Applying Basic Number Properties

Before we can begin moving numbers around, we must understand how mathematical operations are bundled. Mathematics is read much like a language, but instead of punctuation marks like commas and periods, it utilizes structural brackets.

Grouping symbols such as parentheses dictate which arithmetic operations must be performed first within an algebraic expression. When you encounter an expression like $3 \times (4 + 5)$, the parentheses are an absolute command: evaluate the relationship between 4 and 5 before allowing the multiplication to interact with them. Understanding these boundaries is essential, because the properties we are about to explore are largely tools for either respecting, navigating, or legally circumventing these parentheses.

Source: Order of operations by HB, CC BY-SA 3.0.

The word "commute" means to travel or move around. In mathematics, the commutative property governs whether numbers can physically trade places without altering the final reality of the calculation.

Addition and Multiplication

Nature is highly cooperative when we are simply pooling quantities together or scaling them up.

The Commutative Property of Addition states that changing the order of addends does not change the resulting sum. Algebraic Representation: $a + b = b + a$

If you pour 3 ounces of water into a beaker and then add 5 ounces, you possess 8 ounces. If you pour 5 ounces first and then 3 ounces, you still possess 8 ounces. The commutative property allows numbers in an addition expression to be rearranged to combine compatible numbers that sum to a multiple of ten. For instance, faced with $17 + 46 + 13$, a clever mathematical mind rearranges the expression to $17 + 13 + 46$. The $17$ and $13$ instantly snap together to form $30$, turning a tedious calculation into an effortless $30 + 46 = 76$.

Source: Commutative Addition by Weston.pace . Attribution: Apples in image were created by Melchoir, CC BY-SA 3.0.

Multiplication shares this exact same spatial freedom.

The Commutative Property of Multiplication states that changing the order of factors does not change the resulting product. Algebraic Representation: $a \times b = b \times a$

A rectangle that is 4 units wide and 5 units long contains exactly the same 20 square units of area as a rectangle that is 5 units wide and 4 units long. You are merely viewing the same rectangle from a different angle.

The Strict Limits of Subtraction and Division

We must be equally precise about where these freedoms fail. The operation of subtraction does not satisfy the commutative property. If you have $10 in your bank account and write a check for $2, you have $8. If you have $2 and attempt to write a check for $10, you have a financial disaster. Order dictates reality here: $10 - 2 \neq 2 - 10$.

Similarly, the operation of division does not satisfy the commutative property. Sharing 10 pizzas among 2 hungry students yields a joyous 5 pizzas per student ($10 \div 2$). Attempting to share 2 pizzas among 10 students leaves everyone wanting more ($2 \div 10 = 0.2$). The roles of the dividend (what is being broken up) and the divisor (how many ways it is being broken) are entirely non-interchangeable.

While the commutative property is about moving numbers, the associative property is about grouping them. The root word "associate" means to join or partner with. If the order of the numbers remains perfectly static, does it matter which two numbers decide to interact first?

The Associative Property of Addition states that changing the grouping of addends does not change the resulting sum. Algebraic Representation: $(a + b) + c = a + (b + c)$

Imagine a line of three people holding cash: Person A has $5, Person B has $10, and Person C has $15. If Person A and B combine their money first ($15) and then add Person C's money, the total is $30. If A waits while B and C combine theirs first ($25), and then A hands over their $5, the total is still $30. The brackets shift; the sequence of numbers does not.

Source: AdditionAsc, CC BY-SA 3.0.

This works exactly the same way for scaling factors.

The Associative Property of Multiplication states that changing the grouping of factors does not change the resulting product. Algebraic Representation: $(a \times b) \times c = a \times (b \times c)$

This is an immensely powerful tool for mental arithmetic. The associative property allows a string of multiplication operations to be regrouped to form easier partial products without changing the final result. Consider evaluating $14 \times 5 \times 2$. If we group the first two:$(14 \times 5) \times 2$, we are forced to calculate$70 \times 2 = 140. But if we group the last two: \14 \times (5 \times 2), the calculation becomes \14 \times 10. The answer is instantly identifiable as \140$. We manipulated the parentheses to create a friendly multiple of ten.

Where Association Fails

Just as with the commutative property, directional operations like subtraction and division reject associative freedoms. The operation of subtraction does not satisfy the associative property. Take $(10 - 5) - 2$. The parentheses tell us to evaluate $10 - 5$first, yielding$5$. Then$5 - 2 = 3. Now shift the grouping: \10 - (5 - 2)$. Here,$5 - 2 = 3, and \10 - 3 = 7$. Because$3 \neq 7$, the associative property utterly fails for subtraction.

Likewise, the operation of division does not satisfy the associative property. Division demands a strict adherence to left-to-right processing precisely because shifting the parentheses wildly distorts the quotient.

The commutative and associative properties deal with one operation at a time—all addition, or all multiplication. But what happens when multiplication and addition are forced to occupy the same space? Enter the distributive property, arguably the most important mechanical lever in all of algebra.

The distributive property of multiplication over addition allows evaluating the product of a value and a sum by first multiplying the value by each addend separately and adding the resulting products.

Algebraic Representation: $a \times (b + c) = (a \times b) + (a \times c)$

Imagine you are ordering 3 combo meals. Each meal consists of a $5 burger and a $2 soda. You can calculate the cost in two ways:

1. Find the total cost of one combo meal first: $(5 + 2) = 7$. Then multiply by the 3 meals: $3 \times 7 = 21$.
2. Multiply the quantity of each item separately: $3 \times 5 = 15 for the burgers, and \3 \times 2 = 6 for the sodas. Combine them: \15 + 6 = 21$.

The multiplier is distributed to every term inside the parentheses.

This property holds true even when subtracting inside the parentheses. The distributive property of multiplication over subtraction allows evaluating the product of a value and a difference by first multiplying the value by each term separately and subtracting the resulting products. $a \times (b - c) = (a \times b) - (a \times c)$

Applying the distributive property enables mental math calculations by breaking down a difficult multiplication problem into the sum of two easier multiplication problems. If a student encounters $6 \times 98, the traditional [vertical algorithm](https://en.wikipedia.org/wiki/Multiplication_algorithm) takes time and scratch paper. But a mathematician sees \98$as$(100 - 2)$. By applying the distributive property: $6 \times (100 - 2) = (6 \times 100) - (6 \times 2)$ $600 - 12 = 588$ The complex operation has been instantly demystified into a simple mental subtraction.

Every mathematical system requires reference points—elements that allow us to step forward without moving at all. These are known as identity elements.

The additive identity property states that the sum of any given number and zero is the original given number. Zero is the mirror of addition; it reflects back exactly what you gave it. ($a + 0 = a$).

Source: AdditionZero, CC BY-SA 3.0.

The multiplicative identity property states that the product of any given number and one is the original given number. When you multiply by 1, you are taking "one copy" of a number, which leaves its magnitude entirely unchanged. ($a \times 1 = a$).

However, zero plays a drastically different role when exposed to multiplication. The multiplicative property of zero states that the product of any number and zero is exactly zero. Think of multiplication as scaling an object. If you take zero copies of an object, no matter how massive that object initially was, it ceases to exist in your inventory. ($a \times 0 = 0$).

In physics, every action has an equal and opposite reaction. In mathematics, this concept is captured by "inverses." An inverse is a mathematical undo button; it is an operation combined with a specific number designed to return you to the identity elements we just discussed.

Additive Inverses

The additive inverse of a given number is the unique number that yields zero when added to the given number. It destroys the original value to return to the additive identity (zero).

The additive inverse of a positive number is the negative counterpart of the identical magnitude. If you walk 5 steps forward (a value of $+5), the only way to return to your starting point of zero is to walk 5 steps backward (a value of \-5$). $5 + (-5) = 0$

Multiplicative Inverses

Multiplication has its own method of returning to baseline. The multiplicative inverse of a given nonzero number is the unique number that yields one when multiplied by the given number. Notice that it returns us to the multiplicative identity (one), not zero.

How do we turn a number like $4 into \1 purely through multiplication? We multiply it by its [fractional](https://en.wikipedia.org/wiki/Fraction) opposite. **The multiplicative inverse of a mathematical fraction is the [reciprocal](https://en.wikipedia.org/wiki/Multiplicative_inverse) of that specific fraction.** To find the reciprocal, we invert the [numerator](https://en.wikipedia.org/wiki/Fraction) and the [denominator](https://en.wikipedia.org/wiki/Fraction). The number \4$can be written as$\frac{4}{1}$. Its multiplicative inverse is$\frac{1}{4}. $$4 \times \frac{1}{4} = \frac{4}{4} = 1$$ If you have the fraction \frac{2}{3}$, its multiplicative inverse is$\frac{3}{2}$, because$\frac{2}{3} \times \frac{3}{2} = \frac{6}{6} = 1$.

To synthesize these structural laws, review the table below. Notice how the shape of the expression provides the greatest hint about which mathematical law is in play.

By deeply absorbing these properties, you do far more than pass a mathematics exam. You develop an intuitive fluency with numbers. When you recognize that $8 \times 25 is really just \8 \times (20 + 5), or that \12 + 89 + 88$is merely$100 + 89$ in disguise, you stop doing arithmetic and start conducting it.