Addition in the Real Numbers

Addition as an Operation on the Real Numbers

The previous tutorial introduced the field axioms, including the rules governing addition in \(\mathbb{R}\). We now use those rules to study addition itself: how sums can be rearranged, how additive inverses interact with sums, and how adding a fixed number can be undone. These facts are useful both for simplifying expressions and for solving equations.

Addition is an operation that takes two real numbers and produces another real number. Its closure axiom ensures that \(x+y\in\mathbb{R}\) whenever \(x,y\in\mathbb{R}\). The associative law allows parentheses to be changed, while commutativity allows the terms to be reordered. Together, these rules let us interpret a finite sum without depending on an arbitrary choice of grouping or order.

For example, the expression \(a+b+c\) can be read as either \((a+b)+c\) or \(a+(b+c)\), because addition is associative. It can also be reordered, such as \(c+a+b\), because addition is commutative. When several terms appear, these laws permit a convenient calculation strategy: collect terms whose sum is easy to evaluate. No multiplication rule is needed for this kind of rearrangement.

The Additive Identity and Additive Inverses

The additive identity is the real number \(0\), which leaves every real number unchanged when added. The additive inverse of \(x\) is written \(-x\); it is the number that sums with \(x\) to give \(0\). The previous tutorial established that the identity and each element’s additive inverse are unique. Thus, \(-x\) always refers to one definite real number.

These definitions give a precise meaning to subtraction without requiring a separate subtraction operation. In particular, subtracting \(y\) means adding the additive inverse of \(y\). Parentheses matter when a negative sign applies to a whole sum: \(-(x+y)\) denotes the additive inverse of the number \(x+y\), not merely a change to one term.

Proof. Since \(0+0=0\), the number \(0\) is an additive inverse of \(0\). By uniqueness of additive inverses, \(-0=0\).

The defining property of \(-x\) gives \(x+(-x)=0\). By commutativity, \((-x)+x=0\), so \(x\) is an additive inverse of \(-x\). Uniqueness of the additive inverse of \(-x\) now gives \(-(-x)=x\).

To prove the final identity, use associativity and commutativity to add \(x+y\) to \((-x)+(-y)\):

Therefore \((-x)+(-y)\) is an additive inverse of \(x+y\). The uniqueness of that inverse implies \(-(x+y)=(-x)+(-y)\). All three identities follow. \(\square\)

Adding a Number Can Always Be Undone

A useful way to view addition is as a reversible change to a number. If a fixed real number \(a\) is added to \(x\), adding \(-a\) afterward returns \(x\). This works for every \(a\), including \(a=0\); no nonzero condition is needed. The next theorem states this reversibility as a property of a function.

Proof. Closure of addition ensures that \(T_a(x)\) is real for every \(x\in\mathbb{R}\), so the stated rule has values in its codomain. We show that composing the two translations in either order gives the identity function. For each \(x\in\mathbb{R}\), associativity and the additive inverse and identity rules give

Likewise, for every \(y\in\mathbb{R}\),

Thus \(T_{-a}\circ T_a=\operatorname{id}_{\mathbb{R}}\) and \(T_a\circ T_{-a}=\operatorname{id}_{\mathbb{R}}\). By the inverse-function result established earlier in the course, a function with an inverse is bijective. Therefore \(T_a\) is bijective and its inverse is \(T_{-a}\). \(\square\)

This theorem packages two important conclusions together. First, adding \(a\) never identifies two different inputs: if \(x+a=y+a\), then the outputs are equal only when the inputs are equal. Second, every real number \(b\) is reached as an output, because \(T_a(b+(-a))=b\). The theorem proves both conclusions through an explicit undoing operation.

Subtracting Sums and Keeping Signs Straight

The basic identities for additive inverses explain how a negative sign distributes over a sum. They also allow a difference of sums to be rewritten in a form where opposite terms can cancel. The key is to treat subtraction as addition of an inverse, then use the field’s addition rules.

Proof. By the definition of subtraction, \((a+b)-(c+d)=(a+b)+(-(c+d))\). The identity for the negative of a sum gives \(-(c+d)=(-c)+(-d)\). Substituting this expression proves the formula. \(\square\)

This formula does not mean that a negative sign can be distributed over addition without changing both terms. In particular, \(-(c+d)\) is \((-c)+(-d)\), not \((-c)+d\). Writing subtraction as addition of an inverse makes the sign change explicit and avoids that common error.

Using Addition Rules Carefully

The field axioms justify the manipulations above, but each manipulation has a specific basis. Associativity changes grouping; commutativity changes order; the identity property removes a zero term; and the inverse property replaces a pair such as \(x+(-x)\) by zero. These rules apply to sums of real numbers because the real numbers are closed under addition.

A common pitfall is to infer a sign rule from a visual pattern without checking what the negative sign applies to. For instance, the inverse of \(x+y\) must be the number that, when added to \(x+y\), gives zero. The theorem proves that this number is \((-x)+(-y)\). Testing the proposed expression confirms the point:

Another useful check is to undo an operation rather than move a term by an unexplained rule. In an equation \(x+a=b\), adding \(-a\) to both sides is justified by the field addition rules, and it produces \(x=b+(-a)\). Translation bijectivity additionally tells us that this solution exists and is unique. This reasoning depends only on addition and its inverse; multiplication will be studied separately.