Number System In Maths

Number Systems: Naturals, Integers, Rationals, Irrationals, Reals, and Beyond

The Natural Numbers

The natural (or counting) numbers are 1,2,3,4,5,1,2,3,4,5, etc. There are infinitely many natural numbers. The set of natural numbers, {1,2,3,4,5,…}{1,2,3,4,5,…}, is sometimes written NN for short.

The whole numbers are the natural numbers together with 00.

(Note: a few textbooks disagree and say the natural numbers include 00.)

The sum of any two natural numbers is also a natural number (for example, 4+2000=20044+2000=2004), and the product of any two natural numbers is a natural number (4×2000=80004×2000=8000). This is not true for subtraction and division, though.

Sets

In modern mathematics, just about everything rests on the very important concept of the set .

A set is just a collection of elements, or members. For instance, you could have a set of friends:

F=F= {Abdul, Gretchen, Hubert, Jabari, Xiomara}

or a set of numbers:

Y={−3.4,12,9999}Y={−3.4,12,9999}

There are two methods of representing a set :

(i) Roster or tabular form

(ii) Set-builder form.

Roster or tabular form: In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces { }.

For Example:

Z=the  set  of  all  integers={…,−3,−2,−1,0,1,2,3,…}Z=the  set  of  all  integers={…,−3,−2,−1,0,1,2,3,…}

Set-builder form: In the set builder form, all the elements of the set, must possess a single property to become the member of that set.

For Example:

Z={x:x  is  an  integer}Z={x:x  is  an  integer}

You can read Z={x:x  is  an  integer}Z={x:x  is  an  integer} as “The set ZZ equals all the values of xx such that xx is an integer.”

M={x | x>3}M={x | x>3}

(This last notation means “all real numbers xx such that xx is greater than 33 .” So, for example, 3.13.1 is in the set MM , but 22 is not. The vertical bar | means “such that”.)

You can also have a set which has no elements at all. This special set is called the empty set, and we write it with the special symbol ∅∅ .

If xx is a element of a set AA , we write x∈Ax∈A , and if xx is not an element of AA we write x∉Ax∉A .

So, using the sets defined above,

−862∈Z−862∈Z , since −862−862 is an integer, and

2.9∉M2.9∉M , since 2.92.9 is not greater than 33 .

The Integers

The integers are the set of real numbers consisting of the natural numbers, their additive inverses and zero.

{…,−5,−4,−3,−2,−1,0,1,2,3,4,5,…}{…,−5,−4,−3,−2,−1,0,1,2,3,4,5,…}

The set of integers is sometimes written JJ or ZZ for short.

The sum, product, and difference of any two integers is also an integer. But this is not true for division… just try 1÷21÷2.

The Rational Numbers

The rational numbers are those numbers which can be expressed as a ratio between two integers. For example, the fractions 1313 and −11118−11118 are both rational numbers. All the integers are included in the rational numbers, since any integer zz can be written as the ratio z1z1.

All decimals which terminate are rational numbers (since 8.278.27 can be written as 827100827100.) Decimals which have a repeating pattern after some point are also rationals: for example,

0.0833333….=1120.0833333….=112.

The set of rational numbers is closed under all four basic operations, that is, given any two rational numbers, their sum, difference, product, and quotient is also a rational number (as long as we don’t divide by 00).

Rates & Ratios

A ratio is a comparison of two numbers. A ratio can be written using a colon, 3:53:5 , or as a fraction 3535 .

A rate , by contrast, is a comparison of two quantities which can have different units. For example 55 miles per 33 hours is a rate, as is 3434dollars per square foot.

Writing the ratio using a colon, we get 6:216:21 .

Note that this can be reduced, like a fraction, by dividing both numbers by a common factor — in this case, 33 . In simplest form, the ratio is 2:72:7 .

To find the total amount of punch, add 6+21+21=486+21+21=48 .

The ratio of apricot juice to the total amount of punch is 21:4821:48 . But this ratio is probably more clearly written as a fraction, since the apricot juice makes up a fraction of the whole.

21482148

To reduce the fraction, divide both the numerator and the denominator by 33 .

716716

Note that this can be reduced, like a fraction, by dividing both numbers by a common factor — in this case, 33 . In simplest form, the ratio is 2:72:7 .

Note that it doesn’t matter if there are 100100 or 10,00010,000 centipedes; the ratio of legs to eyes will remain the same.

Writing the ratio using a colon, we get 46:846:8 .

Divide both numbers by 22 . In simplest form, the ratio of legs to eyes is 23:423:4 .

Write the rate as a fraction.

170  wingbeats10  seconds170  wingbeats10  seconds

Divide both the numerator and the denominator by ten.

=17  wingbeats1  second=17  wingbeats1  second

So, the rate is 1717 beats per second.

The first job is to figure out the rate per day.

50  meters1  hour⋅8  hours1  day=50(8)metersday50  meters1  hour⋅8  hours1  day=50(8)metersday

=400metersday=400metersday

He is climbing at a rate of 400400 meters per day.

Now divide 32003200 by the daily rate to find the number of days it will take him to reach the top.

3200  meters400metersday=8  days

Terminating and Repeating Decimals

Any rational number (that is, a fraction in lowest terms) can be written as either a terminating decimal or a repeating decimal . Just divide the numerator by the denominator . If you end up with a remainder of 00 , then you have a terminating decimal. Otherwise, the remainders will begin to repeat after some point, and you have a repeating decimal.

The division is as follows:

      0.62585.000        48−−           20           16−−              40              40−−                 0      0.62585.000        48_           20           16_              40              40_                 0

So, 58=0.62558=0.625 . This is a terminating decimal.

The division is as follows:

        0.5833127.0000          6 0−−          1 00              96−−                 40                 36−−                    40                    36−−                       4        0.5833127.0000          6 0_          1 00              96_                 40                 36_                    40                    36_                       4

So:

712=0.583¯712=0.583¯

This is a repeating decimal.

The bar over the number, in this case 33 , indicates the number or block of numbers that repeat unendingly.

The Irrational Numbers

An irrational number is a number that cannot be written as a ratio (or fraction).  In decimal form, it never ends or repeats. The ancient Greeks discovered that not all numbers are rational; there are equations that cannot be solved using ratios of integers.

The first such equation to be studied was 2=x22=x2. What number times itself equals 22?

2√2 is about 1.4141.414, because 1.4142=1.9993961.4142=1.999396, which is close to 22. But you’ll never hit exactly by squaring a fraction (or terminating decimal). The square root of 22 is an irrational number, meaning its decimal equivalent goes on forever, with no repeating pattern:

2√=1.41421356237309…2=1.41421356237309…

Other famous irrational numbers are the golden ratio, a number with great importance to biology:

1+5√2=1.61803398874989…1+52=1.61803398874989…

ππ (pi), the ratio of the circumference of a circle to its diameter:

π=3.14159265358979…π=3.14159265358979…

and ee, the most important number in calculus:

e=2.71828182845904…e=2.71828182845904…

Irrational numbers can be further subdivided into algebraic numbers, which are the solutions of some polynomial equation (like 2√2 and the golden ratio), and transcendental numbers, which are not the solutions of any polynomial equation. ππ and ee are both transcendental.

The Real Numbers

The real numbers is the set of numbers containing all of the rational numbers and all of the irrational numbers.  The real numbers are “all the numbers” on the number line.  There are infinitely many real numbers just as there are infinitely many numbers in each of the other sets of numbers.  But, it can be proved that the infinity of the real numbers is a bigger infinity.

The “smaller”, or countable infinity of the integers and rationals is sometimes called ℵ0ℵ0(alef-naught), and the uncountable infinity of the reals is called ℵ1ℵ1(alef-one).

There are even “bigger” infinities, but you should take a set theory class for that!

The Complex Numbers

The complex numbers are the set {a+bia+bi | aa and bb are real numbers}, where ii is the imaginary unit, −1−−−√−1.

The complex numbers include the set of real numbers.  The real numbers, in the complex system, are written in the form a+0i=aa+0i=a. a real number.

This set is sometimes written as CC for short. The set of complex numbers is important because for any polynomial p(x)p(x) with real number coefficients, all the solutions of p(x)=0p(x)=0 will be in CC.

Monomials, Binomials, Polynomials

A monomial is any product of numbers and variables, like 1717 , or 3xy3xy , or −4x2−4×2 , or

7100a5b6c7d8z9997100a5b6c7d8z999

The only rules are that the variables should be raised to only positive integer powers (no square roots or 1x1x ‘s allowed), and no plus or minus signs.

The coefficient of a monomial is the numerical part. For example, in the above examples, the coefficients are 17,3,−417,3,−4 and 710710 .

A binomial is the sum of two monomials, for example x+3x+3 or 55x2−33y255×2−33y2 or

7100a5b6c7d8z999−13x7100a5b6c7d8z999−13x

A trinomial is the sum of three monomials.

A polynomial is the sum of nn monomials, for some whole number nn . So monomials, binomials and trinomials are all special cases of polynomials. A polynomial can have as many terms as you want.

The degree of a monomial is the sum of the exponents of all its variables. The degree of a polynomial is the term of the polynomial that has the highest degree.

Beyond…

There are even “bigger” sets of numbers used by mathematicians. The quaternions, discovered by William H. Hamilton in 18451845, form a number system with three different imaginary units!

Number System Question