expressions in algebra : streamlined knowledge

Some basics in maths and science are presented in disconnected pieces or as abstracted concepts. The same can be explained with a logical connection which helps to understand and better remember. This streamlined knowledge series brings out such disconnected pieces from books / e-learning sites and connect them as given in nubtrek.com

Note: Algebra at high-school level is slated to be released in nubtrek.com by Aug’18

Expressions

An expression is a statement of a value. For example, all the following are numerical expressions of value 2

  • 1+5–4
  • 130⁰ + 123/123
  • √ 2 × √ 2

Similarly, algebraic expressions are statements of value(s) specified in terms of variable(s).

  • x²+x+1 is {7 when x=2} and {3 when x=1} , etc.

Arithmetics between Expressions

Arithmetics between numerical expressions are nothing but numerical arithmetics and some examples given below.

  • 3+2 added to 1–3 ⇒ 3+2+1–3
  • 1–3 subtracted from 3+2 ⇒ 3+2 — (1–3)
  • 3+2 multiplied by 1–3 ⇒ (3+2)×(1–3)
  • 3+2 divided by 1–3 ⇒ (3+2)÷(1–3)
  • (4+2+12+6) factored ⇒ (1+3)(4+2)
  • 3+2 exponent 2 ⇒ (3+2)²
  • 3+2 root 2 ⇒ √(3+2)

Similarly, arithmetics between algebraic expressions are seen as equivalent to numerical arithmetics.

  • 3x+2 added to 1-x ⇒ 3x+2+1-x
  • 1-x subtracted from 3x+2 ⇒ 3x+2 — (1-x)
  • 3x+2 multiplied by 1-x ⇒ (3x+2) × (1-x)
  • 3x+2 divided by 1-x ⇒ (3x+2) ÷ (1-x)
  • etc.

Manipulation of Expressions

Numerical expressions can be manipulated or modified without changing the value of the expression.

Numerical expressions or the sub-expressions in them can be modified as per PEMA / BOMA rules and CADI properties without changing the value of the expression.

(PEMA / BOMA : Parentheses / Bracket, Exponent / Order, Multiplication, Addition)

(CADI : Closure, Commutative, Associative, Distributive, Identity, Inverse)

  • 3 — (3 + 4×3) simplified by PEMA to 3 — (3+12) = 3–15 = -12
  • In 2–6/2+2 the subexpression -6/2 is a number by closure law
  • 2–6/2+2 by commutative law ⇒ 2+2–6/2
  • 2+(2–6/2) by associative law ⇒ (2+2)-6/2
  • 2÷3(6+3) by distributive law ⇒ 2×(6/3+3/3)
  • 3+4 by additive identity ⇒ (3+4)+5–5
  • 3+4 by multiplicative identity ⇒ (3+4)×5÷5

Similarly, algebraic expressions can be modified without changing the value specified by the expression.

Example: Simplification of (x²-x)+(x-3)

  • as per closure property of addition (x-3) is a number, and in the following step it is modified as a number
  • = x²+(-x+(x-3)) (associative property of addition)
  • = x²+(-x+x)-3 (associative property of addition)
  • = x²+0–3 (additive inverse property)
  • = x²-3 (additive identity property)

This example is quite simple for an elaborate solution given above. This solution is to illustrate modifying algebraic expressions.

Advantages

The transition from numerical arithmetics to algebra is smooth. Students can relate what they are learning very easily. Abstraction (in algebra) is solidly founded on numerical expressions and arithmetics.

In the current form of teaching, students have trouble in manipulation of algebraic expressions and they are usually instructed to do in a particular way without any reasoning.

For example, (3x² — x) / x, students make mistake and simplify this to 3x²-1. They are usually explained as a rule, if it is division, it has to be done this way (3x-1).

The better approach, as given in nubtrek, is to explain that each of the term in that is equivalently a number and distributive law of multiplication over addition is used in this. This can be explained with an example (4–2)/2. The expression evaluates directly to (2/2 = 1). Instead of direct evaluation, the expression is manipulated to two possibilities

  • (4–2)/2 = 4–1 = 3 which is wrong as the value of the expression is changed
  • (4/2–2/2 = 2–2 = 1) which is correct, and we have learned that the distributive law as applied to division holds. The value of the expression remains unchanged in the modifications

With this, it is understood that (3x² — x) / x = (3x-1)

In nubtrek.com, the lessons on numerical arithmetics sets up students for algebra and the realisation that all of algebra is nothing but a natural extension of numerical arithmetics. This realisation helps students to grasp the abstracted concepts in algebra.

This is the third part of 4 part series on algebra. Read the other parts from

Thanks for reading.