Successor and Predecessor in mathematics are terms primarily used in understanding the sequence of whole numbers.
In mathematics, the terms predecessor and successor given n are frequently used in the context of sequences and series of whole numbers. A predecessor refers to the number that comes immediately before aumber, while a successor denotes the number that follows it directly after.
In this article, we will learn about Predecessor Definition, Successor Definition, Examples of Successors and Predecessor, and others in detail.
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What are Successor and Predecessor?
Successor and Predecessor are the numbers just after and before a particular number, respectively. The concept of predecessor and successor can be best understood with the help of a number line.
- On a number line, a successor refers to the number immediately to the right of a given number when moving along the line
- On the number line with 0 as the starting point, moving to the right, 1 is the successor of 0, and moving further, 2 becomes the successor of 1.
- A predecessor is the number immediately to the left.
- Similarly, moving to the left from 1, 0 is the predecessor of 1, and moving further, -1 becomes the predecessor of 0.
- This relationship can be visualized on a number line, where each number has a successor to its right and a predecessor to its left, aiding in understanding the sequential arrangement of numbers.
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Successor
Successor denotes the value succeeding a specific number or term. If 'n' represents any whole number, its successor is 'n+1'. Alternative terms for a successor include following, directly subsequent to, and immediate next value/number. Let's learn the definition of successor.
Successor Definition
Successor of a given number is defined as the number that comes immediately after it in a sequence, especially in the context of the natural numbers.
Examples of Successor
When n is assigned the value 5 the subsequent number after 5 known as the successor is found by adding 1 to 5 resulting in 6.
In another instance when n equals 10 the successor of 10 is determined by adding 1 to 10 yielding 11.
- Similarly, for n = 20 the successor is calculated as 20 + 1 producing 21.
- The pattern continues as for n = 50 and n = 100 where the successors are computed as 51 and 101 respectively.
Consequently, based on these instances it can be inferred that the successor of any given whole number is obtained by adding one to the original number.
Predecessor
As mentioned earlier, a predecessor refers to the value that directly precedes or comes immediately before a specific value. If a value is represented as 'x' (where 'x' is any whole number), then the predecessor of 'x' is calculated as 'x - 1'. Hence, determining the predecessor of any given value involves subtracting 1 from the given value.
Predecessor Definition
Predecessor of a number refers to the number that comes immediately before it in a sequence. In the context of the natural numbers, the predecessor of a number n is n−1.
Examples of Predecessor
If a variable 'x' equals 5 the number just before it known as its predecessor is obtained by subtracting 1 from 5, resulting in 4.
- When 'x' is set to 10, its predecessor is determined by subtracting 1 from 10 yielding 9.
- Similarly, for 'x' values such as 20, 50, and 100, their respective predecessors are 19, 49, and 99, achieved by subtracting 1 from these numbers.
Therefore, the predecessor of any integer can be found by reducing the number by 1.
How to Find Successors and Predecessors?
We can find the successor and predecessor of a number by the following Method:
Finding Successor
To determine the successor of a given integer 'n', add 1 to that number.
- For instance, if 'n' is 5, the successor would be 5+1=6.
Finding Predecessor
To find the predecessor of a given integer 'n', subtract 1 from that number.
- For example, if 'n' is 10, the predecessor would be 10−1 = 9.
These operations work for positive integers in a number line, providing the next or previous number in the sequence.
Successor and Predecessor of Integers
Understanding successors and predecessors in integers is fundamental in mathematics.
Successor: The successor of any given integer is the immediate number that comes after it in the number line. For instance, if we take the integer 5, its successor is 6. This relationship can be expressed as n+1, where 'n' represents the given integer. In the context of positive integers, the successor is always one more than the given number.
Predecessor: Conversely, the predecessor of an integer is the number that comes immediately before it on the number line. For example, if we consider the integer 10, its predecessor is 9, denoted as n−1. In the realm of positive integers, the predecessor is always one less than the given number.
Here's a simple example,
For the integer 20, the successor is 21, and the predecessor is 19.
Understanding the successor and predecessor helps in comprehending number sequencing and their positions in the numerical continuum. These concepts are essential in various mathematical operations and sequences.
Difference Between Successor and Predecessor
The difference between Successor and Predecessor is tabulated below:
Successor vs Predecessor | |
|---|---|
Successor | Predecessor |
The successor of a number is the next number after it. For instance, the successor of 5 is 6. | The predecessor of a number is the number that comes before it. For example, the predecessor of 10 is 9. |
It is found by adding 1 to the given number. | It is obtained by subtracting 1 from the given number. |
For any whole number 'n', its successor is 'n + 1'. | For any whole number 'n', its predecessor is 'n - 1'. |
Example: Successor of 25 = 25 + 1 = 26 | Example: Predecessor of 50 = 50 - 1 = 49 |
It represents the next term in the sequence. | It signifies the previous term in the sequence. |
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Successor and Predecessor Examples
Example 1: Find the successor and predecessor of 45?
Solution:
Successor of 45: 45 + 1 = 46
Predecessor of 45: 45 - 1 = 44
Hence, the successor of 45 is 46, and the predecessor is 44.
Example 2: Determine the successor and predecessor of 83?
Solution:
Successor of 83: 83 + 1 = 84
Predecessor of 83: 83 - 1 = 82
Therefore, the successor of 83 is 84, and the predecessor is 82.
Example 3: Calculate the successor and predecessor of 100?
Solution:
Successor of 100: 100 + 1 = 101
Predecessor of 100: 100 - 1 = 99
So, the successor of 100 is 101, and the predecessor is 99.
Example 4: For the number 25, determine its successor and predecessor?
Solution:
Successor of 25: 25 + 1 = 26
Predecessor of 25: 25 - 1 = 24
Thus, the successor of 25 is 26, and the predecessor is 24.
Example 5: Identify the successor and predecessor of 6?
Solution:
Successor of 60: 60 + 1 = 61
Predecessor of 60: 60 - 1 = 59
Hence, the successor of 60 is 61, and the predecessor is 59.
Practice Questions on Successor and Predecessor
1. Find the successor and predecessor of 78?
2. Determine the successor and predecessor of 95?
3. Calculate the successor and predecessor of 120?
4. For the number 56, identify its successor and predecessor?
5. Identify the successor and predecessor of 88?