Today’s soundtrack is Saor: Guardians.
This morning, I’ll be reading the final chapter of Basic Math & Pre-Algebra for Dummies Workbook, and the last two chapters of Basic Math & Pre-Algebra for Dummies. These three chapters are all short and don’t have a worksheet section, so I’m just going to do all of them at once.
Chapter 17: “Ten Curious Types of Numbers”
Square Numbers
Numbers multiplied by themselves any number of times
Triangular Numbers
A “sequence of consecutive positive numbers starting with 1” (p. 288), when added, results in a triangular number. If you represent the numbers with coins, the results will always make a perfect equilateral triangle.
Cubic Numbers
Any number multiplied by itself three times (cubed) adds a third dimension of area.
Factorial Numbers
If we follow a number with an exclamation point, we get a factorial. “[Y]ou read 1! as one factorial” (p. 289).
“1! = 1
2! = 2 x 1 = 2
3! = 3 x 2 x 1 = 6” (p. 289), and so on. So the “sequence of factorial numbers begins as follows: 1, 2, 6, 24, 120, …” (p. 289).
Powers of Two
The sequence of 2 multiplied by itself “gives you the power of two” (p. 290). The “[p]owers of two are the basis of binary numbers” (p. 290), and their sequence begins as follows: 2, 4, 8, 16, 32…
Perfect Numbers
A perfect number is “[a]ny number that equals the sum of its own factors (excluding itself)” (p. 290). The way to find a perfect number is to find all of its factors, then add the factors up. The first perfect number is 6; “[t]he next perfect number is 28” (p. 290). The sequence of perfect numbers begins as follows: “6; 28; 496; 8,128; 33,550,336; …” (p. 290).
Amicable Numbers
These “are similar to perfect numbers, except they come in pairs” (p. 291). Adding the factors of one number will equal the other number, and the factors of the second number will equal the first. The sequence begins as follows: 220 and 284, then “1,184 and 1,210” (p. 291).
Prime Numbers
Prime numbers are only divisible by 1 and itself. The sequence begins as follows: “2, 3, 5, 7, 11, 13, 17, 19, …” (p. 291).
Mersenne Primes
These numbers are one less than a power of two and can only be divided by 1 and itself, and the sequence begins as follows: 3, 7, 31, 127, 8191…
Fermat Primes
To find a Fermat prime, we find a power of two, “use that answer as an exponent on 2” (p. 292), and add 1. These numbers, being prime, can only be divided by 1 and itself. The sequence begins as follows: 3, 5, 17, 257, 65,537...
Chapter 24: “Ten Little Math Demons That Trip People Up”
The multiplication table
Some of the items on the multiplication table are difficult to remember. The ten hardest ones for people to remember are 8x7, 7x9, 6x6, 7x7, 8x8, 9x9, 6x8, 8x9, 9x6, and 7x6 (324).
Adding and subtracting negative numbers
The key here is to remember that non-symbolized numbers are positive, and any positive number goes up, and any negative number goes down.
Multiplying and dividing negative numbers
Two positives make a positive; two negatives make a positive; one positive and one negative make a negative.
Knowing the difference between factors and multiples
A factor is a number that fits inside of another number neatly a certain number of times. A multiple is a number that contains another number a certain number of times. “3 is a factor of 12 and 12 is a multiple of 3” (p. 325).
Reducing fractions to lowest terms
Divide the numerator and denominator “by a common factor” (p. 326), and repeat as needed
Adding and subtracting fractions
Use cross-multiplication if necessary to match the denominators, then add or subtract the numerators
Multiplying and dividing fractions
To multiply: multiply the numerators to get the final numerator; multiply the denominators to get the final denominator; reduce to lowest terms.
To divide: flip the second fraction upside-down, then multiply the two new numerators and the two new denominators; reduce to lowest terms.
Identifying algebra’s main goal
The ultimate goal of algebra is to find the value of the variable.
Knowing the main rule of algebra
Keep both sides of the equation balanced.
Seeing algebra’s main strategy
“The best way to find x is to isolate it” (p. 328). We do this by balancing the equation, moving all the numbers to one side of the equation, and all the variables on the other side.
Chapter 25: “Ten Important Number Sets to Know”
Natural numbers
An ascending set of numbers, starting from 1.
Integers
Natural numbers, negative numbers, and 0
Rational numbers
Integers and fractions
Irrational numbers
Any number that isn’t rational: these “can be approximated only as a nonterminating, nonrepeating decimal” (p. 331).
Algebraic numbers
These are solutions to algebraic equations that don’t require division by a variable, and whose “variables are raised only to positive, whole-number exponents” (p. 332).
Transcendal numbers
These are any numbers that aren’t algebraic. A study of trigonometry goes more into this.
Real numbers
All rational and irrational numbers.
Imaginary numbers
“[A]ny real number multiplied by the square root of negative one.
Complex numbers
“[A]ny real number plus or minus an imaginary number” (p. 334).
Transfinite numbers
These represent “different levels of infinity” (p. 335).
