Mathematics: Real Number System
Is the following number rational or irrational? 4.123696857... _____________
Irrational. 4.123696857... is neither periodic, nor does it terminate. It appears to continue with no apparent pattern.
A ________ number "n" is an integer > 1 that is only divisible by itself and 1.
Prime. Examples of prime numbers are 2, 3, 5, 7, 11 NOTE: 1 is not a prime number as the definition applies to integers > 1
Is the following number rational or irrational? 2.135135. ___________
Rational. 2.315315 has a clear cut repetitive pattern and ends after the second 5.
Note
When a variable (x) is part of the absolute value, you cannot tell the sign of the number or value that is contained in the variable. Since you cannot tell just by looking whether or not the variable contains a positive or negative value, you have to consider both cases. To clear the absolute-value bars, split the equation into two cases, one for each sign:
The ________ ____ ___________ refers to which computations are done first in a compound equation.
Order of Operations. In an equation, certain operations must be performed before others. For example, whatever is in brackets always gets the highest priority. The Acronym BEDMAS is used to remember the order of operations: Brackets Exponents Division Multiplication Addition
Simplify the equation 7 - 6x - 15x2 - 2x3 = 0.
2x3 + 15x2 + 6x - 7=0. Standard form: -2x3 - 15x2 - 6x + 7 = 0 Factor out a -1 so that the highest power has a positive coefficient. -(2x3 + 15x2 + 6x - 7)= 0 2x3 + 15x2 + 6x - 7 = 0
The binary number system, used by computers, is a base ___ number system.
Base 2. The number system we're used to is base 10--there are 10 numbers all other numbers are made of--0,1,2,3,4,5,6,7,8,9. On the other hand, binary is a base 2 system--there are 2 numbers used to make up any value--0 and 1. To make things simpler, binary numbers are broken up into groups of 4, much like decimal numbers are separated into groups of 3 by a comma (123456 = 123,456). For example, 0110000111011011 = 0110 0001 1101 1011. Converting a binary number to a decimal number is easy. Just multiply each digit by its weighted position, and add each of the weighted values together. For example, the binary number 1101 0101 = 1×27 + 1×26 + 0×25 + 1×24 + 0×23 + 1×22 + 0×21 + 1×20 =128 + 64 + 0 + 16 + 0 + 4 + 0 +1 = 213 A simpler example, 101 in binary equals 1× 22 + 0× 21 + 1× 20 = 5 (note that anything to the zero power equals 1). Converting a decimal number to a binary number is a little harder. This is done by dividing the decimal number by 2, paying attention to whether or not there is a remainder of 1. A remainder corresponds to a "1". No remainder corresponds to a "0". Begin with the number furthest to the right and work your way leftward. This process is repeated until there are no more operations to perform.
The least ________ __________ (LCM) of two numbers is the smallest number (not zero) that is a multiple of both.
Common Multiple. The Least Common Multiple for two numbers is the smallest number which both act as a factor for. Example: Find the LCM of 30 and 45. First, find list the prime factors of each number. 30: 2× 3× 5 45: 3× 3× 5
The _____________ value of a number can be thought of as its distance from 0.
Absolute. Look at the number line: The absolute value of x, denoted |x| only indicates how far from 0 the number is - it does not consider the direction; i.e. it doesn't say right = positive, left = negative. The absolute value of 3 = |3| = 3, because 3 is three units to the right of zero, and |-3| = 3, because -3 is three units to the left of zero. In other words, the absolute value of a number is the non-negative version of that number. The absolute value of -220 is 220, the absolute value of 220 is 220.
Writing any composite number as a product of prime factors is called prime _______________.
Factorization. To find the prime factors of a number, divide the number by the smallest possible prime number and work up the list of prime numbers until the result is itself a prime number. Example: Find the prime factors of 160. Since 160 is even, start by dividing it by the smallest prime number, 2. 160 divided by 2 is 80. Then divide 80 by 2 = 40 40 / 2 = 20 20 / 2 = 10
To reduce a fraction to _________ terms you divide the numerator and denominator by their GCF.
Lowest. A fraction is reduced to lowest terms, or simplified, when its numerator and denominator have no common factors; meaning there is no number, except 1, that can be divided evenly into both the numerator and the denominator. Example: Simplify 27/18 First, find the prime factors. 18: 2×3×3 27: 3×3×3 Next, find the factors common to both the numerator and denominator. 3×3 is common to both 18 and 27 Finally, divide the numerator and denominator by all common factors (called canceling). 18/9 = 2 27/9 = 3 18/27 is simplified to 2/3
In decimal form, ____________ numbers neither terminate nor are periodic (no repeating pattern).
Irrational. Example: 3/65 = 0.04651538... √3 = 1.7320508... (the square root of 3 is an infinite decimal)
The greatest _________ __________ (GCF) is the greatest factor that divides two numbers evenly.
Common Factor. The Greatest Common Factor for two numbers is the largest factor that will divide evently into both of the numbers. To find the GCF of two numbers: 1. List the prime factors of each number. 2. Multiply the factors that both numbers have in common. If there are no common prime factors, the GCF is 1.
A ____________ number n is an integer > 1 that is not a prime number (divisible by more than just itself and 1).
Composite. Composite numbers can be divided evenly (without a remainder) by numbers other than 1 and the number. Examples are 12, 49, 55
The ___________ form of a rational number either terminates or is periodic.
Decimal. 3/4 = 0.75 this decimal terminates 1/6 = 0.16666... this is a periodic decimal as it has a repeating pattern: also written 0.16 6/33 = 0.181818... this is also a periodic decimal as it has a repeating pattern: also written 0.18
A _________ is a number that is multiplied to get a product.
Factor. When you factor a number (factoring) you take the number apart to find its factors. A factor can be either a composite or prime number. For example the factors of 16 are: 1, 2, 4, 8, 16 A number's factors are those numbers that will evenly divide into it.
All odd numbers are prime numbers [True or False] _______
False. With the exception of 2, for a number to be prime it must be an odd number (all even numbers are divisible by themselves, 1, and 2), but that doesn't mean that all odd numbers are prime. There are many odd numbers which are not prime--i.e. numbers ending in 5 are all divisible by 5 and a number like 49 is not prime as 7 * 7 = 49 (in other words, it's divisible by 1, 7, and 49). Note that 2 is the only even prime number, all other prime numbers are odd.
An ____________ number cannot be written in the form of a/b, where a and b are integers, and b does not equal 0.
Irrational. Example: 5/0 = undefined
A __________ number is a number that can be written in the form of a/b, where a and b are integers, and b does not equal 0.
Rational. Examples: -3 5/4 = 1.25 1/3 = 0.33 Whole numbers can be written with a denominator of 1, so they count as rational numbers (i.e. -3 = -3/1, 15 = 15/1)
A _________ factor is a factor that is a prime number.
Prime. A prime factor is a factor that has no other factors than 1 and itself. The following is a list of prime numbers up to 100 and as you can see none of these numbers can be factored any further: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97
To solve a polynomial equation it must be written in ___________ form.
Standard. A polynomial equation is one in which you have multiple powers of the variable 'x'. A polynomial in standard form has all the terms (listed from highest to lowest power) on one side of the equation and 0 on the other. Example: To solve 2x³ + 6x² + 12x = -8 It must be put into standard form: 2x3 + 6x2 + 12x + 8 = 0 After that it can be simplified by factoring out the common factors: Factor out a 2: x3 + 3x2 + 6x + 4 = 0 Note: The point of this is simply to show you what standard form is. We will work further with polynomials in later sections.