number: abstract concept of a characteristic of things or ideas. NOT the numeral, which is the symbol that represents the number. Think of quantity.

constant: a symbol that represents a number of known value. Ex: 2, 3, 5, 7, 11, 0, -1,π, √2

variable: a symbol that represents a number of unknown value. Generally a letter of the alphabet is used because people already know how to write and pronounce the symbol. Ex: a, b, c, (used to represent the place of a constant in an algebraic pattern); x, y, and z (used to represent the place of a variable in an algebraic pattern)

set: a collection of items that have something in common.

Sets of Numbers: collections of numbers that fit a common pattern.

Natural, aka counting, numbers: the set of numbers that start with 1 and increase by 1’s
Ex: 1, 2, 3, . . .
Questions: What is the next Natural Number?
Is 15 a Natural Number? Why or why not?
How do you recognize Natural Numbers?
What is the largest Natural Number?

Whole: the set of numbers that start with 0 and increase by 1’s
Ex: 0, 1, 2, 3, . . .
Questions: What is the next Whole Number?
Is 15 a Whole Number? Why or why not?
How do you recognize Whole Numbers?
Can a number be both Natural and Whole?
Are all Whole numbers Natural or are all Natural Numbers Wholes?
What is the largest Whole Number?

Integers: The positive and negative Whole numbers and zero. The set of numbers that starts at zero (0) and both increases by 1’s (positive Integers) and decreases by 1’s (negative Integers).
Ex: . . ., -3, -2, -1, 0, 1, 2, 3, . . .
Questions: What is the next larger Integer? the next smaller?
Is 15 an Integer Number? Why or why not? Is -15 an Integer Number? Why or why not?
How do you recognize Integers?
Can a number be both Natural and Integer? Whole and Integer? All three?
Are all Whole numbers Integers? Are all Integers Wholes?
What is the largest Integer? The smallest Integer?

Rational: any number that can be written in the form of a ratio of two Integers, as long as the second Integer is NOT ZERO. (Think: it looks like a regular fraction and the bottom number can not be zero. See Division by Zero on web site).
Ex:. . ., -1, 3/-4, 2/-3, -1/2, 1/-3, -1/4, 0, 1/4, 1/3, 1/2, 2/3, 3/4, 1, . . .

Questions: What is the next larger Rational? the next smaller?
Is 15 a Rational Number? Why or why not? Is -15/9 a Rational Number? Why or why not?
How do you recognize Rationals?
Can a number be both Integer and Rational? Whole and Rational? All three?
Are all Whole numbers Rational? Are all Integers Rational? Are all Rational Integers?
What is the largest Rational? The smallest Rational?

Irrational: Numbers that can not be written as Rational Numbers,
Ex. π, √2, -π and -√7
Questions: What is the next larger Irrational? the next smaller?
Is 22/7 an Irrational Number? Why or why not? Is -15/9 an Irrational Number? Why or why not?
How do you recognize Irrationals?
Can a number be both Rational and Irrational?
Are any Rational numbers Irrational?
What is the largest Irrational? The smallest Irrational?

Real: the set made by combining all of the Rational and Irrational numbers. The union of the Rationals and Irrationals into one set.
Ex. all the numbers represented by the continuous number line.
Questions: What is the next larger Real after 2? the next smaller before -p? Is it possible to list all of the Real Numbers?
Is 22/7 a Real Number? Why or why not? Is -√7 a Real Number? Why or why not?
How do you recognize Real Numbers?
Can a number be both Rational and Real? both Irrational and Real?
Are any Rational numbers Real? Are any Irrational numbers Real?
What is the largest Real Number? The smallest Real Number?

Unless specified, number in the following definitions means Real Number.

fractions: A fraction represents a portion of a thing, rather than an entire unit. We will use three forms of fractions:

1. rational: what we generally think of as a fraction; the ratio of Integers, the bottom number NOT ZERO. See definition of Rational Numbers above. Top number is called the numerator, bottom number the denominator.
2. decimal: written as a ratio of an Integer to a power of 10. Think of cents written in dollar form: $2.35, where the part that represents the 35 cents is the decimal fraction.
3. percent (%): a ratio of an Real Number to 100. Percent is from the Latin per centum: out of 100. An entire unit is 100% of the item.

mixed number: a number that has both Whole and fractional parts, such as $2.35, read as two dollars and thirty-five cents, or 3, read as 3 and two sevenths. The word and in reading mixed numbers specifies that what follows is a fraction. Numbers such as 247 are read correctly as two hundred forty-seven. Two hundred and forty-seven (hundredths) is written as 200.47

evaluate: substitute the given value for the variable and compute the result using the order of operations.

expression, algebraic expression: any collection of constants, variables. operation signs and grouping symbols that does NOT have an equal sign or inequality sign. Expressions can be simplified and evaluated, but CANNOT be solved. Does not make a complete statement, but represents the same thing a noun does in written language.

translate to an algebraic expression: write the algebraic version of an idea expressed in words. Similar to a noun phrase in grammar.

operation: How two numbers can be combined. Forms shortcuts to counting. In algebra, only two operations, addition and multiplication, because of the use of signed numbers.

sum: the result of combining two numbers that have the same sign by addition.

difference: the result of combining two numbers that have opposite signs by addition (we called it subtraction in arithmetic).

product: The result of combining two numbers by multiplication.

quotient: the result of multiplying a number times the reciprocal of another number.

reciprocal: the multiplicative inverse of a number. The product of a number and its reciprocal is ALWAYS +1. Ex. the reciprocal of 3 is 1/3 , and the reciprocal of 1/3 is 3. How do you generate the reciprocal of a number?

term: the parts of an algebraic expression being added or subtracted. The result of combining terms is called the sum.

like terms are two terms that have the same place value. Algebraically, two terms are alike when their variable factors are identical.
Ex. in the Integers 234 and 759, the 2 and the 7 are like because they are both in the hundreds place, the 3 and the 5 are alike because they are both in the tens place, and the 4 and the 9 because they are both in the ones place.
The algebraic terms 3x and 7x are alike because the variable factor of x is identical in both.
The terms 4xy and -9xy are like terms. 3x and 4y are not like terms. Are 5x and 5xy like terms?

a factor: a number being multiplied. The result of combining factors is called the product. A group may be a factor, for example, in the expression 3(x + 2) the factors are 3 and the group (x + 2).

to factor: to write a number as the product of its Integer factors. Ex. 12 can be written as 1 * 12, 2 * 6, or 3 * 4. The number 3x has a factor of 3 and a factor of x, so its factored form is 3 * x.

distribute: the process used to multiply a single digit number times a two or more digit number.

Distributive Law: a(b + c) = ab + ac
Since this Law of Real Numbers is expressed as an equation, it works in both directions. As stated above, it is used to multiply. Stated as ab + ac = a(b + c), it is used to factor.
Ex. to multiply:
3(2x + 5) = 6x + 15
4(7x - 3) = 28x - 12
to factor:
27x + 18 = 9(3x + 2)
16x - 24 = 8(2x - 3)

Order of Operations: mnemonic: Please/ Excuse/ My Dear /Aunt Sally. ALWAYS applied from left to right.

GCF: Greatest Common Factor: the largest Integer that evenly divides all the numbers in question. Ex. the GCF of 16, 24, and 12 is 4. May be written as GCF(16, 24, 12) = 4

prime numbers: any Integer greater than 1 whose only Whole number factors are 1 and itself.
Ex. 2, 3, 5, 7, 11, . . .
What is the next prime number?
What are the first 25 prime numbers?
How many prime numbers are also even?

composite numbers: any Integer greater than 1 that is not prime; that is, a number that has more than one set of Integer factors.
Ex. 24 is composite because it has 2 * 12 as a set of Integer factors.
Is 51 prime or composite?

even numbers, odd numbers: Integers that are multiples of two are even, those that are not multiples of two are odd.

lowest terms: a fraction is in lowest terms when its numerator and denominator have no common Integers factors other than 1; it can not be reduced any more.

simplest form: an expression is in simplest form when all grouping symbols have been removed and all like terms have been combined by following the Order of Operations and using the Distributive Law.

number line, direction and distance: A continuous horizontal line that represents the set of Real Numbers. Every point on the number line has two characteristics, direction from zero and distance. Points to the right of zero are positive; points to the left are negative. When we know the direction and distance of a number, then we can find its coordinate, the point on the number line that best visualizes that number.