Arithmetic
Mental Arithmetic: Basic Tricks
The Human Computer: Shakuntala Devi vs. the Machine
In **1977**, the Indian woman **Shakuntala Devi** extracted the 23rd root of a 201-digit number - in **50 seconds**. The UNIVAC computer needed a full minute just to verify her answer. She was called a "human computer".
Without mathematics, life is incomplete. - Shakuntala Devi
Devi began performing at **age 3**. Her father - a circus artist - noticed that his daughter memorized numbers in card tricks faster than he could. He became her first teacher. By age 6 she was touring with mathematics shows. Talent? Partly. But the main ingredient was **thousands of hours of practice** from early childhood.
A waiter at a restaurant instantly totals the bill. A market vendor quotes a discount faster than you can pull out your phone. They're not geniuses - they know tricks. 78×5? In a second: 78×10÷2 = 390. Now you can do it too.
- **Finance:** quick estimation of discounts, tips, percentages
- **Commerce:** instant calculation of change and totals
- **Programming:** estimating algorithm complexity
The Complement Method
The **complement method** simplifies calculations by rounding one number to the nearest multiple of 10 or 100. Instead of a hard addition, we do an easy one.
**Complement principle:** To add a + b, find how much b needs to reach a round number: a + b = a + (10 - c) = (a - c) + 10 **Complements to 10:** • 7 + 3 = 10 • 8 + 2 = 10 • 9 + 1 = 10
The complement method turns any addition or subtraction into work with round numbers, which the brain processes instantly.
How do you calculate 67 + 28 using the complement method?
Doubling and Halving
**Doubling and halving** is an ancient multiplication method (the Egyptian method). It replaces complex multiplication with a series of doublings.
**Principle:** a × b = (a × 2) × (b ÷ 2) Doubling one factor and halving the other leaves the product unchanged. **When useful:** • Multiplying by 4, 8, 16... (multiple doublings) • Multiply by 5 = ×10 ÷ 2 • Multiply by 25 = ×100 ÷ 4
The Egyptians had no multiplication table, yet solved problems by doubling. This method underlies binary multiplication in computers.
What is the easiest way to compute 44 × 25?
Working with Near-Round Numbers
**Closeness to a round number** is a powerful trick: if a number is near 10, 100, or 1000, represent it as a difference or sum.
**Principle:** • 98 = 100 - 2 • 997 = 1000 - 3 • 102 = 100 + 2 **Multiplication:** a × 99 = a × 100 - a a × 101 = a × 100 + a
Numbers near round values are a gift for mental arithmetic. Remember: 98×97 = 9506, and practice the trick on similar examples.
What is 95 × 97 (both numbers close to 100)?
Cross Multiplication
**Cross multiplication** is a universal method for multiplying two-digit numbers. We split into tens and units, then multiply "cross-wise".
**Formula:** (10a + b) × (10c + d) = 100ac + 10(ad + bc) + bd **Steps:** 1. Multiply tens: a × c → hundreds 2. Cross: a×d + b×c → tens 3. Multiply units: b × d
The cross method is the foundation for multiplying multi-digit numbers of any length. Practice on simple examples, and soon you'll multiply two-digit numbers instantly.
Mental arithmetic requires an innate gift
Mental arithmetic is a set of techniques anyone can learn
All the methods described here are algorithms trained through practice. Ancient merchants without calculators used these same tricks. Start with simple examples and gradually increase difficulty. With a month of regular practice, speed improves 3–5 times.
What is 34 × 21 using the cross method?
Key Ideas
- Complement: round to the nearest round number
- Doubling/halving: ×5 = ×10÷2, ×25 = ×100÷4
- Near 100: 98×97 = 95|06 = 9506
- Cross method: ac | ad+bc | bd
Related Topics
Mental arithmetic exploits number properties:
- Vedic Mathematics — Ancient Indian sutras
- Order of Operations — Foundation for rearranging
- Modular Arithmetic — Checking results
Вопросы для размышления
- Which method seemed most useful in everyday life?
- Why is working with round numbers easier for the brain?
- How would you teach these tricks to a child?