“우리 대부분은 초라한 옷차림과 엉터리 가구들을 부끄럽게 여기지만, 초라한 생각과 엉터리 철학을 부끄럽게 여길 줄 알아야 한다.” – Albert Einstein

00250101
algebra

https://www.eeweb.com/tools/algebra-reference-sheet

Arithmetic Operations

The basic arithmetic operations are addition, subtraction, multiplication, and division. These operators follow an order of operation.

Addition

Addition is the operation of combining two numbers. If more than two numbers are added this can be called summing. Addition is denoted by + symbol. The addition of zero to any number results in the same number. Addition of a negative number is equivalent to subtraction of the absolute value of that number.

Subtraction

Subtraction is the inverse of addition. The subtraction operator will reduce the first operand (minuend) by the second operand (subtrahend). Subtraction is denoted by – symbol.

Multiplication

Multiplication is the product of two numbers and can be considered as a series of repeat addition. Multiplication of a negative number will result in the reciprocal of the number. Multiplication of zero always results in zero. Multiplication of one always results in the same number.

Division

Division is the method to determine the quotient of two numbers. Division is the opposite of multiplication. Division is the dividend divided by the divisor.

Arithmetic Properties

The main arithmetic properties are Associative, Commutative, and Distributive. These properties are used to manipulate expressions and to create equivalent expressions in a new form.

Associative

The Associative property is related to grouping rules. This rule allows the order of addition or multiplication operation on numbers to be changed and result the same value.

latex!encoded:base64,QSAqIChCKkMpID0gKEEqQikgKiBD
Commutative

The Commutative property is related the order of operations. This rule applies to both addition and subtraction and allows the operands to change order within the same group.

latex!encoded:base64,QSArIEIgKyBDID0gQiArIEEgKyBD
Distributive

The law of distribution allows operations in some cases to be broken down into parts. The property is applied when multiplication is applied to a group of division. This law is applied in the case of factoring.

latex!encoded:base64,QSAqIChCK0MpID0gQSAqIEIgKyBBICogQw==

Arithmetic Operations Examples

latex!encoded:base64,YWIrYWMgPSBhIChiK2Mp
latex!encoded:base64,YSBcbGVmdCAoIFxmcmFje2J9e2N9IFxyaWdodCApID0gXGZyYWN7YWJ9e2N9
latex!encoded:base64,XGZyYWN7XGxlZnQgKCBcZnJhY3thfXtifSBccmlnaHQgKX17Y30gPSBcZnJhY3thfXtiY30=
latex!encoded:base64,XGZyYWN7YX17XGxlZnQgKCBcZnJhY3tifXtjfSBccmlnaHQgKX0gPSBcZnJhY3thY317Yn0=
latex!encoded:base64,XGZyYWN7YX17Yn0gKyBcZnJhY3tjfXtkfSA9IFxmcmFje2FkICsgYmN9e2JkfQ==
latex!encoded:base64,XGZyYWN7YX17Yn0gLSBcZnJhY3tjfXtkfSA9IFxmcmFje2FkIC0gYmN9e2JkfQ==
latex!encoded:base64,XGZyYWN7YS1ifXtjLWR9ID0gXGZyYWN7Yi1hfXtkLWN9
latex!encoded:base64,XGZyYWN7YStifXtjfSA9IFxmcmFje2F9e2N9ICsgXGZyYWN7Yn17Y30=
latex!encoded:base64,XGZyYWN7YWIrYWN9e2F9ID0gYiArIGMsIGEgXG5lcSAgMA==
latex!encoded:base64,XGZyYWN7XGxlZnQgKCBcZnJhY3thfXtifSBccmlnaHQgKX17XGxlZnQgKCBcZnJhY3tjfXtkfSBccmlnaHQgKX0gPSBcZnJhY3thZH17YmN9

Exponent Properties

latex!encoded:base64,YV5uIGFebSA9IGFee24rbX0=
latex!encoded:base64,KGFebilebSA9IGEgXm5ebQ==
latex!encoded:base64,KGFiKV5uID0gYV5uYl5u
latex!encoded:base64,YV57LW59ID0gXGZyYWN7MX17YX0=
latex!encoded:base64,XGxlZnQgKCBcZnJhY3thfXtifSBccmlnaHQgKV57LW59ID0gXGxlZnQgKCBcZnJhY3tifXthfSBccmlnaHQgKV5uID0gXGZyYWN7Yl5ufXthXm59
latex!encoded:base64,XGZyYWN7YV5ufXthXm19ID0gYV57bi1tfSA9IFxmcmFjezF9e2Fee20tbn19
latex!encoded:base64,YV4wID0gMSwgYSBcbmVxIDA=
latex!encoded:base64,XGxlZnQgKCBcZnJhY3thfXtifSBccmlnaHQgKV5uID0gXGZyYWN7YV5ufXtiXm59
latex!encoded:base64,XGZyYWN7MX17YV57LW59fSA9IGFebg==
latex!encoded:base64,YV5cZnJhY3tufXttfSA9IFxsZWZ0ICggYV5cZnJhY3sxfXttfSBccmlnaHQgKV5uID0gXGxlZnQgKCBhXm4gXHJpZ2h0ICleXGZyYWN7MX17bX0=

Properties of Radicals

latex!encoded:base64,XHNxcnRbbl17YX0gPSBhXlxmcmFjezF9e259
latex!encoded:base64,XHNxcnRbbV17XHNxcnRbbl17YX19ID0gXHNxcnRbbW5de2F9
latex!encoded:base64,XHNxcnRbbl17YWJ9ID0gXHNxcnRbbl17YX0gXHNxcnRbbl17Yn0=
latex!encoded:base64,XHNxcnRbbl17XGZyYWN7YX17Yn19ID0gXGZyYWN7XHNxcnRbbl17YX19e1xzcXJ0W25de2J9fQ==
latex!encoded:base64,XHNxcnRbbl17YV5ufSA9IGEsIFx0ZXh0cm17IGlmIFx0ZXh0c2x7bn0gaXMgb2RkfQ==
latex!encoded:base64,XHNxcnRbbl17YV5ufSA9IFxsZWZ0IHwgIGFccmlnaHQgfCwgXHRleHRybXsgaWYgXHRleHRzbHtufSBpcyBldmVufQ==

Properties of Inequalities

latex!encoded:base64,XHRleHRybXsgaWYgfSBhIDwgYiBcdGV4dHJteyB0aGVuIH0gYSArIGMgPCBiICsgYyBcdGV4dHJteyBhbmQgfSBhIC0gYyA8IGIgLSBj
latex!encoded:base64,XHRleHRybXsgaWYgfSB7YTxifSA=
latex!encoded:base64,XHRleHRybXsgYW5kIH0ge2M+MH0gXHRleHRybXsgdGhlbiB9IGFjIDwgYmMgXHRleHRybXsgYW5kIH0gXGZyYWN7YX17Yn0gPCBcZnJhY3tifXtjfQ==
latex!encoded:base64,XHRleHRybXsgaWYgfSB7YTxifSBcdGV4dHJteyBhbmQgfSB7YzwwfSA=
latex!encoded:base64,XHRleHRybXsgdGhlbiB9IGFjPmJjIFx0ZXh0cm17IGFuZCB9IFxmcmFje2F9e2J9PlxmcmFje2J9e2N9

Properties of Absolute Value

latex!encoded:base64,XGxlZnQgfCBhIFxyaWdodCB8ID0gXGxlZnRce1xiZWdpbnttYXRyaXh9CmEsICYgXHRleHRybXsgaWYgfSBhIFxnZXEgMCBcXCAKLWEsICYgXHRleHRybXsgaWYgfSBhIDwgMApcZW5ke21hdHJpeH1ccmlnaHQu
latex!encoded:base64,XGxlZnQgfCBhIFxyaWdodCB8ID0gXGxlZnQgfCAtYSBccmlnaHQgfA==
latex!encoded:base64,XGxlZnQgfCBhIFxyaWdodCB8IFxnZXEgMA==
latex!encoded:base64,XGxlZnQgfCBhYiBccmlnaHQgfCA9IFxsZWZ0IHwgYSBccmlnaHQgfCBcbGVmdCB8IGIgXHJpZ2h0IHw=
latex!encoded:base64,XGxlZnQgfCBcZnJhY3thfXtifSBccmlnaHQgfCA9IFxmcmFje1xsZWZ0IHwgYSBccmlnaHQgfH17XGxlZnQgfCBiIFxyaWdodCB8fQ==
latex!encoded:base64,XGxlZnQgfCBhK2IgXHJpZ2h0IHwgXGxlcSBcbGVmdCB8IGEgXHJpZ2h0IHwgKyBcbGVmdCB8IGIgXHJpZ2h0IHw=

Complex Numbers

Definition of Complex Numbers

Complex numbers are an extension of the real number system. Complex numbers are defined as a two dimension vector containing a real number and an imaginary number. The imaginary unit is defined as:

latex!encoded:base64,aSA9IFxzcXJ0LTE=

The complex number format where a is a real number and b is an imaginary number is defined as:

latex!encoded:base64,YSArIGJp

Unlike the real number system where all numbers are represented on a line, complex numbers are represented on a complex plane, one axis represents real numbers and the other axis represents imaginary numbers.

Properties of Complex Numbers

latex!encoded:base64,aSA9IFxzcXJ0LTE=
latex!encoded:base64,YSArIGJp
latex!encoded:base64,aV57Mn0gPS0xCg==
latex!encoded:base64,XHNxcnR7LWF9ID0gaVxzcXJ0e2F9LCBcIGFcZ2VxIDA=
latex!encoded:base64,XGxhcmdlIChhK2JpKSsoYytkaSk9YSArIGMgKyAoYitkKWk=
latex!encoded:base64,XGxhcmdlIChhK2JpKS0oYytkaSk9YSAtIGMgKyAoYi1kKWk=
latex!encoded:base64,KGEgKyBiaSkoYytkaSkgPSBhYyAtIGJkICsgKGFkICsgYmMpaQ==
latex!encoded:base64,KGErYmkpKGEtYmkpID0gYV57Mn0rYl57Mn0=
latex!encoded:base64,fGEgKyBiaXwgPSBcc3FydHthXnsyfStiXjJ9fQ==
latex!encoded:base64,XG92ZXJsaW5leyhhK2JpKX0gPSBhIC0gYmk=
latex!encoded:base64,XG92ZXJsaW5leyhhK2JpKX0oYStiaSkgPSB8YSArIGJpfF57Mn0=

Logarithms

Definition of Logarithms

A logarithm is a function that for a specific number returns the power or exponent required to raise a given base to equal that number. Some advantages for using logarithms are very large and very small numbers can be represented with smaller numbers. Another advantage to logarithms is simple addition and subtraction replace equivalent more complex operations. The definition of a logarithms is:

latex!encoded:base64,eSA9IGxvZ197Yn14
latex!encoded:base64,XGxhcmdlIHg9Yl57eX0=
latex!encoded:base64,XGxhcmdlIHg+MA==

, where   and   

Definition of Natural Log
latex!encoded:base64,bG5cIHggPSBsb2dfe2V9eA==
latex!encoded:base64,ZT0yLjcxODI4MTgyODQ1OQ==

,  where  

Definition of Common Log
latex!encoded:base64,bG9nXCB4ID0gbG9nX3sxMH14

Logarithm Properties

latex!encoded:base64,bG9nX3tifWIgPSAx
latex!encoded:base64,bG9nX3tifTE9MA==
latex!encoded:base64,bG9nX3tifWJee3h9ID0geA==
latex!encoded:base64,XGxhcmdlIGJee2xvZ197Yn14fSA9IHg=
latex!encoded:base64,bG9nX3tifSh4XntyfSk9cmxvZ197Yn0gXCB4
latex!encoded:base64,bG9nX3tifSh4eSkgPSBsb2dfe2J9eCArIGxvZ197Yn15
latex!encoded:base64,bG9nX3tifShcZnJhY3t4fXt5fSkgPSBsb2dfe2J9eC1sb2dfe2J9eQ==

Factoring

Polynomials

A polynomial is an expression made up of variables, constants and uses the operators addition, subtraction, multiplication, division, and raising to a constant non negative power. Polynomials follow the form:

latex!encoded:base64,Zih4KSA9IGFfe259eF57bn0rYV97bi0xfXhee24tMX0rLi4uK2FfezJ9eF57Mn0rYV97MX14K2FfezB9

The polynomial is made up of coefficients multiplied by the variable raised to some integer power. The degree of a polynomial is determined by the largest power the variable is raised.

Quadratic Equation

A quadratic equation is a polynomial of the second order.

latex!encoded:base64,YXheezJ9K2J4K2M9MA==

The solution of a quadratic equation is the quadratic formula. The quadratic formula is:

latex!encoded:base64,eD1cZnJhY3stYlxwbSBcc3FydHtiXnsyfS00YWN9fXsyYX0=

Common Factoring Examples

latex!encoded:base64,eF4yIC0gYV4yID0gKHgrYSkoeC1hKQ==
latex!encoded:base64,eF4yICsgMmF4ICsgYV57Mn0gPSAoeCthKV57Mn0=
latex!encoded:base64,eF57Mn0gLSAyYXggKyBhXnsyfSA9ICh4LWEpXjI=
latex!encoded:base64,eF57Mn0rKGErYil4ICsgYWIgPSh4K2EpKHgrYik=
latex!encoded:base64,eF57M30gKyAzYXheezJ9ICsgM2FeezJ9eCArYV57M30gPSAoeCthKV57M30=
latex!encoded:base64,eF57M30gKyBhXnszfSA9ICh4ICsgYSkoeF57Mn0gLSBheCArIGFeezJ9KQ==
latex!encoded:base64,eF57M30gLSBhXnszfSA9ICh4IC0gYSkoeF57Mn0gKyBheCArYV57Mn0p
latex!encoded:base64,eF57Mm59IC0gYV57Mm59ID0gKHhee259IC0gYV57bn0pKHhee259ICthXntufSk=

Square Root

The square root is a function where the square root of a number (x) results in a number (r) that when squared is equal to x.

latex!encoded:base64,XHNxcnQgeCA9IHIg
latex!encoded:base64,cl57Mn0gPSB4

  and 

Also the square root property is:

latex!encoded:base64,eF57Mn0gPSBh
latex!encoded:base64,eD0gXHBtXHNxcnQgYQ==

if  then 

Absolute Value

latex!encoded:base64,fHh8ID0gYlxyaWdodGFycm93IHggPSBi
latex!encoded:base64,eCA9IC1i

  or  

latex!encoded:base64,fHh8IDxiIFxyaWdodGFycm93ICAtYiA8eCA8Yg==
latex!encoded:base64,fHh8PmJccmlnaHRhcnJvdyAgeDwtYiA=
latex!encoded:base64,eD5i

  or  

Completing the Square

Completing the square is a method used to solve quadratic equations. Algebraic properties are used to manipulate the quadratic polynomial to change its form. This method is one way to derive the quadratic formula.

latex!encoded:base64,YXheMitieCtjPWEoLi4uKV4yK2NvbnN0YW50

The steps to complete the square are:

  1. Divide by the coefficient a.
  2. Move the constant to the other side.
  3. Take half of the coefficient b/a, square it and add it to both sides.
  4. Factor the left side of the equation.
  5. Use the square root property.
  6. Solve for x.

Functions and Graphs

Expressions evaluated at incremental points then plotted on a Cartesian coordinate system is a plot or graph.

Constant Function

When a function is equal to a constant, for all values of x, f(x) is equal to the constant. The graph of this function is a straight line through the point (0,c).

latex!encoded:base64,Zih4KT1j

Linear Function

A linear function follows the form:

latex!encoded:base64,Zih4KT1teCti

The graph of this function has a slope of m and the y intercept is b. It passes through the point (0,b). The slope is defined as:

latex!encoded:base64,bT1cZnJhY3t5X3syfS15X3sxfX17eF97Mn0teF97MX19PVxmcmFje3Jpc2V9e3J1bn0=

An addition form for linear functions is the point slope form:

latex!encoded:base64,eT15X3sxfSArIG0oeC14X3sxfSk=

Parabola or Quadratic Function

A parabola is a graphical representation of a quadratic function.

latex!encoded:base64,Zih4KT1heF4yK2J4K2M=

The graph of a parabola in this form opens up if a>0 and opens down if a<0. The vertex of the parabola is located at:

latex!encoded:base64,XGxlZnQgKCAtXGZyYWN7Yn17MmF9LCBcbGVmdCBmKCAtXGZyYWN7Yn17MmF9IFxyaWdodCApIFxyaWdodCAp

Other forms of parabolas are:

latex!encoded:base64,Zyh5KSA9IGF5XnsyfSArYnkrYw==

The graph of a parabola in this form opens right if a>0 or opens left if a<0. The vertex of the parabola is located

latex!encoded:base64,XGxlZnQgKCBnXGxlZnQgKCAtXGZyYWN7Yn17MmF9IFxyaWdodCApLC1cZnJhY3tifXsyYX0gXHJpZ2h0ICk=

Circle

The function of a circle follows the form:

latex!encoded:base64,KHgtaCleMisoeS1rKV4yPXJeMg==

Where the center of the circle is (h,k) and the radius of the circle is r.

Ellipse

The function of an ellipse follows the form:

latex!encoded:base64,XGZyYWN7KHgtaCleMn17YV4yfStcZnJhY3soeS1rKV4yfXtiXjJ9PTE=

Where the center of the ellipse is (h,k)

Hyperbola

The function of a Hyperbola that opens right and left from the center follows the form:

latex!encoded:base64,XGZyYWN7KHgtaCleMn17YV4yfS1cZnJhY3soeS1rKV4yfXtiXjJ9PTE=

The function of a Hyperbola that opens up and down from the center follows the form:

latex!encoded:base64,XGZyYWN7KHktayleMn17Yl4yfS1cZnJhY3soeC1oKV4yfXthXjJ9PTE=

Where the center of the hyperbola is (h,k), with asymptotes that pass through the center with slopes of:

latex!encoded:base64,bSA9IFxwbSBcZnJhY3tifXthfQ==

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