Numbers
The set of natural numbers (positive integers) is {1;2;3;4;5;...}.
The set of counting numbers (d non-negative integers) is {0;1;2;3;4;5;...}.
The set of negative integers is {...; −5; −4; −3; −2; −1}.
Combining the negative integers and the non-negative integers (or zero plus the positive integers) yields the set of
integers: {··· −3; −2; −1; 0; 1; 2; 3; ·
There are in fact an number of points between two integer markings. All these points on the number line (that is, the
solid line) represent the set of real numbers.
And variables
A variable is a symbol for a number we do not know
Multiplication of numbers by variables can be represented in more than one way, for example:
• With a multiplication sign: 4×a
• With a dot: 4·a
• With brackets: 4(a)
• Or straight: 4a
An expression is a combination of numbers, operators, brackets and variables. Two examples of expressions are 3x−6
and 2×5. 2
An equation is a statement that says that what is on the left of the equal sign (“=”) is equal to what is on the right of
it. Two examples of equations are 7x−4 = 1 and 2+4 = 6.
Basic operations on numbers
The four basic operations are addition, subtraction, multiplication and division.
When multiplying with plus and minus signs, the rule is as follows:
• If we multiply/divide two numbers with the same signs, the answer is a positive number, that is
+×+ = + and − × − = +
• If we multiply/divide two numbers with different signs, the answer is a negative number, that is
+× − = − and − ×+ = −
Note that we usually do not write the positive sign, +. A number without a sign always indicates a positive number.
,Basic operations on variables
Take note of two more general notes concerning notation:
• Note that when we have more than one variable in a factor, we arrange the variables alphabetically. For example,
we write 4ab, rather than 4ba.
• If the coefficient is 1, we usually do not write it. For example, 1y is written as y, or 1xy is written as xy, and −1x is
written as −x.
Addition and subtraction
When adding or subtracting, we only add or subtract like terms. Terms are alike when they have exactly the same
combination of letter variables.
Multiplication and division
When we want to multiply
6c with 8a, we first multiply the numbers (the coefficients): 6×8 = 48. (Always keep the sign rule in mind.) Then
multiply
Now what about 6c × 6c? We follow the same method as above and first multiply the coefficients: 6 × 6 = 36. Then
multiply the variables and remember the special notation for repeated factors:
c×c = c 2 . Thus 6c×6c = 36c 2
When we divide:
−6a 2a We divide the numbers −6÷2 = −3. The a’s cancel out, that is, a÷a = 1
When you multiply or divide expressions, always use the following order:
• Establish the sign of the answer by using the applicable sign rule.
• Multiply or divide the numbers (coefficients).
• Multiply or divide the variables.
, Powers & Exponents
A power consists of a base and an exponent for example, 82 is a power, the second power of 8.
In the expression 82 , the base is 8, the exponent is 2, and the power is the whole expression, 82 .
Remember that the value of any base to the power one is always just the base.
For example, 5 1 = 5 and x 1 = x.
What about a number or variable to the power zero? By definition, any number or variable to the power zero is one,
so that 100 = 1,4 0 = 1 and a 0 = 1.
negative exponents
Law of exponents
The set of natural numbers (positive integers) is {1;2;3;4;5;...}.
The set of counting numbers (d non-negative integers) is {0;1;2;3;4;5;...}.
The set of negative integers is {...; −5; −4; −3; −2; −1}.
Combining the negative integers and the non-negative integers (or zero plus the positive integers) yields the set of
integers: {··· −3; −2; −1; 0; 1; 2; 3; ·
There are in fact an number of points between two integer markings. All these points on the number line (that is, the
solid line) represent the set of real numbers.
And variables
A variable is a symbol for a number we do not know
Multiplication of numbers by variables can be represented in more than one way, for example:
• With a multiplication sign: 4×a
• With a dot: 4·a
• With brackets: 4(a)
• Or straight: 4a
An expression is a combination of numbers, operators, brackets and variables. Two examples of expressions are 3x−6
and 2×5. 2
An equation is a statement that says that what is on the left of the equal sign (“=”) is equal to what is on the right of
it. Two examples of equations are 7x−4 = 1 and 2+4 = 6.
Basic operations on numbers
The four basic operations are addition, subtraction, multiplication and division.
When multiplying with plus and minus signs, the rule is as follows:
• If we multiply/divide two numbers with the same signs, the answer is a positive number, that is
+×+ = + and − × − = +
• If we multiply/divide two numbers with different signs, the answer is a negative number, that is
+× − = − and − ×+ = −
Note that we usually do not write the positive sign, +. A number without a sign always indicates a positive number.
,Basic operations on variables
Take note of two more general notes concerning notation:
• Note that when we have more than one variable in a factor, we arrange the variables alphabetically. For example,
we write 4ab, rather than 4ba.
• If the coefficient is 1, we usually do not write it. For example, 1y is written as y, or 1xy is written as xy, and −1x is
written as −x.
Addition and subtraction
When adding or subtracting, we only add or subtract like terms. Terms are alike when they have exactly the same
combination of letter variables.
Multiplication and division
When we want to multiply
6c with 8a, we first multiply the numbers (the coefficients): 6×8 = 48. (Always keep the sign rule in mind.) Then
multiply
Now what about 6c × 6c? We follow the same method as above and first multiply the coefficients: 6 × 6 = 36. Then
multiply the variables and remember the special notation for repeated factors:
c×c = c 2 . Thus 6c×6c = 36c 2
When we divide:
−6a 2a We divide the numbers −6÷2 = −3. The a’s cancel out, that is, a÷a = 1
When you multiply or divide expressions, always use the following order:
• Establish the sign of the answer by using the applicable sign rule.
• Multiply or divide the numbers (coefficients).
• Multiply or divide the variables.
, Powers & Exponents
A power consists of a base and an exponent for example, 82 is a power, the second power of 8.
In the expression 82 , the base is 8, the exponent is 2, and the power is the whole expression, 82 .
Remember that the value of any base to the power one is always just the base.
For example, 5 1 = 5 and x 1 = x.
What about a number or variable to the power zero? By definition, any number or variable to the power zero is one,
so that 100 = 1,4 0 = 1 and a 0 = 1.
negative exponents
Law of exponents
