PROOF 90

Prove.

Only like terms can be added

On the real numbers ℝ.

A way to read this proof

First separate what is assumed from what is to be proved. At each step, ask which definition or earlier result permits the move. If it is unclear, return to that premise rather than skipping to the conclusion.

Definition D1

A term in the variable x is an expression of the form cx, and c is called its coefficient. Two terms with the same variable part are called like terms.

Proof.

(1)x itself is a term with c = 1.convention, A2, A3

(2)3x and 2y have different variable parts, so they are not like terms.convention

∎

Theorem T1

For all a, b, x, ax + bx = (a + b)x

Proof.

(1)(a + b)x = x(a + b)A2

(2)x(a + b) = xa + xbA1

(3)xa = ax, xb = bxA2

(4)(a + b)x = ax + bx(1), (2), (3)

(5)ax + bx = (a + b)x(4)

∎

Theorem T2

If x and y are two independent variables, no number c writes 3x + 2y as a single term.

Proof.

(1)Suppose there is a number c with 3x + 2y = cx for all x, y.assumption

(2)It holds in particular at x = 0, y = 1.(1), A5

(3)3 × 0 = 0, 2 × 1 = 2A4, A3

(4)Left side = 0 + 2 = 2(3), A6

(5)Right side = c × 0 = 0A4

(6)2 = 0(2), (4), (5)

(7)2 ≠ 0, a contradiction.(6), A7

(8)Supposing 3x + 2y = cy and substituting x = 1, y = 0 gives 3 = 0 in the same way, a contradiction.A3, A4, A5, A6, A7

(9)Therefore 3x + 2y has neither the form cx nor the form cy.(7), (8), D1

∎

Corollary C

'Add only like terms' is precisely the range in which the distributive law applies.

Proof.

(1)The proof of T1 uses only A1 and A2.T1

(2)To use A1 that way, the two terms must share the common factor x.A1

(3)3x and 2y lack that common factor, and T2 showed they cannot be combined.(2), T2

(4)Adding like terms is not a new rule but a consequence of the distributive law.(1), (2), (3)

∎

Remark

Can A1 be removed from this proof?

Check whether the combinable side and the uncombinable side collapse together.