Sigma (Σ) in mathematics is a symbol that represents summation, which is a way to concisely write the sum of a sequence of numbers. It's often called sigma notation or summation notation.
Here's how to use sigma in maths:
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The Basic Structure:
- Σ (sigma) is the symbol for summation.
- Below the sigma, you write the index of summation (often denoted by i or k), along with its starting value.
- Above the sigma, you write the final value of the index.
- To the right of the sigma, you write the expression you want to sum.
For example, the expression:
∑_(i=1)^5 i^2Represents the sum of the squares of integers from 1 to 5.
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Expanding the Sum:
- To expand the sum, you substitute the index (i) with each value from its starting point to its final point, adding each result.
- In the example above, we would get:
∑_(i=1)^5 i^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25 = 55
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Common Applications:
- Series: Sigma notation is used to represent infinite series (like the sum of all natural numbers).
- Calculating Averages: To find the average of a set of numbers, you can use sigma to sum the values and divide by the total number of values.
- Statistics: Sigma notation is fundamental in statistical calculations, like calculating variance and standard deviation.
Examples:
- Sum of the first ten even numbers:
∑_(i=1)^10 2i = 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 = 110 - Sum of the squares of the first five odd numbers:
∑_(i=1)^5 (2i - 1)^2 = 1^2 + 3^2 + 5^2 + 7^2 + 9^2 = 1 + 9 + 25 + 49 + 81 = 165
In summary, Sigma notation is a powerful tool to express sums concisely and efficiently in mathematical expressions. It helps us perform calculations and understand complex formulas in various fields, from statistics and calculus to physics and engineering.
