# The answer to 5 − 5 × 5 + 5

$5 - 5 \times 5 + 5$ is calculated as follows:

According to the order of operations (PEMDAS/BODMAS), multiplication comes before addition and subtraction.

So, $5 \times 5 = 25$.

Now substitute back into the expression:

$5 - 25 + 5$Perform the operations from left to right:

- $5 - 25 = -20$
- $-20 + 5 = -15$

**Answer:** $-15$.

## What is PEMDAS/BODMAS ?

### PEMDAS

**P**: Parentheses**E**: Exponents**MD**: Multiplication and Division (left to right)**AS**: Addition and Subtraction (left to right)

In this order, calculations are completed as:

- Parentheses (brackets).
- Exponents (powers and roots).
- Multiplication and Division (from left to right).
- Addition and Subtraction (from left to right).

### BODMAS

**B**: Brackets**O**: Orders (another term for exponents, including powers and roots)**DM**: Division and Multiplication (left to right)**AS**: Addition and Subtraction (left to right)

This order is the same as PEMDAS but uses different terms (Brackets instead of Parentheses, Orders instead of Exponents).

Both help to prioritize operations correctly in expressions so everyone gets the same result.

Here’s an example for both PEMDAS and BODMAS to show how they work in solving mathematical expressions.

### Example 1: Using **PEMDAS**

Expression:

$3 + 6 \times (5 + 4)^2 - 3$

Steps:

$5 + 4 = 9$**P**(Parentheses): Solve inside parentheses first:So the expression becomes:

$3 + 6 \times 9^2 - 3$

$9^2 = 81$**E**(Exponents): Calculate the exponent:Now we have:

$3 + 6 \times 81 - 3$

$6 \times 81 = 486$**MD**(Multiplication and Division, left to right):The expression is now:

$3 + 486 - 3$

$3 + 486 = 489$ $489 - 3 = 486$**AS**(Addition and Subtraction, left to right):

**Final answer**: $486$

### Example 2: Using **BODMAS**

Expression:

$8 + (12 \div 4) \times (3 + 5) - 6$

Steps:

**B**(Brackets): Solve inside each bracket:- $12 \div 4 = 3$
- $3 + 5 = 8$

The expression becomes:

$8 + 3 \times 8 - 6$**O**(Orders/Exponents): There are no exponents, so skip this step.

$3 \times 8 = 24$**DM**(Division and Multiplication, left to right):Now the expression is:

$8 + 24 - 6$

$8 + 24 = 32$ $32 - 6 = 26$**AS**(Addition and Subtraction, left to right):

**Final answer**: $26$

In both examples, following the PEMDAS/BODMAS order ensures correct results.

The order of operations conventions, commonly known as **PEMDAS** or **BODMAS**, were not invented by a single person at a specific time but rather developed gradually over centuries as mathematical notation and practices evolved.

## When does these Systems of Calculation started and by Whom ?

### Early Roots in Algebraic Rules

**Ancient Mathematics**: Early forms of mathematics, such as those practiced by the Egyptians, Greeks, and Babylonians, followed specific methods for arithmetic operations, but there were no universally agreed-upon rules for the order of operations.**Medieval Mathematicians**: By the Middle Ages, mathematicians in Islamic and European cultures began formalizing algebra, which influenced the adoption of rules for simplifying expressions. Scholars like**Al-Khwarizmi**(9th century), the "father of algebra," laid foundational concepts that would later be refined into more formalized algebraic methods.

### Development of Modern Notation (1500s - 1700s)

**Robert Recorde**(1512–1558), an English mathematician, introduced the equals sign**(=)**in 1557, and mathematicians like**François Viète**and**René Descartes**introduced variables and other symbolic notations, creating a need for standard rules for operations.**Mathematical Notation**: As algebraic notation was standardized during the 17th century, especially in Europe, so were the implicit rules for order of operations, although they were not strictly formalized in the way we recognize them today.

### Formalization of PEMDAS/BODMAS (1800s - 1900s)

**1800s Textbooks**: By the 19th century, as algebra was increasingly taught in schools, textbooks began to include specific rules to guide students on how to interpret mathematical expressions systematically. This included prioritizing operations within parentheses, exponents, and then multiplication, division, addition, and subtraction.**20th Century Standardization**: In the early 1900s, the use of mnemonics like**PEMDAS**in the United States and**BODMAS**in Britain and Commonwealth countries became popular in educational settings. These were formalized through textbooks and curricula, ensuring that students were taught consistent methods across educational systems.

## Is the System also used in Vedic Maths ?

In **Vedic Mathematics**, the approach to solving mathematical operations differs from traditional PEMDAS/BODMAS rules but still emphasizes logical order and simplification. Vedic Mathematics is based on **16 Sutras** (formulas or aphorisms) derived from ancient Indian texts, aiming to simplify and speed up calculations. These sutras often lead to quick mental solutions, especially for arithmetic, algebra, and geometry.

### Similarities in Order and Simplification

While Vedic Mathematics doesn’t specifically follow PEMDAS/BODMAS, it includes principles that encourage tackling complex calculations in a structured way. For example:

**Simplification through Sutras**:- The sutra “Ekadhikena Purvena” (meaning “By one more than the previous one”) helps simplify squaring numbers ending in 5.
- The sutra “Urdhva Tiryak” (meaning “Vertically and Crosswise”) allows for rapid multiplication of numbers, often eliminating the need for sequential multiplication and addition.

**Prioritizing Certain Operations for Efficiency**:- Vedic methods often prioritize steps that reduce larger operations into smaller, simpler mental calculations, thus implicitly organizing the order of operations to achieve quick results.
- For instance, when multiplying or adding a series of numbers, Vedic Mathematics uses methods that consolidate steps and reduce the need for detailed steps like in PEMDAS.

### Example: Multiplication in Vedic Math

In standard arithmetic, solving a multiplication problem like $98 \times 97$ would involve breaking it down traditionally. However, using the Vedic sutra **“Nikhilam Navatashcaramam Dashatah”** (meaning “All from 9 and the last from 10”), you can quickly find the answer:

- Both 98 and 97 are close to 100 (a base).
- Subtract each number from 100:
- $98$ is $100 - 2$
- $97$ is $100 - 3$

- Multiply the differences: $(-2) \times (-3) = 6$.
- Cross-subtract any of the numbers with the difference of the other:
- $98 - 3 = 95$ (or $97 - 2 = 95$)

- Combine the results:
**9506**.

So, while Vedic Mathematics doesn’t formally follow PEMDAS/BODMAS, it achieves simplification by using sutras to prioritize operations that streamline calculations—often much faster than traditional methods.

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