We Distribute, Yet Things Multiply

Vocabulary and key identities regarding the distributive property and algebraic products from Chapter 6.

Distributive property of multiplication over addition

A property relating multiplication and addition captured by the identity $a(b + c) = ab + ac$.

Commutativity of multiplication

A property where the order of two numbers does not change their product, expressed as $ab = ba$.

Identity

A mathematical statement that expresses the equality of two algebraic expressions for any values substituted for their letter-numbers.

Like terms

Terms in an algebraic expression that have exactly the same letter-numbers, allowing them to be simplified and added or subtracted into a single term.

Identity 1

The general expansion for the product of two binomials: $(a + m)(b + n) = ab + mb + an + mn$.

Identity 1A

The algebraic identity for the square of the sum of two numbers: $(a + b)^2 = a^2 + 2ab + b^2$.

Identity 1B

The algebraic identity for the square of the difference of two numbers: $(a - b)^2 = a^2 - 2ab + b^2$.

Identity 1C

The algebraic identity showing the product of a sum and a difference: $(a + b)(a - b) = a^2 - b^2$.

khaᅇᅅa-guᅇanam

A term used by Brahmagupta in the Brahmasphuᅦasiddhānta meaning 'multiplication by parts,' referring to the use of the distributive property.

ista-gunana

A term used by Brahmagupta to describe methods for fast multiplication using the distributive property.

Sridharacharya's square method

A method to quickly compute squares using a modified form of Identity 1C: $a^2 = (a + b)(a - b) + b^2$.

Product increase (one number by 1)

When one number, say $b$, in the product $ab$ is increased by 1, the product increases by $a$, such that $a(b + 1) = ab + a$.

Product increase (both numbers by 1)

When both numbers in the product $ab$ are increased by 1, the product increases by $a + b + 1$, such that $(a + 1)(b + 1) = ab + (a + b + 1)$.

Multiplication by 11 (Rule)

A fast multiplication method using the distributive property $dcba \times (10 + 1) = dcba \times 10 + dcba$, resulting in a sum where digits are added in a specific sequence.

Multiplication by 101 (Rule)

A fast multiplication method using the distributive property $dcba \times (100 + 1) = dcba \times 100 + dcba$, represented as $d, c, (b+d), (a+c), b, a$ in place values.