## Saturday, March 10, 2018

### Basic Mathematics

Basic Mathematics

The time has certainly come to write more information about the beautiful subject of mathematics. For thousands of years, humanity has utilized mathematics for a diversity of purposes. Some wanted to create complex structures globally. Some wanted to count objects, to device formulas, and to improve society in a myriad of positive ways. Learning math can bring excellent career choices from engineer to computer software analysts. Not to mention that mathematics can spark up extra human creativity in a glorious fashion. That is why math is always important. From Euclid’s Elements to the mathematical views of the Greek philosopher Pythagoras, we witness the glory of mathematics. The greatness of mathematics deals with learning about factors, prime numbers, and the concept of i. It is seeing young people and older people search and find solutions to problems that deals with science, economics, and other spheres of human endeavors. Math can be simplistic or complex in dealing with trigonometric or calculus. Also, many cultures brought us contributions in mathematical development. For example, Muslim mathematicians during the Middle Ages brought us the decimal point, Arabic numerals in notation, many trigonometric functions. Ancient Babylonian mathematics was written using a sexagesimal (base-60) numeral system.  From this derives the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle, as well as the use of seconds and minutes of arc to denote fractions of a degree. It is likely the sexagesimal system was chosen because 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30. All of these numbers are factors of the number 60. So, we witness bioformatics being used now and we see people of all ages enjoying the wonder of mathematics.

Numbers

Mathematics deal with numbers. Numbers are digits used in computations, analysis, and other forms of developing our world. From telling time, calculating temperatures, and finishing up equations, numbers are very vital part of math. They help us to count, measure, and label so many items. There are many types of them too. Natural numbers include 1,2,3,4, 5, and so forth into infinity. There is the million which is has 6 zeroes on it. One billion has 9 zeroes. The trillion has 12 zeroes, then there is the quadrillion, quintillion, sextillion, and all the way to the Googol, which has 100 zeroes and the Googolplex which is bigger than the Googol too. There are negative numbers and rational numbers. Rational numbers include numbers like ½ and -2/3. A rational number is any number that can be expressed as the quotient or a fraction like p/q or two integers with a numerator like p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. In general a/b = c/d if and only if a x d = c x b. A real number that isn’t rational is called irrational. Irrational numbers include the  √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. An integer is a number that can be written without a fractional component. They include 21, 4, 0, and -2048. 9.75, 5 1⁄2, and √2 are not integrers. Complex numbers is a number that can be expressed in the  form a + bi, where a and b are real numbers. The number of  i is a solution of the equation x2 = −1, which is called an imaginary number because there is no real number that satisfies this equation. For the complex number a + bi, a is called the real part, and b is called the imaginary part.  In mathematics, the absolute value or modulus |x| of a real number x is the non-negativevalue of x without regard to its sign. Namely, |x| = x for a positive x, |x| = −x for a negative x(in which case −x is positive), and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.

Prime and Composite Numbers

One major type of numbers is called prime numbers. A prime number is any number that is a natural number greater than one that can't be formed by 2 natural numbers (these numbers can't be one). The number 5 is prime, because it can only be formed by 1 X 5 or 5 X 1. So, prime numbers are only divisible by itself and one. Natural numbers are positive integers. Therefore, prime numbers are: 2,3, 5, 7, 11, 13, 17, 19, 23, etc. The ancient world knew of prime numbers for years and years. A composite number is different. A composite number is a whole number that can be divided by itself, the number 1, and other numbers. They have much more factors than prime numbers. A factor is a number that you can multiply in order to get into a new number. For example, the simple math problem of 2 X 3=6 has the factors of 2 and 3. More examples of composite numbers include: 4, 6, 8, 9, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, etc. Prime and composite numbers are very diverse. The numbers of 0 and 1 are neither prime nor composite because of many reasons. 0 is not a product of 2 factors. The number 1 is not part of any factors and it has an infinite number of divisors. Any prime number must be greater than one, so one isn't a prime number. Number 1 is a unit. An unit is a special class of numbers. Natural numbers are counting numbers from 1, 2, 3, etc. Whole numbers include the pattern of 0, 1, 2, 3, etc.

Decimals

Decimals are very unique in mathematics. A decimal is a fraction whose denominator is a power of ten. They exist to the right of the decimal point. One example is that the number 1984.56 has decimals. 0.56 is known as 56 hundredths. To the right of the decimal point is tenths, hundredths, thousandths, ten-thousandths, hundred-thousandths, millionths, etc. There is a link between decimals and fractions too. 0.1 is the same as 1/10. 0.25 is the same as 25/100 or 1/4. You can break down fractions in using the lowest common denominator. So, 3/6 can break down to 1/2. In using addition, subtraction, multiplication, and dividing any number, the decimal points must be in the right place, so mathematical accuracy can be complete.

A fraction is parts to one whole. It is not part of the complete whole. For example, 1/2 of 1 is not 1, but part of 1. The top number is the numerator and the bottom number is the denominator. If the numerator is greater than the denominator, then that fraction is more than one. If it is the opposite, then that number is less than one. Adding fractions with the same denominator is easy since you only need to add the numerators. For example, 2/3 + 1/3 equals to 3/3 or one. When the numerator and denominator is the same, then the answer is always one. To add, subtract, or multiply plus divide fractions with different denominators, then people must use alternative methods like finding the least common denominator or the common denominator. In dividing fractions, you can invert the second fraction as a reciprocal because division is the opposite of multiplication. For example,

1/2 ÷ 1/6 is the same as:

1/2 X 6/1=

6/2=3

3 is the answer as any fraction must be simplified.

In many mathematical problems, there is addition, subtraction, multiplication, and division. The numbers added in addition are called the addends like 6 and 6 in 6+6 are the addends. The solution to 2 addends being added is called the sum. In subtraction, the largest number in the problem is the minuend, the smaller number is the subtrahend, and the answer is the difference. So, in 4-1=3, the minuend is 4, the subtrahend is 1, and the difference is 3. In multiplication, the 2 numbers being used in multiplying are the factors. The answer is the product. So, in 2 X 15 =30, the 2 factors are 1 and 15 while the product is 30. THere are four basic terms to be known in division. THe dividend is the number divided. The divisor is the number that the dividend is divided by. The quotient is the number of times the divisor will go into the dividend. The remainder is a number that is less than the divisor and is too small to be divided by the divisor to form a whole number. So, in a problem with 49474 / 7 = 7067 R 5. In this example 7 is the divisor, 49474 is the dividend, 7067 is the quotient and 5 is the remainder of the division.

In terms of the measuring mass in the metric system, 1 milligram (mg) is 0.001g, 1 centigram is 0.01 g, I decigram is 0.1 gram, 1 dekagram is 10 grams, 1 hectogram is 100 grams, and 1 kilogram is 1,000 g. kilo- is a prefix that means 1,000 times larger than a base unit, hector refers to 100 times larger, deci means 10 times smaller than a base unit, centi- means 100 times smaller than a base unit, and milli- means 1,000 times smaller than the base unit.  In terms of volume, 4 liters is a litter more than 1 gallon. 1 quart is 2 pints. 4 quarts is equal to 1 gallon.

The Order of operations

The Order of operations is one of the most important parts of Elementary Mathematics. It is a simplistic method to solve long mathematical problems. In order to use the order of operations, rules must be utilized. First, you have to look at the problem from left to right. Then, you solve the problem first with roots and exponents, then parenthesis, then multiplication and division, and lastly with addition and subtraction. For example, if a problem exists with the following:

√(1+3) + 5, then the answer is

√4  +5

2 +5

Another problem is :

3 + 6 x (5 + 4) ÷ 3 - 7

You solve first by handling the parenthesis, so the that would cause the problem to look like this:

3 + 6 X 9 ÷ 3 - 7

Then you go left and right to solve multiplication first:

3 + 54 ÷ 3 -7

Then, comes division

3 + 18 -7

21  -7

Involving numbers, there are tons of divisibility rules. One is that if a digit ends with the numbers of 0, 2, 4, 6, and 8, then it is an even number. If the sum of the digits of a number is divisible by 3, then it is an even number. There are other rules too. Math has many properties. Math properties include the associative, commutative, identity, and distributive properties.

*The commutative property is about how changing the order of addends or factors does not affect the sum or product.

Some examples include the following:

a x b = c

b x a = c

5 x 7 = 35

7 x 5 = 35

a + b =c

b + a = c

12 + 6 =18

6 + 12 = 18

*The associative property is that the order in which numbers are grounded does not affect the sum or product. Here are some examples of this:

(a + b) + c = d

a + (b+c) = d

(3+5) + 2 =10

3 + (5+2) =10

(a x b) x c = d

a + (b x c) = d

(4x7) x 3 -84

4 x (7x3) = 84

*The distributive property is when adding two or more numbers together, then multiplying the sum by a factor is equal to multiplying each number alone by the factor first, and then adding the products.

One example is:

a (b + c) = (a x b) + (a x c)

4 (1+8) = (4 x 1) + (4X8)

4 X 9 = 4 + 32

36 = 36

*The Identity property is when the additive identity is zero, then you can add zero to the addend and the sum will equal to that addend. In Multiplication, the multiplicative identity is one. If you multiply a factor by one, the product is equal to that factor.

Examples are:

a + 0 =a

8+0=8

a x 1 = a

25 X 1 = 25

Problem solving deals with math too. It deals with everyday life, because we have to problem solve in terms of buying items, planning events, and learning new information as well. One word problem that deals with the Order of Operations is the following:

Adam has \$450. He spends \$210 on food. Later he divides all the money into four parts and gives three parts to his mom and one part he keeps for himself. Then he found \$20 in one of his Minecraft books being used as a bookmark. Write an expression that shows all of this determine how much money he has left?
First, it is time to organize the information. Adam has \$450, show you start with that. Later, you must know that he spends \$210 on food, so that deals with subtraction. Later, he divided the money among 4 people. Every time, you see the word divide or share that involves division. Therefore the equation should be:
(450-210) ÷ 4 +20

That is the expression. The answer should be:

(450-210) ÷ 4 +20
240÷4+20
60+20
80 or \$80 left.

Another problem in dealing with the Order of Operations includes the following:

Sam has \$1,000 to be distributed among two groups equally. Later, the first part is divided among five children and second part is divided among two brothers. Give the expression that represents how the money distribution between two groups was dispersed?

First, 1,000 dollars is what is needed to be known first. Second, the groups are divided into 2, so addition is important to known. Also, each group will have \$500, because each group will have \$1,000 equally distributed. The first group will have 500 ÷ 5 per \$100 each. The second group will have 500 ÷ 2 or \$250 each.

To make an equation, the answer involves \$1,000 = (5 X 100) + (2 X 250).

By Timothy