The precise nature of the weave is knowledge that evades the minds of mortals. The majority of wizards attempt to focus their manipulation of this mysterious force through the conctrated study of a single school-but the rare few aschew this convention in order to better understand the weave. Self proclaimed as Arithmagicians, these wizards believe that logic, mathmatics, and equations form the fabric and threads of the weave itself.

Mind for Figures

Beginning when you select this school at 2nd level,  whenever you make an  ability check related to performing equations or mental puzzles or to recall history surrounding them, you are considered proficient in the relevant skill and add double your proficiency bonus to the check, instead of your normal proficiency bonus.

Additionally, you may use an abacus as your arcane focus.

Arcane Arithmatic

Beginning at 2nd level, you can utilize your arcane powers in a way which creates outcomes more than the sum of their parts. When you cast a spell using a first level spell slot or higher, you may expend an additional spell slot. If you do, you may resolve the spell at a level equal to the sum of both expended spell slots.

For example, if you cast Blindness Deafness with a 2nd level spell slot and expend an additional 1st level spell slot, you may cast it at the 3rd level.

Correct for Miscalculation

Starting at 6th level, you can grant yourself a second shot when your expirements don't go as planned. When you cast a spell using at least one spell slot and at least half of the creatures targeted succeeds on their saving throw, you can chose to refund the spell slots expended.

Once you use this feature, you must finish a short or long rest before you can use it again

Arcane Calculus

Beginning at 10th level, you can cast spells of 5th level or lower by expending any number of spell slots whose sum is greater than or equal to the spell's level plus one. A spell cast in this way is resolved at the level of the sum of the slots expended minus one, to a maximum of 8th level.

For example, you can cast Cone of Cold by expending both a 4th level spell slot and a 2nd level spell slot, two 3rd level spell slots, four 1st level spell slots and a 2nd level spell slot, or any other combination that equals six. If you choose to cast this spell by expending two 4th level spell slots, it is instead cast at the 7th level.

Arcane Thesis

Starting at 14th level, your understanding of the basic formula of the universe has led you to your masterwork-the ability to better manipulate the weave itself.When you use your Arcane Recovery feature, you may regain an additional number of spell slots equal to your Intelligence modifier. Additionally, you may chose to regain spell slots of 8th level or lower.