Successor and Predecessor
Think of standing in a queue. The person standing just behind you is like your predecessor, and the person standing just in front of you is like your successor. Numbers work the same way.
Every whole number has a number that comes just before it and a number that comes just after it. The number that comes just after is called the successor, and the number that comes just before is called the predecessor.
This is a simple but important idea from the Whole Numbers chapter in Class 6 Maths. It helps you understand how numbers are arranged in order and how counting works.
What is Successor and Predecessor - Grade 6 Maths (Whole Numbers)?
Definition: The successor of a whole number is the number obtained by adding 1 to it.
- Successor of n = n + 1
- Example: Successor of 15 = 15 + 1 = 16
Definition: The predecessor of a whole number is the number obtained by subtracting 1 from it.
- Predecessor of n = n − 1
- Example: Predecessor of 15 = 15 − 1 = 14
Key facts:
- Every whole number has a successor. The successor of 0 is 1, the successor of 1 is 2, and so on.
- Every whole number except 0 has a predecessor. The predecessor of 1 is 0, the predecessor of 0 would be −1 (which is not a whole number).
- There is no largest whole number because you can always find a successor by adding 1.
- The predecessor of the successor of a number is the number itself. If you add 1 and then subtract 1, you get back to where you started.
Successor and Predecessor Formula
Formulas:
Successor of n = n + 1
Predecessor of n = n − 1
Where:
- n is any whole number (0, 1, 2, 3, ...)
- The successor is always 1 more than the number.
- The predecessor is always 1 less than the number.
On a number line:
- The successor is the number immediately to the right.
- The predecessor is the number immediately to the left.
Derivation and Proof
Understanding through a number line:
Imagine the number line:
... 3 — 4 — 5 — 6 — 7 ...
For the number 5:
- The number just to the left is 4 → Predecessor of 5 = 4 (that is, 5 − 1 = 4)
- The number just to the right is 6 → Successor of 5 = 6 (that is, 5 + 1 = 6)
Why does 0 have no predecessor in whole numbers?
- Whole numbers are: 0, 1, 2, 3, 4, ...
- The predecessor of 0 would be 0 − 1 = −1.
- But −1 is a negative number, not a whole number.
- So within the set of whole numbers, 0 has no predecessor.
Connection to natural numbers:
- Natural numbers start from 1 (1, 2, 3, 4, ...).
- In natural numbers, 1 has no predecessor (since 0 is not a natural number).
- In whole numbers, 1 has predecessor 0, but 0 has no predecessor.
Types and Properties
Types of successor and predecessor problems:
Type 1: Finding the successor of a given number
- Just add 1 to the number.
- Successor of 999 = 1,000. Successor of 99,999 = 1,00,000.
Type 2: Finding the predecessor of a given number
- Just subtract 1 from the number.
- Predecessor of 1,000 = 999. Predecessor of 10,00,000 = 9,99,999.
Type 3: Finding a number when successor or predecessor is given
- If the successor of a number is 450, the number is 450 − 1 = 449.
- If the predecessor of a number is 7,999, the number is 7,999 + 1 = 8,000.
Type 4: Identifying numbers with no predecessor
- In whole numbers, 0 has no predecessor.
- In natural numbers, 1 has no predecessor.
Type 5: Special numbers at boundaries
- Successor of 99 = 100 (2-digit to 3-digit)
- Successor of 9,999 = 10,000 (4-digit to 5-digit)
- Predecessor of 100 = 99 (3-digit to 2-digit)
- These boundary cases are interesting because the number of digits changes.
Solved Examples
Example 1: Example 1: Successor of Simple Numbers
Problem: Find the successor of: (a) 23 (b) 78 (c) 0
Solution:
- (a) Successor of 23 = 23 + 1 = 24
- (b) Successor of 78 = 78 + 1 = 79
- (c) Successor of 0 = 0 + 1 = 1
Example 2: Example 2: Predecessor of Simple Numbers
Problem: Find the predecessor of: (a) 50 (b) 100 (c) 1
Solution:
- (a) Predecessor of 50 = 50 − 1 = 49
- (b) Predecessor of 100 = 100 − 1 = 99
- (c) Predecessor of 1 = 1 − 1 = 0
Example 3: Example 3: Successor at Number Boundaries
Problem: Find the successor of: (a) 99 (b) 999 (c) 9,999 (d) 99,999
Solution:
- (a) Successor of 99 = 99 + 1 = 100 (2-digit → 3-digit)
- (b) Successor of 999 = 999 + 1 = 1,000 (3-digit → 4-digit)
- (c) Successor of 9,999 = 9,999 + 1 = 10,000 (4-digit → 5-digit)
- (d) Successor of 99,999 = 99,999 + 1 = 1,00,000 (5-digit → 6-digit, equals 1 lakh)
Example 4: Example 4: Predecessor at Number Boundaries
Problem: Find the predecessor of: (a) 1,000 (b) 10,000 (c) 1,00,000
Solution:
- (a) Predecessor of 1,000 = 1,000 − 1 = 999 (4-digit → 3-digit)
- (b) Predecessor of 10,000 = 10,000 − 1 = 9,999 (5-digit → 4-digit)
- (c) Predecessor of 1,00,000 = 1,00,000 − 1 = 99,999 (6-digit → 5-digit)
Example 5: Example 5: Finding a Number from Its Successor
Problem: The successor of a number is 5,001. What is the number?
Solution:
- If successor = 5,001, then the number = successor − 1
- Number = 5,001 − 1 = 5,000
Answer: The number is 5,000.
Example 6: Example 6: Finding a Number from Its Predecessor
Problem: The predecessor of a number is 24,999. What is the number?
Solution:
- If predecessor = 24,999, then the number = predecessor + 1
- Number = 24,999 + 1 = 25,000
Answer: The number is 25,000.
Example 7: Example 7: Successor and Predecessor of the Same Number
Problem: Find both the successor and predecessor of 4,500.
Solution:
- Successor of 4,500 = 4,500 + 1 = 4,501
- Predecessor of 4,500 = 4,500 − 1 = 4,499
So the three numbers in order are: 4,499, 4,500, 4,501.
Example 8: Example 8: True or False
Problem: State True or False:
(a) The successor of the predecessor of 100 is 100.
(b) 0 is the predecessor of 0 in whole numbers.
(c) Every natural number has a successor.
Solution:
- (a) Predecessor of 100 = 99. Successor of 99 = 100. True.
- (b) Predecessor of 0 = −1, which is not a whole number. So 0 does not have a predecessor in whole numbers. The statement is False.
- (c) Yes, for any natural number n, the successor n + 1 is also a natural number. True.
Example 9: Example 9: Largest and Smallest Whole Numbers
Problem: Is there a largest whole number? Is there a smallest whole number?
Solution:
- Smallest whole number: Yes, it is 0. No whole number is smaller than 0.
- Largest whole number: No. For any whole number n, the successor n + 1 is also a whole number and is bigger. So you can never reach a "largest" whole number.
Example 10: Example 10: Cricket Runs — Word Problem
Problem: Virat scored 99 runs in one match and 100 runs in the next match. Are these scores successor and predecessor of each other?
Solution:
- Predecessor of 100 = 100 − 1 = 99
- Successor of 99 = 99 + 1 = 100
- Yes, 99 and 100 are predecessor and successor.
Answer: Yes. 99 is the predecessor of 100, and 100 is the successor of 99.
Real-World Applications
Real-life uses of successor and predecessor:
- Counting: When you count 1, 2, 3, 4, ..., each number is the successor of the one before it. Counting is simply finding successors one after another.
- Page numbers: In a book, page 45 comes after page 44 (successor) and before page 46 (predecessor). If a page is torn and you see pages 112 and 114, the missing page is 113 (successor of 112, predecessor of 114).
- Calendar dates: The predecessor of January 1 is December 31 (of the previous year). The successor of March 31 is April 1.
- Roll numbers: In a class, if your roll number is 27, the student just before you is 26 (predecessor) and just after is 28 (successor).
- House numbers: On a street, house number 15 is the successor of 14 and predecessor of 16.
- Building the number system: The idea that every number has a successor is what makes the set of whole numbers infinite — you can always add 1 more.
Key Points to Remember
- The successor of a whole number n is n + 1.
- The predecessor of a whole number n is n − 1.
- Every whole number has a successor.
- Every whole number except 0 has a predecessor (in the set of whole numbers).
- 0 is the smallest whole number — it has no predecessor among whole numbers.
- There is no largest whole number because every number has a successor.
- The successor of the predecessor of a number is the number itself: (n − 1) + 1 = n.
- The predecessor of the successor of a number is the number itself: (n + 1) − 1 = n.
- When the successor of a number like 99, 999, or 9,999 is found, the number of digits increases by 1.
- When the predecessor of a number like 100, 1,000, or 10,000 is found, the number of digits decreases by 1.
Practice Problems
- Find the successor of: (a) 1,234 (b) 9,99,999 (c) 49,999
- Find the predecessor of: (a) 8,000 (b) 1,00,001 (c) 5,00,000
- The predecessor of a number is 7,649. What is the number?
- The successor of a number is 30,000. What is the number and its predecessor?
- Write three consecutive whole numbers whose middle number is 10,000.
- Is 0 the predecessor of 1? Is −1 the predecessor of 0 in whole numbers? Explain.
- What is the successor of the largest 4-digit number?
- Find a whole number whose successor and predecessor are both 2-digit numbers.
Frequently Asked Questions
Q1. What is the successor of a number?
The successor of a number is the number that comes just after it. To find the successor, add 1. Successor of 56 = 56 + 1 = 57.
Q2. What is the predecessor of a number?
The predecessor of a number is the number that comes just before it. To find the predecessor, subtract 1. Predecessor of 56 = 56 − 1 = 55.
Q3. Does 0 have a predecessor in whole numbers?
No. The predecessor of 0 would be −1, but −1 is not a whole number. So within the set of whole numbers (0, 1, 2, 3, ...), 0 has no predecessor. In integers (..., −2, −1, 0, 1, 2, ...), 0 does have a predecessor: −1.
Q4. Is there a whole number that has no successor?
No. Every whole number has a successor. For any number n, n + 1 is also a whole number. This is why there are infinitely many whole numbers.
Q5. What is the successor of 99?
Successor of 99 = 99 + 1 = 100. Notice that 99 is a 2-digit number but its successor 100 is a 3-digit number.
Q6. What is the predecessor of 1,000?
Predecessor of 1,000 = 1,000 − 1 = 999. The number of digits decreases from 4 to 3.
Q7. What is the difference between successor and predecessor?
The successor is the number obtained by adding 1 (comes after). The predecessor is the number obtained by subtracting 1 (comes before). They are opposite operations. The difference between the successor and predecessor of any number is always 2.
Q8. Can two different numbers have the same successor?
No. Each whole number has a unique successor. If the successor of a is 10 and the successor of b is also 10, then a = b = 9. No two different numbers can have the same successor.
Q9. What is the smallest natural number that has a predecessor which is also a natural number?
The number 2. Its predecessor is 1, which is a natural number. The number 1 has predecessor 0, but 0 is NOT a natural number.
Q10. How are successor and predecessor related to counting?
Counting is the process of finding successors one after another. When you count 1, 2, 3, 4, 5, ..., each number is the successor of the previous one. Counting backwards (5, 4, 3, 2, 1) means finding predecessors.
