therefore:
3
(+)(+)=+
(-)(-)=+
Examples:
5times3=15
(-5)(-3)=15
(+)(-) = -
(-)(+) = -
Example:
(-5)(3)=-15
The same sign rule applies.
(-20)/(-5)=4
frac{(-20)}5=-4
A) -5
B) 0
C) 7
D) -2
Answer: C
A) 1
B) 2
C) 0
D) 5
Answer: C
A) 5
B) 0
C) -7
D) 10
Answer: C
A) Every integer is a whole number
B) Every whole number is an integer
C) Every integer is a natural number
D) Every natural number is negative
Answer: B
A) -10
B) -8
C) 8
D) 10
Answer: B
A) -8
B) -10
C) 8
D) 10
Answer: B
35-15+1=21
Answer: 21
15-(-15)+1
=31
Answer: 31
10-(-10)-1
=19
Answer: 19
A) -12
B) -5
Answer: B
Because -5 is to the right of -12 on the number line.
Let them be:
n, n+1, n+2
Then:
n+n+1+n+2=72
3n+3=72
3n=69
n=23
Therefore:
23,24,25
Let:
n,n+1,n+2
3n+3=0
n=-1
Numbers:
-1,0,1
Let the middle number be n.
Then:
n-2,n-1,n,n+1,n+2
Their sum is:
5n=50
n=10
Therefore:
8,9,10,11,12
frac{25(26)}2
=25times13
325
n^2
20^2=400
n(n+1)
20(21)=420
A) 1
B) -1
C) 0
D) 10
Answer: C — 0
Strictly between:
200-100-1
99
200-100+1
101
A) n,n+2,n+4,n+6
B) n,n+1,n+2,n+3
C) n,n+3,n+6,n+9
D) n,n+4,n+8,n+12
Answer: B
| Concept | Key point |
|---|---|
| Natural numbers | 1,2,3,... |
| Whole numbers | 0,1,2,3,... |
| Integers | ...,-2,-1,0,1,2,... |
| Relationship | N⊆ W⊆ Z |
| 0 | Whole & integer; neither positive nor negative |
| Successor | n+1 |
| Predecessor | n-1 |
| Consecutive integers | Difference = 1 |
| Consecutive even/odd | Difference = 2 |
| First n natural sum | frac{n(n+1)}2 |
| First n odd sum | n^2 |
| First n even sum | n(n+1) |
| Numbers from a to b, inclusive | b-a+1 |
| Integers strictly between a,b | b-a-1 |
