Mathematical evidence that 1 calorie equal 4.18 joule

If you want to go back to the previous definition of a calorie, the amount of heat required to raise 1 gram of water by 1 degree C, you can derive that ~4.2 J/Calorie figure by unit conversions. You need to know that a Joule is measured experimentally as 4200 Joules to raise 1 kilogram of water 1 degree C.