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Key Concepts
3.1
Prime and Composite Numbers
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The natural numbers can be categorized as 1, prime numbers, and composite numbers.
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Prime numbers have as their only factors 1 and themselves.
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Composite numbers have at least three distinct factors.
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Composite numbers can be written in their prime factorization form, which is found by repeatedly factoring prime factors from the number.
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The greatest common divisor (GCD) of a set of numbers is the largest integer that divides all of the numbers in the set. The prime factorizations of the numbers can be used to identify the greatest common divisor.
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The least common multiple (LCM) of a set of numbers is the smallest integer that is divisible by all of the numbers in the set. The prime factorizations of the numbers can be used to identify the least common multiple.
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There are various ways that the GCD and LCM are applied.
3.2
The Integers
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A set of numbers that can be built from the natural numbers are the integers, which consist of the natural numbers, zero (0), and the negatives of the natural numbers.
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Integers are often graphed on a number line, which helps display the relative positions and values of those numbers.
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The number line can be used to visualize when one integer is larger than or smaller than another integer.
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Arithmetic operations with integers are similar to the operations with natural numbers, except that the sign (positive or negative) of the numbers will determine the sign (positive or negative) of the result.
3.3
Order of Operations
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Establishing shared rules on which arithmetic operations are calculated first is necessary. Without them, different people may find different values for the same expression.
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The highest precedence is with expressions in parentheses. This allows parts of an expression to be calculated in an order different than the basic order of operations.
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The lowest precedence is addition and subtraction, as they are the basis for all other calculations.
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Multiplication and division have precedence over addition and subtraction, as they are representations of repeated addition or subtraction.
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Exponents have precedence over multiplication and division, as they represent repeated multiplication and division.
3.4
Rational Numbers
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Rational numbers are fractions of integers, and can always be written as an integer divided by an integer.
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The numerator and denominator of a fraction may have common factors. In such cases, the fraction can be reduced by canceling common factors. When the numerator and denominator of a fraction have no common factors, the fraction is said to be reduced.
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An improper fraction is one with a numerator larger than the denominator. Such a fraction can be rewritten as an integer plus a proper fraction. This is called a mixed number.
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Using division and remainder, an improper fraction may be written as a mixed number.
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A mixed number can be converted to an improper fraction by reversing the process for changing an improper fraction to a mixed number.
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The arithmetic operations or addition, subtraction, multiplication and division can all be performed on rational numbers.
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Addition and subtraction of rational numbers can be performed after a common denominator has been identified, and the fractions have been converted to forms having the common denominator.
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Multiplication and division of rational numbers can be performed without regard to common denominators.
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Between any two rational numbers, there is always another rational number. This is the density property of the rational numbers.
3.5
Irrational Numbers
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Irrational numbers are numbers that cannot be written as an integer divided by another integer. One example is pi, denoted
. Another collection of irrational numbers are natural numbers that are not perfect squares.
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Some irrational numbers can be written as a rational part multiplied by an irrational part. If two irrational numbers have the same irrational parts, they can be added or subtracted.
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When irrational numbers are similar, on can multiply and divide the numbers without a calculator.
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Since
, and
, products and quotients of square roots can be determined.
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Because
and
, it is possible to simplify square root expressions so the radicand contains no perfect square factors.
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When a fraction has an irrational number as its denominator, it is possible to convert the denominator into a rational number using its conjugate. Doing so involves multiplying the numerator and denominator by the conjugate of the denominator, and then applying the difference of squares formula.
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With a single square root term
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Using conjugate numbers for two term denominators
3.6
Real Numbers
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Real numbers is the collection of all rational and irrational numbers. Conceptually, it is the collection of all values that can be represented on a number line, or, as a length along with sign.
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The subsets of the real numbers include the natural numbers, integers, rational numbers and irrational numbers. The natural numbers are a subset of the integers, which is a subset of the rational numbers. The rational and irrational numbers are disjoint sets.
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The real numbers, due to order of operation rules and that performing arithmetic operations on real number always results in a real number, have arithmetic properties that apply in all cases. There include the distributive property, the commutative property, and the associative property. Also, every real number has an additive inverse and, except for zero (0), have a multiplicative inverse.
3.7
Clock Arithmetic
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Clock arithmetic uses the idea that after 12 o’clock comes 1 o’clock. For clock arithmetic, this means that every time 12 is passed in an arithmetic process, the next number is 1, not 13.
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To determine the clock result of an arithmetic operation, divide the final result by 12 and keep the remainder. If the remainder is 0, then the time is 12 o’clock.
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Clock arithmetic is technically called modulo 12 arithmetic. To perform modulo 12 arithmetic, calculate the expression, then divide the result by 12. The modulo 12 result is the remainder.
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Days, in our system, pass in groups of seven. To calculate in day arithmetic, modulo 7 is used. To perform modulo 7 arithmetic, calculate the expression, then divide the result by 7. The modulo 7 result is the remainder.
3.8
Exponents
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Exponents are used to express multiplying a number by itself a number of times. The number being multiplied by itself is the base. The number of times it is multiplied by itself is the exponent, which is often referred to as the power.
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Understanding that exponents represent repeated multiplication of a base makes it possible to establish some rules for combining exponential expressions, using the product rule, the quotient rule, and the power rule. Additionally, it allows us to formulate distributive rules for exponents.
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Any non-zero number raised to the 0th power is 1. This makes the definition of the 0th power consistent with the division rule for exponents.
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For consistency, negative exponents represent the reciprocal of the base raised to the power, so that
, provided that
.
3.9
Scientific Notation
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Some numbers are so large or so small that writing the number out is clumsy and make it difficult to determine the true size of the number. Scientific notation makes the number more readable and make the relative size of the number immediately apparent.
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A number written in scientific notation is a number at least 1 and smaller than 10 multiplied by 10 raised to an exponent. Converting between scientific notation and standard notation involves correctly applying multiplication and division by powers of 10, which in practice equates to understanding how moving the decimal point of a number impacts the exponent of 10.
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Adding and subtracting numbers in base 10 requires the exponent of 10 in each number be the same. Once the numbers are converted to have the same exponent with the ten, then the numbers are added or subtracted as indicated, with the power of 10 remaining the same. If the result is not in scientific notation (for instance, the number has exceeded 10), then then number must be converted into scientific notation.
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Multiplying and dividing numbers in scientific notation is done by multiplying or dividing the number parts, then multiplying or dividing the 10 raised to the power parts, then multiplying those two results. If the new number is not in scientific notation, then the result must be converted into scientific notation.
3.10
Arithmetic Sequences
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A sequence is a list of numbers. Any individual number in that list, or sequence, is a term of the sequence. A specific term of a sequence is denoted by the sequence symbol with a subscript indicating where the term in the sequence is.
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A special form of a sequence is an arithmetic sequence. Each arithmetic sequence is determined by its first term and its constant difference. Any term in an arithmetic sequence is determined by adding the constant difference to the preceding term.
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If the first term and the constant difference of an arithmetic sequence are known, then any term of the sequence can be found directly.
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Because arithmetic sequences follow such a strict pattern, the sum of the first
terms of an arithmetic sequence can be determined with the formula
.
3.11
Geometric Sequences
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A special form of a sequence is a geometric sequence. Each geometric sequence is determined by its first term and its constant ratio. Any term in a geometric sequence is determined by multiplying the constant ratio to the preceding term.
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If the first term and the constant ratio of a geometric sequence are known, then any term of the sequence can be found directly.
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Because geometric sequences follow such a strict pattern, the sum of the first
terms of a geometric sequence can be determined with the formula
.
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Finding the sum of a finite geometric sequence
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Applying arithmetic sequences
3.11
Geometric Sequences
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Geometric sequence.
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Finding an arbitrary term in a geometric sequence.
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Constant ratio.
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Finding the sum of a finite geometric sequence.
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Applying arithmetic sequences.