The laws of mathematics

We all have a mathematical instinct. You can follow along with any calculator if you wish. You will soon learn you can out guess it. Calculator keypads numbers are an example of defining what  integers are. There is only ten including the  0 to 9 that make an infinite combination of numbers.


Algebra teaches us the law of mathematics confirmed by any calculator 12, plus, 13, equals 25, as alternatively 13, plus 12 equals 25. Alternatively 25, take away 12, equals, 13, as, 25 take away, 13. equals, 12 respectively.

If you are a student following along with a calculator agrees. They confirm Algebra's rule it doesn't matter which way we add or take away any numbers the answers are always the same. Algebra teach us taking away is only the opposite of addition and addition the opposite of subtraction.

Algebra says 12 and 13 are the roots of 25. If we follow with any calculator we can find many other examples teaching us the the art of solving equations.

Calculators confirm the Algebraic expression 25 take away, 8, equals,17. To prove it, plus, 8, equals, 25. Algebra teaches us we can alternatively enter 25 take away, 17, equals. Calculator read outs confirm 8. To prove this, 8 plus, 17, equals 25 respectively. Algebra says they are alternate roots to 25 as equally as 13 and 12 are.

We can find many more alternative examples. If you are following along in a calculator enter 25, take away, 6, equals. The read out confirms 19 as, alternatively 25, take away, 19, equals 6. To prove it, plus, 19, equals 25, just as alternatively 19, plus, 6 does.

6 and 19 are alternative roots of 25 just as 12, 13, 8, and 17 are alternate roots the key to understanding Algebra. Lets see what happens when we apply multiplication.

Algebra teaches us the laws of mathematics tells us if we look up in the 4 times table in a children's times table four 5's is 20. If we look up in the 5 times we find the same sum confirming five 4's  equals 20; confirmed by any calculator.

Our mathematical instinct agrees with algebra equal to 5, plus, 5, plus, 5, plus, 5, equals 20. confirmed by any calculator. Alternatively 4, plus 4, plus 4, plus 4, plus 4, equals 20 Once again confirmed confirmed by any calculator. In other words 20 equals 5, plus 5, four  times Alternatively 20 equals 4 plus 4 five  times confirmed by any calculator.

Like wise if we look up in the 6 times table six 7's is 42 just as we would find looking up in the 7 times table seven 6's is 42. Once again confirmed by any calculator.

The sum equals 42 equals 6 plus 6 seven times. Algebra teaches us alternatively 42 equals 7 plus 7 six times. In other words 42 equals 6 plus 6 seven times and alternatively 42 equals 7 plus 7 six times. Once again confirmed by any calculator.

If you are following along 12, times, 13, equals, 156. Algebra teaches us alternatively 156 divide 13 equals 12 as 12 times 13 equals 156 divide 12 equals 13 respectively another example of the key to solving equations.

It is traditional in math's division is pronounced as an over. In mathematical shorthand  a number over a dividing line of a bottom number. If we take 13 equals a 156 over 12 and alternatively 12 equals a 156 over 13 respectively is a clear example. Calculators confirm like addition it doesn't matter which way we multiply or divide 12 and 13 the answer is always the same.

Over (divide if you like) is only the opposite of multiplication and multiplication the opposite to division. They are opposites to each other. If 12 and 13 are anything to go by are valid roots of a 156. In the same way as addition way we can find many other examples.

If you are following along with a calculator confirms 156, divide, 4, equals, 39, just as, Algebra tells us 39 times, 4, equals 156. If we alternatively divide the 156 by 4 equals 39 as times  4, equals 156. We can break down 156 in further options by dividing by  2, equals 78. To confirm it, times, 2 equals a 156 or alternatively divided by 78 equals 2.

78, 4, 2, and 39 are equally alternate roots of 156.  Algebra teaches us mathematical shorthand with the small Greek letters of the alphabet standing in for the alternate numbers in solving equations. The multiply sign is omitted because of the confusion issue between the Greek letter x. For example the Greek letters, a plus b equals c and a  equals b times c alternatives in 1 statement. We have a Algebraic equation c = a + b, and with no function signs ( c equals a times b) c = a b.

The common children's time table is an example of a list of numbers that divide evenly called composites. If we look up in the 5 times table five 3's equals 15, you will note 15 divide 5 equals 3. Your calculator confirms it. If we look up in the times 3 table, three  5's equals 15, just as 15 dived 3 equals 5. Your mathematical instinct agrees with  your calculator confirming both versions are equally valid.

Alternatively if we look up in the times 6 table, six 3's equals 18, just as 18 divide 3 equals 6 Alternatively if we look up in the 3 times table, 3 times 6 equals 18, just as 18 divide 6 equals 3 respectively. Once again any calculator confirms this.

Note the big jump between five 3.s and six 3's from 15 to 18. In between is 16, and 17. Sixteen is a composite because we find it in the times 2 table. But we can't divide evenly 17 as it always divides with a fraction over.

Taking the numbers in the key pad there is only 9 integers that make up any array of numbers out of those 1 to 9 numbers, including the 0. They confirm our mathematical instinct agrees 1 divide 1 can only be 1 other wise can only be divided with fractions over. So too 2 divided by 1 confirms it divides evenly other wise it too can only be divided with fractions over.

We are accentually asking ourselves now many 1's go into 1. Our mathematical instinct agrees can only be one 1. Of course one 1's are 1. If we ask how many 2's go into 2 can only be 1 two, respectively. One 2's are 2 the same as asking how many 7's in 7 can only be 1 seven.

Here we are asking our selves how many 1's or the same number goes into itself.  All calculators confirm 1 times 7 equals 7. In other words seven 1's are 7 equal to one 7 respectively. The same applies to any integer of you keypad.

Our mathematical instinct aggress how many 5's go into 5 and how many 1's go into it. It aggress there can only be five 1's in 5 and there is only one 5 in 5 confirming five 1's.

But if we apply 1 dived 3 doesn't divide evenly. 4 however divided by 2 confirms the 2 times 2 table equals two 2's are 4 as 4 divides evenly by 2. But not 5. Six dived 2 confirms six 2's are 36 divided by 2 evenly.

But not 7. Eight divided by 2 confirms the 8 times 2 table equals 16 divided by 2 evenly. 9 divided by 3 confirms the 9 times 3 table three 3's are 9 divides by 3 evenly.

Calculators confirms the first 5 numbers 1, 2, 3, 5 and 7 in your calculator key pad integers  always divide with fractions over. You will never find these numbers in any times table called primes.

If you follow along in any calculator confirms the prime number 3 divide 1 equals confirms 3 as 3 divide 3 equals 1, just as 7 divide 1 equals 7 as 5  divide 1 equals 5 respectively.

Your calculator and the times table confirms the definition of primes are the numbers that never appear in the times table because they never divide evenly except by 1 and alternatively as in all numbers by themselves.

Studying primes you will never find, 1, 2 ,3 5, 7,  11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59. 61, 67, 71, 73, 79, 83, 89, and 97 in any times table as the first 25 prime numbers of out of a infinite number of primes.  It's what makes the math's so intriguing specially those who love challenging mathematics. By challenging it we learn the art of solving equations skillfully.

The great 300BC mathematician Euclid reluctantly concluded primes must go on on for ever. Ever since mathematicians have been trying to find the last prime for centuries. Up to this day the largest is as many as 12 million 978 thousand and a 189 digit number long and suspect doesn't end there.

Algebra teaches the law of mathematics we can break down equations with this method. To prove it lets examine 17 equals 13 plus 12 take away 8 expression. If you are following along in any calculator makes this clear. If you enter 13, plus, 12, equals 25, take away, 8 equals 17, just as 13, plus, 12, take away 8 equals 17.

To prove it, in your calculator enter 25 plus, 8, equals 33, take away 17, equals, 8, just as 25, take away  equals 8 17, just as 13 equals 12, just as you enter 25 plus, 8, equals 33, take away 17 equals 25, take away 12 equals 13.

In Algebraic shorthand notation expression 17 equals 12 the small Greek letter x take away 8. It shows alternatively 17 equals 13 the small Greek letter x take away 8. We can find x in any calculator. Enter 17 plus 8 equals 25. Your calculator confirms 25 take away 12 equals confirms 13. If 13x we take away 13 from 25 confirming 12 and alternatively if 12x we take away 12 from 25 equals 13 respectively.

The same applies to multiply. In a Algebraic expression 164 equals 12 the small Greek letter x plus 8. It shows alternatively 164 equals 13 the small Greek letter x plus 8. We can find the missing number represented by the x in any calculator.

Enter 164 take way 8. Your calculator confirms 156. The we can apply taking away 12 equals. The read out confirms 13. If 13x we take away 13 from 156 confirming 12. Operating on the principle we can find solutions to equations with an infant array of numbers.

Calculators confirm when we add two of the same numbers is doubled equals to times 2. The same applies to taking away we are left with the same number. When we dived in half we we are dividing by 2.

When we multiply any same numbers together is called a square. Most basic calculators allow us to hit the equals key twice. If we enter 12, times, times, equals confirms a 144, as 13, times, times, confirms 169.

If we divide 144 by 12  you calculator confirms the square root of a 144. The same applies to entering 169 dived by 13 confirms 13 is the square root of a 169. All calculators including basics have a tick like symbol key for square roots.

If we enter 144 pressing the square root key confirms 12 clear the screen as entering 169 pressing the square root key equals confirms 13. We can apply the same principle to any number in the calculators key pad.

Scientific models have square keys the small Greek letter alpha the looks like an English x with a little razed 2 just to it's right a mathematical shorthand notion used in Algebra in this case for a square. If you have a smart phone you can use the menu to call up the scientific calculator mode or if you have access to a scientific calculator or alternatively if a basic is all you have you are stuck with times, times for a square.

Lets try 12 squared plus 13 squared

Enter 12, if you have a basic calculator 12, times, times, equals, confirms 144. If a scientific enter 12 pressing the square key. The calculator confirms 144. Clear the screen. Now 13 squared equals confirms a 169 pressing the square root key confirms 13.

Note we can apply this to time. We know there is 60 seconds in a minute and 60 minutes in an hour. Our mathematical instinct agrees it is only a mater of entering in a basic calculator 60, time, times, equals confirms 60, as 60 times, 60, equals confirms 60 squared is 3 thousand 600 seconds in a 1 hour long TV program.

If using a scientific 60 pressing the square key  confirms 3 thousand 600 seconds in an hour. Writing down in shorthand notation we write the seconds of an hour equals 60 with a little razed 2 just the right of the 60.

The wonder of light speed coasts in the zero resistance of out of space a distance of a 186 thousand miles by the time the second hand of wrist watches and clocks move a division  including a digital second.

What of kilometers? Every mile equals 1.609km. Our mathematical instinct agrees it is only a mater of 1.609 times, 186000 equals confirms 299 thousand 274km per second we round off to 300,000 just as dividing 1.609 equals confirms 186000 miles per second as entering 299,274km per second divided by 186,000 miles per second equals confirms 1.609km per second.

We agree as there is 60 seconds in a minute every second a distance of a 186 thousand miles 60 times a 186 thousand equals in any calculator equals the  distance light traveled in a minute.

In and hour enter 60 squared, equals 3 thousand 600 seconds in an hour times 186,000 equals confirms a distance of 669 million 600 thousand miles in that time. It has converted miles per second to mph.

If we 60 squared equals confirms 3 thousand 600 seconds in an hour, times 24 confirms 86 thousand 400 seconds in 24 hours. The distance light travels times 186,000 equals 669 million 600 miles, times, 1.609km equals confirms a distance of 1 thousand and 77 million 386 thousand 400km. Another way of saying the same thing converting mph to kmph. Your calculator has confirmed light travelers a distance of 669 million 600 thousand miles times 1.609 in km by a 1 hour long TV program.

Calculators tell us 1 plus a 0 equals 10. If you are following along in a calculator enter 1 followed by a 0 becomes 10. Your calculator confirms 1 times 10 equals 10 respectively

If you enter another 0 becomes a 100. If you enter another 0 becomes a thousand. 1, 2, and 3 zeros equals a 1,000. Enter in your calculator 1, times, 10, confirms 10, times,  confirms a 100, times 10, confirms a 1,000 and so on. Every time you entered the multiply key confirms adding a zero as 1 times 1000 euals. If you enter in your calculator 1 with 4 zeros  confirms 10 thousand.

Entering another 0 confirms 5 zeros a hundred thousand. Entering another 0 confirms 6 zeros a million. Entering another 0 confirms 7 zeros a hundred million the maximum of basic calculators.

Your calculator confirms 1 times,10, confirms 10. Times,10, confirms, 100. Times, 10, confirms a thousand. Times, 10, confirms 10 thousand. Times, 10 confirms a 100 thousand. Times, 10, confirms a million. Times 10 confirms 10 million.  Ten million is 1 multiplied by 10 7 times. In other words 1 multiplied by 10 multiple times. Shorthand notation is 10 to the known as powers.

In written shorthand notation 10 with a little razed 7 is 10 to the 7. Every time you entered the multiply key a 0 was added until the end of basic calculators. In scientific shorthand notation is pounced, 10 to the 8 equals 1, multiplied by 10, eight times confirming a hundred million.
 A common term is called a powers where a number is multiplied by 10 multiple times

10 to the 9 equals 1 multiplied by 10 nine times equals a billion

10 to the 10 equals 1 multiplied by 10 ten times equals 10 billion

10 to the 11 equals 1 multiplied by 10 eleven times equals a 100 billion

10 to the 12 equals 1 multiplied by 10 twelve times equals a trillion

10 to the 13 equals 1 multiplied by 10 thirteen times equals 10 trillion

10 to the 14 equals 1 multiplied by 10 fourteen times equals a 100 trillion

10 to the 15 equals 1 multiplied by 10 fifteen times equals a thousand trillion

If you using a scientific calculator enter 5.62 EXP 2 equals the calculator will confirm 562. If you enter  6.738 EXP 3 equals will confirm 6,738. If you enter 156 EXP 12 equals conforms 1,560,000,000,000.(1 trillion 560 million).

For minus powers there is a +/- key. If you enter 5.62 EXP 2 pressing the +/- key equals will conform decimal fraction 0.0562. As 6.735 EXP 3 pressing the +/- key equals conforms 0.006735 respectively. 

We are familiar with the standard metric system prefixes including k, kilo- for a thousand, M mega for a million, G-, giga for a billion and T-, Tera for a trillion.

One. Ten, a hundred. Ten is two zeros followed by to the 3 equals 1k. Ten to the 4, equals 10k. Ten, to the 5, equals a 100k.

Ten to the, 6, equals 1M. Ten, to the 7, equals, 10M. Ten to the 8, equals a 100M. Ten, to the 9 equals, 1G.

Ten to the 10 equals 10G. Ten to the 11, equals a 100G.

Ten to the 12 equals 1T. Ten to the 13 equals 100T respectively. We are particularly familiar with these large numbers with electronic computer and mobile phone memory in bytes and resolution of our video and photo image in pixels.

If you enter in a thousand in a scientific calculator pressing the log key equals confirms 3. If you enter a million (1000000) pressing the log key confirms 6. if you test a billion (100000000) confirms the log of a billion is 9 respectively. In other words a log is the number of times is 1 multiplied by 10. These are constant numbers called constants as each equation in the classic children's times table are.

In scientific calculators we can multiply 1 by 10 as many times as we like remaining a constant like clockwork  There is no limit to the number of 0's. In mathematics they are referred to as powers expressed as ten to the.... called exponents. All scientific calculators have an exponent key EXP.

For example in smart phone scientific calculator mode or any scientific calculator mathematicians  express 1 thousand 200, entering 1.2, EXP pressing the 3 key equals confirms 1,200. The same is true with 1 thousand 300. Scientific calculators confirm 13, EXP 3, equals 1,300. The same is true with 1,200.

The read out confirms shorthand notation 1.2e+0. The e is for exponent and the plus the positive number 1.2 to the 0 until you enter 3 equals. The read out will confirm 1,200 or 1.2e+3 pronounced as 1.2 times ten to 3.

When we come to adding such large numbers calculator confirms we add proportionately. They confirm if we 12,000, add,13,000, equals 12, plus, 13, equals, 25, keying in 3 zeros

Scientific calculators prove this. Enter, 12, EXP, 3, plus, 13, EXP, 3, equals. The read out will either confirm 25,000 or 25e+3. Pounced as 25 times 10 to the 3, as, 13, EXP, 3, plus, 12, EXP, 3, equals. The read out will either display 25,000 or 2.5e+4. The same applies to 13,000, take away, 12,000, equals, 1,000. Your calculator read out will confirm 1 EXP 3 10 to the 3) equal 1,000.

When we multiply we add the number of zeros. Scientific calculators confirm 1.2, EXP, 3, times, 1.3, EXP, 3, equals. The read out confirms 1,560, pronounced 1.56 times 10 to the 3.

They confirm in division we take away the zeros. Enter 1.56, EXP, 6, dived, 1.3, EXP, 3 ,equals. The read out confirms 1,200 or 1.2e+3. Enter, 1.56, EXP, 6, dived,1.2, EXP, 3, equals. The read out confirms 1,200 or 1.3e+3 (1.2 times 10 to the 3) respectively. Scientific calculators confirm we can enter a infinite amount of numbers in exponents. ideal for working out the vast distances of the universe.

If we 1 divided by 10 we did with the positive numbers you have decimal fractions. They are best described as percentage 1 out of a 100. Any calculator includes a percentage key %.

Calculators conform 10% is 1, divided, 10 equals  0.1, divided, 10, equals, 0.01, equals 1% as 1, divide a 100  equals 1%.  Calculators confirm 1 divide a quarter of a 100 equals 1 dived 25 divide equals 0.25, as a half is 50%, as 1 divide 4 is 75% (three quarters) of 100%. Any decimal fraction up to 0.99 is out of a 100% For example 30 percent is 0.3. Sixty percent is 0.6.

Calculators confirm 0.25 times 13 equals 3.25% of 13, as 13 times 0.25 equals 3.25% of 13, as 0.25 times 12 equals 3%, 12 time 0.25 equals 3% of 12. The same is true with a half and three quarters of 13 and 12.

Calculators clearly define a reciprocal  as 1 divided by any positive number. For example your calculator will confirm, 1, divided,10 thousand is 10 to the 4 equals thousandth. (10 to the minus 4) respectively. It is 1% (0.1) divided by 10. Your calculator will confirm the log of 10 thousand  reciprocal equals decimal 0.0001 as 1, divided by 10 thousand respectively.

The reciprocal of light speed for example. It is a 168 thousand times as fast as every second. Enter in your calculator 1 divide 186 thousand. Calculator read outs display 5 point large decimal places to the minus 6. It is a 186 thousandth second.

Our mathematical instinct agrees 186 thousand times 1.609km every 1 mile. Using this your calculator confirms 1.609km times a 186000 miles equals. The read out confirms 229 thousand 274 in km's.

Calculators confirm our insight into light traveling a distance of 1 mile in a 186 thousandth second. Enter 1 for 1 mile divided by 186 thousand mile per second equals 1 mile in a 186 thousandth second. If we times a 186 thousand equals confirms a 186 thousand miles per second.

Your calculator confirms in km's light travels 1.609km the same amount of time as 1 mile travels 186 thousandth second. In other words Both 1.609km and 1 mile travel 186 thousand times your wrist watch second hand can move a division.

Calculators confirm 1.609km traveling the same velocity as 1 mile equals 1 divide light speed equals 5 point large decimal places to the minus 6 as a 186 thousandth second. If we times 186 thousand equals confirms 1 second.

This confirms reciprocal of light speed in miles for example is 1 divide 186 thousand times as fast as every second for both 1 mile and 1.609km. Calculators confirm 1 divide light speed equals 5 point large decimal places to the minus 6 as a 186 thousandth second. It will confirm times 186 thousand equals 1 second light travels 1.609km a 186 thousand times as fast that time.

In other words the reciprocal of light speed in miles equals times 1,609 in km's. It satisfies our mathematical instinct light travels both 1.609km and 1 mile in a 186 thousandth of a second.

Calculators confirm this. Enter 1.609 times 186 thousand miles confirms 229 thousand 274km in 1 second as divided by 1.609 equals 186 thousand miles per second.

One of the familiar unit is math's are decimal fraction. Calculators confirm 1, divided by 4, equals decimal 0.25 confirming a 1/4. 1, divided by 2 equals confirms decimal 0.5 conforming a 1/2 and 3 divided by 4, equals decimal 0.75 confirming 3/4.

They confirm a 1, divide 6 equals a recurring 0.3333333333333 for ever equals a third just over a 1/4 just as 1, divided by 6, equals another recurring decimal fraction 0.16666666666666 for ever.

We are familiar with the standard metric system prefixes including Mil-, (mill lee) for 1 divide a kilo, including the small letter of the Greek alphabet for 1 divide mega (a million), p-, pica for 1 divide a Giga and ato for 1 dived a Tera respectively. Calculators confirm if we dived 1 by any positive prefix number confirm reciprocals. The following are reciprocals of each other

1 dived 10. Equals 10 to the minus 1. One zero and 1. Equals a 100 milli- of 1k

1 divide a 100. Equals 10 to the minus 2. Two 0's and 1. Equals 10 milli- of 1k

1 divide 1k. Equals 10 to the minus 3. Three 0's and 1. Equals to 1 milli- of 1k

1 divide 10K. Equals 10 to the minus, 4. Four 0's and 1, Equals a 100 micro. of 1k

1 divide, a 100k. Equals 10 to the minus 5. Five 0's and 1. Equals 10 micro- of 1k

1 divide 1 Mega. Equals 10 to the minus 6. Six 0's and 1. Equals a 1 micro- of 1k

1 divide, 10 mega. Equals 10 to the minus 7. Seven 0's and 1. Equals a 100 Pico-  of 1k

1 divided by 100 mega. Equals 10, to the minus 8, eight 0's and 1, equals 10 Pico. equals to of 1k

1 divide 1G. Equals 10 to the minus 9. Equals. Nine 0's and 1. Equals  1 Pico of 1k

1 divide 10G. Equals 10 to the minus 10. Ten 0's and 1. Equals 100 Tera of 1k

1 divide a 100G. Equals 10 to the minus 11. Eleven 0's and 1. Equals 10 Tera of 1k

1 divide 1 Tera. Equals 10 to the minus 12. Twelve 0's and 1. Equals 1 Tera of 1k

Our mathematical instinct agrees two 1/4 equals a half and two halves equals a whole. Calculators confirm, 0.25, times, 2, equals, 0.5. And times, 2, equals,1. So doubling, 0.75 equals, amounts to times, 2, equals, Calculators confirm 3 divided by 4, times 2 equals 1.5.

They confirm when we are doubling a decimal fraction squares it. They confirm, 0.8, times, 2, equals, confirms, 0.8 squared equals, 1.6, respectively.

Calculators confirm 25 percent of a 100 is 100 divided by 4, equals 25, as times 4 equals a 100. They confirm 50 percent is a 100 divided by 2, equals 50, as  times 2, equals 100 percent because a 100 is a composite.

Calculators confirm 10 divide 4, equals 2.25 we'd think 10 is a prime number. But it is found in the 2 times 5 and 5 times 2 times tables confirmed by any calculator so is not a prime. 10 divide 4 is a prime factor. Proof 10 is a composite divided by 2 equals 5, as divided by 5 equals 2 respectively.

Calculators confirm 1 divide by 4 equals 0.25 as times 4 equals 1 because 1 is acting as a  prime not a composite. The same is true with 1 divide 2 equals 0.5, as times 2 equals 1, as, 1 divide 5 equals 0.25, as times 5 equals 1.

Calculators confirm a 100 divided 75 equals. Read outs display 0.01 with endless 3's. 100 divided by 75 is a prime factor of a 100. Prime factors crop up in mathematical equations all the time.

They confirm 25 percent of 13 is 1 divide 4 equals, times 13, equals 3.25 percent, as 0.25 times 13. The same is true with any. Alternatively the decimal point of the percentage times 12 or the 13 projects the percentage.

Algebra teaches us 25 equals 100 take away x or any other small letter of the Greek alphabet representing the unknown number. Alternatively the lower case n for number or the Greek letter p is common. All calculators confirm the lower case (small letters) of the Greek alphabet equals 100 take away 25, equals 75, as a 100, take away 75 equals 25, respectively.