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Basic Concepts of Algebra

Algebra Basic Operation Symbol

Basic operation symbol including addition, subtraction, multiplication, division, power, and square root.

Examples

three more than x can be written as: x + 3,

two less than x can be written as: x - 2,

two times x can be written as: 2x,

devide one by two can be written as: 1/2,

two to the three can be written as 2³, which is 2 × 2 × 2,

square root of 2 can be written as: Sqrt 2.

square root of 2 can be written as Sqrt 2

An algebra expression is an expression that is constructed from number and variables using the basic operation symbol.

Examples

a single number is an algebra expression, such as: 2; 3; 9...

a single variable is also an algebra expression, such as: x; y; z...

x² - 2xy + y² use the operation symbol of power, subtraction, multiplication, addition, then it is an algebra expression.

Value of an Algebra Expression

By substitute variable with number and follow the order of the operation, the final result is the value of the algebra expression.

Examples

find the value of algebra expression: 5x² - 7x + 6 when x = -1.

Solution

step 1: Substitute x with the number -1 in the algebra expression,

Step 2: Follow the order of operation to get the final result which is the value of the algebra expression.

5x² - 7x + 6

= 5(-1)² - 7(-1) + 6

= 5(1) + 7 + 6

= 18

Rational

Rational include natural number, positive number, negative number, integer and action.

Examples

Natural Number: 1,2,3,4,5,6,7,8,9,...;

Positive Number: All numbers larger than zero, ie, 1, 2.5, 125,...;

Negative Numbers: All numbers less than zero, ie, -1, -2.3, -123,...;

Integer: -n, ...,-3, -2, -1, 0, 1, 2, 3, ..., n;

Fraction: a/b (b is not zero) -2/3, 5/6,...;

Number Axis

Number axis is a line which has a reference point, unit length, and positive direction. All rational number can be a point on the number axis.

Examples

0 is the reference point,

all numbers less than 0 are located on the left side of the reference point,

all number larger than 0 are located on the right side of the reference point.

The length of the distance between 0 and 1 is the unit length,

the unit length repeated through positive and negative direction of the number axis.

The letter x indicate the positive direction in the number axis.

Label Points in Number Axis

Examples

Label the following points in number axis: -2.1, -1.2, 0.75, 2.6

Rules

Points in left side is always smaller than the point in right side in the number axis,

-2.1 less than -1.2 and -1.2 less than 0.75 and 0.75 less than 2.6.

All positive number are larger than 0 and all negative number are smaller than 0.

Understand Negative Number

After defined the reference point as 0 in number axis, all points below the reference point are negative numbers.

Examples

In temprature system, we defined zero degree as 0, then one degree below zero is -1, one degree above zero is +1, more often, we write +1 as 1.

Absolute Value

Absolute value of a number is the distance between the point of the number and the point of zero in number axis. The absolute value of a number is always positive.

Examples

the absolute value of number -2.1 is the distance between the point of -2.1 and the point of zero in number axis, |-2.1| = 2.1, so, the absolute value of a negative number is the number with a positive sign. The absolute value of a positive number is itself. For example, |0.75| = 0.75; |a| = a (if a >= 0), |a| = -a (if a < 0)

Opposite Number

If the two number have the same absolute value but one is positive and another is negative, then the two number are opposite number.

Examples

+1 and -1 are opposite number. Two opposite numbers are symmetry to zero point in number axis. The sum of two opposite numbers is zero.

Commutative Law of Addition

a + b = b + a

Associativity Law of Addition

(a + b) + c = a + (b + c)

Commutative Law of multiplication

a × b = b × a

Associative Law of multiplication

(a b) c = a (b c)

Distributive Law of multiplication

a (b + c) = a b + a c

Examples

Addition Law

Case 1:

If two numbers have the same sign, the sum of them has the same sign as them, and the value is the sum of absolute value of one number and absolute value of another number.

Case 2:

If two numbers have different sign, the sum of them has the same sign as the number that has larger abosolute value, and the value is the number that has a larger absolute value subtract the number that has a small absolute value.

Examples

Subtraction Law

a - b = a + (-b) ; a - (-b) = a + b

Examples

Compare Two Fractions

Case 1:

When two fractions have the same denominator, the fraction with larger numerator is larger than the fraction with small numerator.

Case 2:

When two fractions have the same numerator, the fraction with the small denominator is large than the fraction with larger denominator.

Case 3:

when two fractions have different denominator and numerator, find the common denominator first, then compare their numerator.

Rule:

At the condition of the same common denominator, the fraction with the large numerator is large the fraction with small numerator.

Examples

Compare Two Negative Fractions

The negative fraction that has a small absolute value is larger than the negative fraction that has a large absolute value.

Example

To compare (-3/4) with (-4/5), since the absolute value of (-3/4) is smaller than the absolute value of (-4/5), then (-3/4) is larger than (-4/5).