 maths > algebra

Numerical Expressions

what you'll learn...

overview

In this lesson, properties of expressions are explained.

•  An expression is a statement of a value

•  the value of an expression remains unchanged when the expressions are modified per PEMA / CADI

It is very important to go through this once to understand in-equalities in algebra.

'to express' means to 'say'

A numerical expression can be described as a collection of numbers and arithmetic operations between them forming a quantity. In other words, a numerical expression is a statement of a quantity with numbers and arithmetic operations.

Some example of numerical expression are

$2+2+84-2$$2 + 2 + 84 - 2$

$23×123+8$$23 \times 123 + 8$

$\left(32-23\right)+43×2-4$$\left(32 - 23\right) + 43 \times 2 - 4$

one or the other

A numerical expression is a statement of quantity with numbers and arithmetic operations. This can be evaluated to a number.

For example, all the following are numerical expressions evaluating to $2$$2$.

•  $1+5-4$$1 + 5 - 4$ (expression involving addition and subtraction)

•  ${130}^{0}+\frac{123}{123}$${130}^{0} + \frac{123}{123}$ (expression involving exponent and division)

•  $\sqrt{2}×\sqrt{2}$$\sqrt{2} \times \sqrt{2}$ (expression involving multiplication and root)

•  $2$$2$ ($2$$2$ is technically an expression and also the simplified form of the expressions given above.)

In other words, the numerical expression can be used in place of the number it evaluates to. Consider the number $2$$2$ and the equivalent numerical expressions $1+5-4$$1 + 5 - 4$. The following two statements are identical for any practical purposes.

•  The length is $2$$2$.

•  The length is $1+5-4$$1 + 5 - 4$.

The numerical expression and the number can be used interchangeably.

That is because, the numerical expression $1+5-4$$1 + 5 - 4$ can be simplified to $1+5-4=1+1=2$$1 + 5 - 4 = 1 + 1 = 2$

The numerical expressions can be simplified as per PEMA rules.

more than one

Consider two numerical expressions $3+2$$3 + 2$ and $1-3$$1 - 3$. These represent two quantities and the sum of these two are required.

Normally, these are simplified first to $5$$5$ and $-2$$- 2$ respectively and then these are added.

But, these numerical expressions can be directly added : $3+2+1-3$$3 + 2 + 1 - 3$

Two or more numerical expressions can be added.

Similarly, the numerical expressions be multiplied : $\left(3+2\right)×\left(1-3\right)$$\left(3 + 2\right) \times \left(1 - 3\right)$

expressions are numbers

Arithmetics with Numerical Expressions :
Numerical expressions can be

•  simplified as per PEMA precedence
eg: ${3}^{2}+{3}^{3}+4×{3}^{2}$${3}^{2} + {3}^{3} + 4 \times {3}^{2}$ simplified to $5×{3}^{2}+{3}^{3}$$5 \times {3}^{2} + {3}^{3}$

eg: $3+2$$3 + 2$ added to $1-3$$1 - 3$ gives $3+2+1-3$$3 + 2 + 1 - 3$

•  subtracted
eg: $1-3$$1 - 3$ subtracted from $3+2$$3 + 2$ gives $3+2-\left(1-3\right)$$3 + 2 - \left(1 - 3\right)$

•  multiplied
eg: $3+2$$3 + 2$ multiplied by $1-3$$1 - 3$ gives $\left(3+2\right)×\left(1-3\right)$$\left(3 + 2\right) \times \left(1 - 3\right)$

•  divided
eg: $3+2$$3 + 2$ divided by $1-3$$1 - 3$ gives $\left(3+2\right)÷\left(1-3\right)$$\left(3 + 2\right) \div \left(1 - 3\right)$

•  factorized
eg: $\left(4+2+12+6\right)$$\left(4 + 2 + 12 + 6\right)$ factored gives $\left(1+3\right)\left(4+2\right)$$\left(1 + 3\right) \left(4 + 2\right)$

Note: Exponents, roots, and logarithm will be explained in higher level mathematics

expressions are numbers

By closure law of addition and multiplication, any numerical expression simplifies to a number.

That also implies that any subexpression is a number by itself.

Any subexpression in a numerical expression can be modified as per PEMA Precedence / CADI Laws and Properties of Arithmetics.

For example, consider the numerical expression $2-6/2+2$$2 - 6 / 2 + 2$.

By commutative law of addition, the expression can be written as $2+2-6/2$$2 + 2 - 6 / 2$
because, $-6/2$$- 6 / 2$ is a number.

By associative law of addition, the expression can be written as $2+\left(-6/2+2\right)$$2 + \left(- 6 / 2 + 2\right)$

By additive identity property, the expression can be written as $2+2-6/2+0$$2 + 2 - 6 / 2 + 0$

summary

Laws and Properties of Arithmetics -- Numerical Expressions : The value of a numerical expression remains unchanged in the following.

•  any subexpression can be considered a number by closure law of addition and multiplication

•  position of two subexpressions can be changed by commutative laws of addition and multiplication

•  order of operations can be modified as per associative laws of addition and multiplication

•  multiplication by a subexpression can be distributed over addition as per the distributive law of multiplication over addition

•  additive identity can be added or multiplicative identity can be multiplied

•  additive identity in an expression can be modified into sum of a number and its additive inverse as per the additive inverse property

•  multiplicative identity in an expression can be modified into product of a number and its multiplicative inverse as per the multiplicative inverse property.

Important note: Why are the PEMA / CADI properties of numerical arithmetics relevant for algebra?
The algebraic expressions are modified as per the exact same laws and properties given above. For example: $3\left(x+2\right)=3x+3×2$$3 \left(x + 2\right) = 3 x + 3 \times 2$ as per distributive law of multiplication over addition.

Outline

The outline of material to learn "Algebra Foundation" is as follows.

→   Numerical Arithmetics

→   Arithmetic Operations and Precedence

→   Properties of Comparison

→   Properties of Multiplication

→   Properties of Exponents

→   Algebraic Expressions

→   Algebraic Equations

→   Algebraic Identities

→   Algebraic Inequations