At higher levels of secondary and tertiary education discrete mathematics , is often more challenging than the mathematics of continuous functions. With continuous functions, a small change in the input variable leads to a small change in the output variable. Smooth, continuous functions lead on to most of the functions students meet at secondary school, including calculus at the senior secondary school level. The numbers we meet at school are generally represented by using combinations of ten number symbols also called numerals or digits plus the symbols ".
All of these symbols also represent the numbers one, two, three, It is called zero, nil, nought etc. It is also a place-holder. The number above might be written. It is not necessary to do this, but it can be kinder to the reader. It is more comfortable to read large numbers in groups of three digits.
The commas or spaces are conveniently positioned to separate thousands, millions, billions, trillions etc. The convention of using commas or spaces is not the same all over the world. In the Netherlands, for example, dots are used instead. Our example would therefore be written 1. In the UK, a dot is used to denote a decimal point when writing a fraction of a number see our pages on Fractions and Decimals , but in the Netherlands they use a comma for this purpose.
Always be careful to check the convention of the country you are in—it could mean the difference between getting a bag or a truck full of potatoes! Integers can be either positive or negative. See our pages Fractions and Decimals for more information. There are two ways of displaying fractional values in mathematics. When dealing with a decimal place we can use the same columns as we do when dealing with whole numbers integers ; we simply continue the columns to the right, as each number is smaller than the one before.
So Minus 1. For example, 3. If a 0 occurs before the end of the number then this must be kept, so 5. In maths we call any kind of calculation an operation. Fractions are written in this way. Similarly, an addition operation involves adding numbers together and a subtraction operation involves taking one number away from another.
Actually, what we mean is that we are doing some maths calculations. Roman numerals are still used in some disciplines but most commonly to count or show numbers of years. We often also see them on clock faces. A mixed decimal is one consisting of an integer and a decimal fraction, e.
All prime denominators produce repeating decimals. Fractions with the same denominator often produce decimals with the same period and period length but with the digits starting with a different number in the period. Other denominators produce two or more repeating periods in different orders. Investigate those with prime denominators of 11, 13, 17, 19, etc.
What do you notice? Another interesting property of repeating decimals of even period length is illustrated by the following. The repeating period is Split the period into two groups of three digits and add them together. The result is Do the same with any other repeating decimal period and the result will always be a series of nines. Adding and gives you All repeating decimals, regardless of the period and length, are rational numbers.
This simply means that it can be expressed as the quotient of two integers. A question that frequently arises is how to convert a repeating decimal, which we know to be rational, back to a fraction.
An easier way to derive the fraction is to simply place the repeating digits over the same number of 9's. For example, the repeating decimal of. Another trick where zeros are involved is to place the repeating digits over the same number of 9's with as many zeros following the 9's as there are zeros in the repeating decimal.
For example,. Deficient numbers are part of the family of numbers that are either deficient, perfect, or abundant. In the language of the Greek mathematicians, the divisors of a number N were defined as any whole numbers smaller than N that, when divided into N, produced whole numbers.
Equivalently, N is also deficient if the sum, s N , of all its divisors is less than 2N. The divisors of a pefect or deficient number is deficient. The digital root of a number is the single digit that results from the continuous summation of the digits of the number and the numbers resulting from each summation. For example, consider the number The summation of its digits is The summation of 2 and 4 is 6, the digital root of Digital roots are used to check addition and multiplication by means of a method called casting out nines.
For example, check the summation of and The summation of and is The DR of is 6. With the two final DR's are equal, the addition is correct.
Egyptian fractions are the reciprocals of the positive integers where the numerator is always one. They are often referred to as unit fractions. They were used exclusively by the Egyptians to represent all forms of fractions.
In the year , Leonardo Fibonacci proved that any ordinary fraction could be expressed as the sum of a series of unit fractions in an infinite number of ways. He used the then named greedy method for deriving basic unit fraction expansions.
He described the greedy method in his Liber Abaci as simply subtracting the largest unit fraction less than the given non unit fraction and repeating the process until only unit fractions remained. An example will best illustrate the process. Dividing the fraction yields. Alternatively, divide the numerator into the denominator and use the next highest integer as the new denominator. Much more information regarding unit fractions can be found at. While the number of unit fractions derivable for any given fraction is therefore infinite, there is apparently no known procedure for deriving a series with the least number of unit fractions or the smallest largest denominator.
A shape is called equable if its area equals its perimeter. There are exactly five equable Heronian triangles: the ones with side lengths 5,12,13 , 6,8,10 , 6,25,29 , 7,15,20 , and 9,10, Equivalent numbers are numbers where the aliquot parts proper divisors other than the number itself are identical.
For instance, , and are equivalent numbers since their aliquot parts sum to Probably the easiest number to define, an even number is any number that is evenly divisible by 2.
By signing up, you agree to receive useful information and to our privacy policy. Shop Math Games. Skip to main content. Search form Search. Compiled by William Tappe. Introduction Those ten simple symbols, digits, or numbers that we all learn early in life that influence our lives in far more ways than we could ever imagine. Numbers - The Basics Integers - Any of the positive and negative whole numbers, Whole Numbers - the natural numbers plus the zero. Some examples of irrational numbers are Transcendental Numbers - any number that cannot be the root of a polynomial equation with rational coefficients.
Real Numbers - the set of real numbers including all the rational and irrational numbers. Irrational numbers are numbers such as Rational numbers include the whole numbers 0, 1, 2, 3, Draw a line. Put on it all the whole numbers 1,2,3,4,5,6, Then put in all of the decimals [ some decimals aren't fractions ] Now you have what is called the "real number line" The way to get a number that is not "real" is to try to find the square root of - 1 can't be 1 because 1 squared is 1, not -1 can't be -1 because the square of -1 is 1, not -1 So there is no number on your number line to be and new numbers would need to be put somewhere.
Every even integer greater than 46 is expressible by the sum of two abundant numbers. Every integer greater that 83, is expressible by the sum of two abundant numbers. Any multiple of an abundant number is abundant. Other amicable numbers are: There are more than known amicable pairs. For values of n greater than 1, amicable numbers take the form: given that x, y, and z are prime numbers. The sum of the first 36 positive numbers is which makes it the 36th triangular number.
The sum of the squares of the first seven prime numbers is To find the number of permutations of "n" dissimilar things taken "r" at a time, the formula is: Example: How many ways can you arrange the letters A, B, C, and D using 2 at a time? Without getting into the derivation, Example, how many different permutations are possible from the letters of the word committee taken all together? The specific Catalan numbers are 1, 1, 2, 5, 14, 42, , , , , 16, and so on deriving from This particular set of numbers derive from several combinatoric problems, one of which is the following.
To find the number of combinations of "n" dissimilar things taken "r" at a time, the formula is: which can be stated as "n" factorial divided by the product of "r" factorial times n - r factorial. We have: How many handshakes will take place between six people in a room when they each shakes hands with all the other people in the room one time? Here, Notice that no consideration is given to the order or arrangement of the items but simply the combinations. Therefore, which results in Using the committee of 3 out of 12 people example from above, Consider the following: How many different ways can you enter a 4 door car?
Another way of expressing this is: If we ignore the presence of the front seats for the purpose of this example, how many different ways can you exit the car assuming that you do not exit through the door you entered? This too can be expressed as: Carrying this one step further, how many different ways can you enter the car by one door and exit through another? Therefore, the total number of ways of entering and exiting under the specified conditions is: Another example of this type of situation is how many ways can a committee of 4 girls and 3 boys be selected from a class of 10 girls and 8 boys?
It is thought that there are an infinite number of circular primes but has not yet been proven. If a positive number N is evenly divisible by any prime number less than , the number N is composite. The first ten cubes are 1, 8, 27, 64, , , , , and The cube of any integer is the difference of the squares of two other integers. Every cube is either a multiple of 9 or right next to one. A perfect cube can end in any of the digits 0 through 9.
Three digit numbers that are the sum of the cubes of their digits: , , , The sum of any two cubes can never be a cube. The sum of a series of three or more cubes can equal a cube. Note that each cube of the numbers 1 through 10 ends in a different digit: n Given the cube of a number between 1 and , say , Eliminating the last 3 digits of the cube leaves the number The number lies between the cubes of 6 and 7 in our listing above.
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