Math fundamentals

Summary of notes I took while working through this playlist on youtube.

Basic Operations

Both operands in addition are called addends, they result in a sum. The left operand of a subtraction is called the minuend, the right one the subtrahend. The result is the difference. In multiplication both operands are called the factors of a product (the result). Operands and result of a division, from left to right, are called dividend, divisor and quotient.

Addition and multiplication possess commutative and associative properties. The former means the addends or factors can be placed in any order. The latter signifies that they can be grouped in any order. e.g.

Commutative Property

a+b=b+aa + b = b + a
a×b=b×aa \times b = b \times a

Associative Property

(a+b)+c=(a+c)+b(a + b) + c = (a + c) + b
(a×b)×c=b×(a×c)(a \times b) \times c = b \times (a \times c)

Multiplication further possesses the distributive property:

Distributive Property

a×(b+c)=(a×b)+(a×c)a \times (b + c) = (a \times b) + (a \times c)

Rounding and Estimating

Generally only the place value to the right of the place value we want to round to is relevant in rounding to a value. If it is greater than 5 the place value we are rounding to is increased by 1. e.g

278,751278,751 rounded to the ten thousand 280,000\approx 280,000
29.58429.584 rounded to the hundredth 29.58\approx 29.58

The place values to the right of the one we round to are either omitted or turned to 0s.

When estimating individual values are rounded first before they are added or subtracted.

Remainder

In arithmetic the remainder is what's left after a division operation, e.g 939/4=234r3939 / 4 = 234r3. The modulo operation is used to find the remainder of a division which would otherwise wind up with a decimal. The remainder can be traced from a decimal quotient by the following steps:

939/4=234.75939 / 4 = 234.75 | subtract the whole number 234.75234=0.75234.75 - 234 = 0.75 | multiply by the original divisor (4) r=0.75×4r = 0.75 \times 4
r=3r = 3

Average

Given a list of numbers, the average value is found by adding the numbers and dividing the sum by the number of numbers. This is also called the mean.

n=2,8,10,4n = { 2, 8, 10, 4 } average value: (2+8+10+4)/4=6(2 + 8 + 10 + 4) / 4 = 6

The median is the value in the middle of a sorted set of values. Given a set of 1,2,3,5,7,9,10{ 1, 2, 3, 5, 7, 9, 10 }, the median is the value 55. Given a set of 0,1,5,6,8,14{ 0, 1, 5, 6, 8, 14 }, the median is derived from the set 5,6{ 5, 6 } because the numbers of values in the set is even. In the latter case the median is calculated by adding both median values and dividing them by 2: (5+6)÷2=5.5(5+6)\div2 = 5.5.

The mode is the most commonly occuring value in a set. A set may have multiple or no modes. The order of values in the set doesn't matter to the mode. Example: The mode of 14,10,10,12,15,15,15,7,8{ 14, 10, 10, 12, 15, 15, 15, 7, 8 } is 1515.

Exponents

2×2×2×2×2=252 \times 2 \times 2 \times 2 \times 2 = 2^5
The 22 in 252^5 is called the base, 55 is the exponent.

222^2 is read as "2 squared"
232^3 is read as "2 cubed"
242^4 is read as "2 to the power of 4" (all exponents greater than 3 follow this rule)

When paired with paranthesis the result of the inside operation has to be multiplied with itself n1n-1 times. e.g (86)2=42=4×4(8 - 6)^2 = 4^2 = 4 \times 4.

Terms, Expressions and Equalities

Terms are the components of an operations that are added to or subtracted from another. Expressions are made up of two or more terms. Equalities are a set of expressions that are equal in value. Terms may have variables, letters that are meant to be substituted with numbers. Evaluating an expression means to plug in values for its variables and calculating the result. Equalities can be solved for a variable to find a variable's value.

Integers and Absolute Value

Integers are whole numbers ranging from negative infinite to positive infinite including the 0.

(x)=x-(-x) = x
(y)=y-(y) = -y
7(9)=7+97 -(-9) = 7 + 9

A number's absolute value is its distance from 0. It's written as a number between pipes: 5=5|-5| = 5, 7=7|7| = 7.

Rules for integer addition and subtraction:
Subtraction problems are first turned into addition problems by putting the subtracted amount into paranthesis and prefixing those with a plus: 34=3+(4)3 - 4 = 3 + (-4). If both operands share the same sign, they are to be added and the sign kept. If their signs differ, the smaller number is to be subtracted from the larger one and the larger number's sign is kept for the result.
4+(18)=144 + (-18) = -14
2+(5)=7-2 + (-5) = -7

If an expression consists of more than two terms, individual terms are to be worked from one after another. e.g.
(2)+4+(11)(-2) + 4 + (-11)
=2+(11)= 2 + (-11)
=9= -9

Multiplication and division have to be calculated first:
7+3(2)=7+3×2-7 + 3(2) = -7 + 3 \times 2
7+6=1-7 + 6 = -1

Alternatively one can sum up all positive and negative integers individually before subtracting the latter from the former: 7+(12)+(3)+8=15157 + (-12) + (-3) + 8 = 15 - 15

Rules for integer multiplication and division: If both operands share the same sign, the result ist positive. Otherwise the result is negative. e.g.
(2)×(2)=4(-2) \times (-2) = 4
2×5=102 \times -5 = -10 (the same applies for division problems)

Unary minus is basically short for n×1n \times -1:
(6)2=(6)(6)=36(-6)^2 = (-6)(-6) = 36
62=1×62=36-6^2 = -1 \times 6^2 = -36
(9)=1×9=9-(-9) = -1 \times -9 = 9

Primes

Prime numbers are numbers that are divisible into a whole number only by 1 and itself. e.g. 6 is not a prime number because it can be divided cleanly by 1, 6, 3 and 2. 5 on the other hand is a prime number because it can only neatly divided by 1 and 5. 2 is the only even number prime number because every other even number is divisible by 2.
Composite numbers are all numbers that are not prime. They can be interpreted as products of various factors.

Prime factorization

Composite numbers can be decomposed into factors of smaller integers, e.g 30=2×1530 = 2 \times 15. Further decomposing composite factors yields a tree with a prime number in each leaf node: 15=3×515 = 3 \times 5. That process is called prime factorization. Every composite number can be written as a product of its prime factors, e.g. 2×3×52 \times 3 \times 5.

Fractions

Fractions are two numbers written vertically around a horizontal bar. The upper number is called the numerator, the lower number is called the *denominator. Fractions signify the relationship between two numbers. In counting they show a portion of a whole.
Proper fractions have a numerator that is lesser than their denominator: 12\frac 1 2, 38\frac 3 8. Improper fractions have a numerator that is greater or equal in value to their denominator: 74\frac 7 4, 55\frac 5 5. They can be written as mixed numbers, e.g. 73=213\frac 7 3 = 2 \frac 1 3. To find a mixed number, divide the numerator of an improper fraction by its denominator. The quotient becomes the whole number part of the mixed number and the remainder becomes the new numerator, while the denominator stays the same.
Therefore the remainder of a division can be written as a fraction. In reverse an improper fraction can be derived from a mixed number by multiplying the whole number with the denominator, adding the numerator to the resulting product and using the sum as the new numerator: 456=2964 \frac 5 6 = \frac{29}{6}. Again the denominator doesn't change.

Dividing the numerator and denominator of a fraction by their highest common factor results in an equal fraction that is in its simplest form: 48÷4=12\frac 4 8 \div 4 = \frac 1 2, 2135=35\frac {21}{35} = \frac 3 5. Fractions are equal when they share a common value although their denominators are not the same, e.g. 55=99\frac 5 5 = \frac 9 9, 12=48\frac 1 2 = \frac 4 8. When a fraction's parts are divided by a number that is not their highest common factor the resulting quotient will be further reducible.

Common factors in fractions cancel each other out: 1220÷4=35\frac {12} {20} \div 4 = \frac 3 5 => 4×34×5=1220=35\frac{4 \times 3}{4 \times 5} = \frac {12} {20} = \frac 3 5.

Variables are left alone when reducing: 42x66=7x11\frac {42x} {66} = \frac {7x} {11} (has the highest common factor 66).

Prime factorization can be utilized to simplify fractions whose highest common factor is not easy to identify: 30108=2×3×52×2×3×3×3=2×3×52×2×3×3×3=518\frac {30} {108} = \frac {2 \times 3 \times 5} {2 \times 2 \times 3 \times 3 \times 3} = \frac {\cancel{2} \times \cancel{3} \times 5} {\cancel{2} \times 2 \times \cancel{3} \times 3 \times 3 } = \frac {5} {18}. This also works with exponents: 6x22x3=3x2x3=3×x×xx×x×x=3x\frac {6x^2} {2x^3} = \frac {3x^2} {x^3} = \frac {3 \times \cancel{x} \times \cancel{x}} {\cancel{x} \times \cancel{x} \times x} = \frac {3} {x} (simplify constants first).

A whole number nn as a fraction is n1\frac n 1 (nn parts of 11). So 6x460x=x310=1×x310×1=110x3\frac {6x^4} {60x} = \frac {x^3} {10} = \frac {1 \times x^3} {10 \times 1} = \frac {1}{10}x^3.

To multiply fractions, just multiply their numerators and denominators individually (as above), e.g. 23×57=1021\frac 2 3 \times \frac 5 7 = \frac {10} {21}. A product of fractions can be simplified before multiplying: 67×1427=6×147×27=2×21×9=49\frac 6 7 \times \frac {14} {27} = \frac {6 \times 14} {7 \times 27} = \frac {2 \times 2} {1 \times 9} = \frac 4 9.

To divide fractions multiply the dividend by the reciprocal of the divisor. The reciprocal in terms of fractions is a fraction with numerator and denominator switched. Examples:
ab÷cd=ab×dc\frac a b \div \frac c d = \frac a b \times \frac d c
ab÷c=ab÷c1=ab×1c\frac a b \div c = \frac a b \div \frac c 1 = \frac a b \times \frac 1 c

Always turn mixed numbers into improper fractions before working with them.

To add or subtract fractions they require to have the same denominator. In that case the numerators get added or subtracted, while the denominator stays the same: 2011+611+711=3311=3\frac {20} {11} + \frac {6}{11} + \frac {7}{11} = \frac {33}{11} = 3. If one fraction is negative the minus sign moves to its numerator! 118+68=11+68=58- \frac {11}{8} + \frac 6 8 = \frac {-11 + 6} {8} = \frac {-5}{8}

The lowest common denominator (LCD) of a set of fractions is the smallest number that has all of the denominators as a factor, e.g. the LCD of 78\frac 7 8 and 1116\frac {11}{16} is 1616 (16 divided by 16 and 8 all result in whole numbers). To find LCDs that are not obvious the highest denominator is increased in multiplies until it results in a number that has the lowest denominator as a factor.
The lowest common denominator can also be found by multiplying the denominators together: 3427\frac 3 4 \frac 2 7 have an LCD of 28 (3×43 \times 4).

Assuming one numerator in a set of fractions is missing 35=x10\frac 3 5 = \frac x 10, the missing value can be found by multiplying the factor that multiplies the lower denominator to the higher with the known numerator: 35=3×210\frac 3 5 = \frac {3 \times 2}{10}.

In order to add or subtract fractions with differing denominators the first step is to convert them into fractions with the same denominators: 34+16\frac 3 4 + \frac 1 6

  1. Find the LCD (16)
  2. Multiply numerator and denominator of both fractions by the number required for both denominators to reach the LCD (912+212\frac {9}{12} + \frac {2}{12})
  3. Add/Subtract (1112\frac {11}{12})

If one fraction is negative, problems can be solved easiest by moving the minus sign to the numerator first (118118118-\frac {1}{18} \frac {-1}{18} \frac {1}{-18} are all equal).

3x4x204x=3x204x\frac {3x} {4x} - \frac {20}{4x} = \frac {3x - 20} {4x} is in its simplest form because variables cannot be simplified when numerator or denominator are an expression.

A complex fraction is a fraction with a fraction in either of its values, e.g. 7x1015\cfrac{\frac{7x}{10}}{\frac{1}{5}}. Dealing with them simply involves multiplying the numerator with the reciprocal of the denominator of the main fraction.

Decimals

Place values behind the decimal are called tenths, hundredths, thousandths ...

Decimals can be written as fractions, examples:
43100=0.42\frac {43}{100} = 0.42 (dividing by 100 makes decimal end in the hundreth)
5011000=0.501\frac {501}{1000} = 0.501 (dividing by 1000 makes decimal end in thousandth). 0.2193=2193100000.2193 = \frac {2193}{10000}
2.12=212100-2.12 = \frac {-212}{100}

If a number's place value repeats infinitely it's denoted by a stroke above the number: 23=0.6=6\frac 2 3 = 0.6\={6}

Solving equations

Since both sides of an equation are either equal (true equation) or unequal (untrue equation), performing the same operation on both sides yields in the same answer. Multiplication and division have to be applied to all terms of an expression, while additions and subtractions are only applied once per expression. Examples:
x4=1x - 4 = -1 | +4+4 (isolate x to find its value)
x=3x = 3

When a variable shows up in both expressions, elimite the smaller one through subtraction:
9x=8x+49x = 8x + 4 | 8x-8x
9x8x=49x - 8x = 4
x=4x = 4 (1x=x)(1x = x)

Because 2x=2×x2x = 2 \times x, division has to be used to solve equations like 2x=102x = 10:
2x=102x = 10 | ÷2\div2
x=5x = 5

If the coefficient is negative the expressions should be divided by a negative number:
7y=35(7)y=35-7y = 35 \Harr (-7)y = 35 | ÷7\div-7
y=5y = -5

Considering that fractions signify division we multiply by the denominator:
x5=3\frac x 5 = -3 | ×35\times35 x=3×5x = -3 \times 5
x=15x = -15

x=4-x = -4 | ÷1\div-1
x=4x = 4

Fractions in equations can be removed by following these steps: y23=512y - \frac 2 3 = \frac {5}{12}

  1. Write everything as a fraction. Terms that are not fractions become numerators over 1 (x=x1x = \frac x 1).
    y123=512\frac y 1 - \frac 2 3 = \frac {5}{12}
  2. Find the lowest common denominator (12)
  3. Multiply every term by the LCD (12×y1×112×21×3=12×51×12\frac {12 \times y}{1 \times 1} - \frac {12 \times 2}{1 \times 3} = \frac {12 \times 5}{1 \times 12})
  4. Simplify to remove all fractions (12y8=512y - 8 = 5)

If a fraction serves as coefficient to a variable, the variable is moved to the numerator of the coefficient: 15y=1y5\frac 1 5 y = \frac {1y}{5}, 37x=3x7\frac 3 7 x = \frac {3x}{7}.

Simplifying Expressions

Expressions can be simplified by combining terms that are alike, terms that include the same variable and exponent. The terms of 4y33xy+x94y^3 - 3xy + x -9 are:
4y34y^3
3xy-3xy
xx
9-9

Terms containing varialbes are variable terms. Those without variables are constant terms. Variable terms are made up of the variable part and the coefficient. The coefficient is the multiplicative factor in front of the variable. xx implies 1×x1 \times x and x-x implies 1×x-1 \times x. The expression 13ab+43ab1013ab + 4 - 3ab - 10 has the following sets of like terms:
13ab,3ab{ 13ab, -3ab }
+4,10{ +4, -10 }

All constant terms are like terms.
The terms 3y3y and 2y22y^2 are not like terms!

Like terms are combined by adding their coefficients: 13ab+(3ab)=10ab13ab + (-3ab) = 10ab. A term that simplifies to 00 can be omitted.

Simplifications for multiplications:
5(4x)=(5×4)×x=20x5(4x) = (5 \times 4) \times x = 20x
(3)(4y)=12y(-3)(-4y) = 12y

If the inside of paranthesis is an expression, the multiplication's distributive property helps in simplifying:
2(4+x)=2×4+2×x=8+2x=10x2(4 + x) = 2 \times 4 + 2 \times x = 8 + 2x = 10x
4(2a+3)=4(2a)+(4)3=8a+(12)=8a12-4(2a + 3) = -4(2a) + (-4)3 = -8a + (-12) = -8a -12 (x+y)=1×x+1×y=x+y-(x+y) = -1 \times x + -1 \times y = -x + -y

In complex expressions distribution comes before simplification of like terms. In equations, like terms can only be combined if they are on the same side of the equation:
x7=26x7=8x - 7 = -2 - 6 \Harr x - 7 = -8

(y2+3)(y+5)(y^2 + 3) (y + 5) | this is also solved by making use of the distributive property.

(y2y)+(y2+5)+(3y)+(15)(y^2 * y) + (y^2 + 5) + (3y) + (15)

y3+y2+5+3y+15y^3 + y^2 + 5 + 3y + 15

y3+y2+20+3yRatioAratiocomparestwonumbers.Everyfractionisaratioifbothnumbersrepresentthesameunit.15milesoutof45milesmaybewrittenas<?ktx1545y^3 + y^2 + 20 + 3y

Ratio

A ratio compares two numbers. Every fraction is a ratio if both numbers represent the same unit. 15 miles out of 45 miles may be written as <?ktx \frac {15}{45} or 1:3. Units are not written and ratios may be simplified.

Rate

Units in rates are different and written with their values. They can be reduced but not written as whole numbers. Example: 30kmhour\frac {30km}{hour}.

Proportions

A proportion is the equality of two ratios (fractions). Among other it's used for scales. A generic proportion is ab=cd\frac a b = \frac c d. Two fractions are proportional if their cross products are equal: a×d=b×ca \times d = b \times c. This makes proportional equations quite easy to solve for one value:

59=x45\frac 5 9 = \frac{x}{45} 5×45=2255 \times 45 = 225
9×x=2259 \times x = 225 | ÷9\div 9
x=25x = 25

This can be applied to many problems. Imagine a certain amount of material costs a certain amount. This techniques calculates how much a different amount of the material would cost.

Percent

Defined as parts of a hundred: 11% = \frac {1}{100} = 0.01.

A decimal is converted to a percent by moving the digits 2 places to the left.
A percent is converted to a decimal by moving the digits 2 places to the right.

A fraction is converted to a percent by multiplying by 100.

Equations dealing with percentages can be solved using the cross product of two equivalent fractions. e.g. Find 2525% of 8484.

isof=percent100\frac {is}{of} = \frac { percent } {100}

x84=25100\frac {x}{84} = \frac {25} {100}

Square Roots

The square root of a number is the inverse of squaring that number. Given 22=42^2 = 4, the square root is 42=2\sqrt[2]{4} = 2. Since a squared number is always positive, the square root of a negative number can't be calculated by ordinary means.

Basic Geometry

The perimeter of any polygon is calculated by finding the sum of the lengths of its sides.

P=A+B+C+D+EP = A + B + C + D + E

The area of a region is defined as its amount of surface. The area for a rectangle is calculated by multiplying its sides:

A=l×hA = l \times h

If the units of ll and hh are cmcm, the unit of AA will be cm2cm^2. In case of a square the area is simply the length of one of its sides squared (A=l2A = l^2)

The distance of a circle is found by the formula C=2πrC = 2\pi r, where rr is the radius - the line from the center of the circle straight to its edge.

π=3.14159\pi = 3.14159 (approximately)

Pythagorean Theorem

Given the lengths of 2 sides of a right angular triangle (a triangle of which one angle is 90 degrees), the length of the third size can be calculated. Given the formula b2+h2=c2b^2 + h^2 = c^2 which assumes bb and hh to be the sides adjacent to the right angle and cc to be the side opposite to the angle, one may find an unknown side length by solving the resulting equation.