maths > algebra

Numbers and Arithmetic Operations

what you'll learn...

overview

This page revises the numbers quickly. It is important to understand the following concepts from this lesson, Laws and Properties of Arithmetic : Numbers and Operations :

•  Ordinal property of numbers

•  Comparison (greater, equal, or lesser)

•  division (inverse of multiplication)

•  exponent (repeated multiplication)

•  root (one inverse of exponent)

•  logarithm (another inverse of exponent)

count or measure

Numbers are
•  a value of measure or count
•  a representation of quantity or amount

The fundamental property of numbers is "numbers are in ordered sequence". The ordered sequence represents the magnitude of quantity represented by the numbers.

The whole numbers are in the ordered sequence

$0,1,2,3,4,5,6,7,8,9,10,11,$$0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 ,$ etc. This is the ordinal property of numbers.

The word "ordinal" means: relating to order or series.

directed whole numbers

"Integers" are negative numbers and whole numbers.

Integers are also called directed whole numbers.

Consider a girl and her brother sharing candies. She takes $2$$2$ candies one day and on another day her brother takes $3$$3$ candies from her.

This represents a directed transaction, "She takes $2$$2$" or "He takes $3$$3$".
The same can be represented as "She takes $2$$2$" or "She gives $-3$$- 3$".

This form of directed representation of numbers is captured with positive and negative numbers under "integers".

Integers have ordinal property and the sequence is $...,-2,-1,0,1,2,...$$\ldots , - 2 , - 1 , 0 , 1 , 2 , \ldots$

part of whole

"Fractions" and "decimals" represent counting parts of whole"

Fractions and Decimals are two forms of numbers representing "part of whole".

Consider a girl and her brother sharing candies. She split a large candy bar into $4$$4$ pieces and gave $3$$3$ pieces to her brother. The pieces are part of whole with place value $\frac{1}{4}$$\frac{1}{4}$ and the count $3$$3$ pieces. The number is represented as $\frac{3}{4}$$\frac{3}{4}$. The count $3$$3$ is the numerator. The place value $\frac{1}{4}$$\frac{1}{4}$ is given by the denominator $4$$4$.

The same can be represented with standardized place-value to $\frac{1}{10}$$\frac{1}{10}$ and the place value need not be specified. The number is represented as $0.75$$0.75$.

These two forms of "part-of-whole" representation of numbers are fractions and decimals respectively.

The ordinal property of fractions and decimals is maintained.

compare

Ordinal Property of numbers is used in comparison of numbers.

$4>2$$4 > 2$ means $4$$4$ is positioned higher in the ordinal to $2$$2$

$4=IV$$4 = I V$ means $4$$4$ & $IV$$I V$ are in the same ordinal position

$3<7$$3 < 7$ means $3$$3$ is positioned lower in the ordinal to $7$$7$

put together

$3+2=5$$3 + 2 = 5$;.

Addition $+$$+$ is one of the arithmetic operations.

take away

Subtracion represents "taking away" part of a quantity. Subtraction is inverse of addition.

$5-2=3$$5 - 2 = 3$;.

Subtraction is one of the arithmetic operations. Subtraction $5-2=3$$5 - 2 = 3$ is the inverse of addition $3+2=5$$3 + 2 = 5$.

Multiplication represents "repeated addition" of a quantity.

$3×2=6$$3 \times 2 = 6$;.

Multiplication is one of the arithmetic operations. Multiplication $3×2$$3 \times 2$ is repeated addition $3$$3$ added twice or $3+3$$3 + 3$.

splitting

Division represents "splitting" a quantity. Division is inverse of multiplication.

$6÷2=3$$6 \div 2 = 3$;.

Division is one of the arithmetic operations. Division $6÷2=3$$6 \div 2 = 3$ is inverse of multiplication $3×2=6$$3 \times 2 = 6$

repeated multiplication

Exponent represents "repeated multiplication" of a quantity.

${3}^{2}=9$${3}^{2} = 9$;.

Exponent is one of the arithmetic operations. Exponent ${3}^{2}=9$${3}^{2} = 9$ is repeated multiplication, $3$$3$ multiplied twice or $3×3=9$$3 \times 3 = 9$

two inverses of exponent

The two forms of inverse of exponents are root and logarithm.

eg:
Exponent form ${3}^{4}=81$${3}^{4} = 81$
Root : Inverse of exponent given the power $\sqrt[4]{81}=3$$\sqrt[4]{81} = 3$
Logarithm : Inverse of exponent given the base ${\mathrm{log}}_{3}\left(81\right)=4$${\log}_{3} \left(81\right) = 4$

$\sqrt{9}=3$$\sqrt{9} = 3$.

Root is one of the arithmetic operations. Root $\sqrt{9}={9}^{\frac{1}{2}}=3$$\sqrt{9} = {9}^{\frac{1}{2}} = 3$ is one form of inverse of exponent ${3}^{2}=9$${3}^{2} = 9$

${\mathrm{log}}_{3}9=2$${\log}_{3} 9 = 2$;.

$\mathrm{log}$$\log$ is one of the arithmetic operations. Logarithm ${\mathrm{log}}_{3}9=2$${\log}_{3} 9 = 2$ is one form of inverse of exponent ${3}^{2}=9$${3}^{2} = 9$

Note that inverse of addition is subtraction and inverse of multiplication is division. But inverse of exponent has two forms: roots and logarithms.

This is because $3+2=2+3=5$$3 + 2 = 2 + 3 = 5$ so inverse to get the left of the operator or right of the operator is defined by a single inverse operator.

But ${3}^{2}\ne {2}^{3}$${3}^{2} \ne {2}^{3}$, and so,
inverse to get the left of exponent is root $\sqrt{9}=3$$\sqrt{9} = 3$ and
inverse to get the right of exponent is logarithm ${\mathrm{log}}_{3}9=2$${\log}_{3} 9 = 2$.

summary

Laws and Properties of Arithmetic : Numbers and Operations :

•  Ordinal property of numbers

•  Comparison (greater, equal, or lesser)

•  division (inverse of multiplication)

•  exponent (repeated multiplication)

•  root (one inverse of exponent)

•  logarithm (another inverse of exponent)

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