Playing with two digit numbers

A two digit number

You are aware of two digit numbers. A two digit number is written in the form ab where ab = 10×a + b. Similarly as we would write 21 = 2×10 + 1. In this post I will teach you how to play with two digit numbers. I will teach fast tricks to do multiplication and finding squares.

Multiplication

You are aware of the formula (a + b)(c + d) = a×c + a×d + b×c + b×d. If you have two digit numbers kl and mn. Then you can represent kl as 10×k + l and mn as 10×m + n.

The product of kl and mn can be represented as
(10k + l)(10m + n) = 100×k×m + 10(k×n + l×m) + l×n

Example
21 × 65
= 100×2×6 + 10(2×5 + 1×6) + 1×5
= 1200 + 10(16) + 5
= 1200+160+5
= 1365

Steps to multiply two numbers
1. Multiply the digits at unit place
2. Multiply the two sets of digits at unit place and digit at tens place. Add. Multiply the result by 10.
   
3. Multiply the two digits at tens place. Multiply the result by 100.
   
4. Add the result obtained in steps 1,2 and 3.

Square

You know that the square of a number is product of a number with itself. So, if the number is kl then the square can be represented as
(10k + l)(10k + l) = 100×k×k + 10(k×l + l×k) + l×l
= 100×k² + 2×10×(k×l) + l²

You can obtain the above formula easily by using the expansion
(a + b)² = a² + 2ab + b²

Example
15×15
= 100×1 + 2×10×5 + 25
= 100 + 100 + 25
= 225

Steps to square a two digit number
1. Square the digit at unit place
2. Multiply the two digits at unit place and at tens place. Multiply the result by 20.
   
3. Square the digit at tens place and multiply by 100.
   
4. Add the result obtained in steps 1,2 and 3.

Squaring a two digit number ending with 5

When the number ends with 5 it can be written as k5.
Its square is 100×k² + 2×10×(k×5) + 5²
Which can be written as 100×k² + 100×k + 25
= 100×(k²+k) + 25
= 100×(k(k+1)) + 25

Rule: Find the product of digit at tens place and (digit at tens place + 1). Place 25 after it.