Whole Numbers and Number Theories

Base 10Non-Positional--> the symbols can be in any order, and the number would still be the same!Symbols represent numbersPlace values are not present

Base 10Positional--> the numbers can only be in one place, otherwise, the number changesex: 234 is not the same as 423Numbers represented are 0,1,2,3,4,5,6,7,8 and 9Place values are 1, 10, 100, 1000, 10000 etc:

Base 60Positional numbersNumbers 1-59 are represented***0's do not exist in this system.Numbers are made up of only two symbols. Place Values are 1, 60, 3600, 21600 etc:

Base 5Positional numbersNumbers 0,1,2,3,4 are representedPlace Values are 1, 5, 25, 125, 625 etc

Modifed Base 20Positional Numbers**Number is written vertically instead of horizontally Symbols represent numbers 0-19Place values are 1, 20, (18*20), (18*20^2) etc1 is on the bottom with 20 on top of that, and (18*20) on top of that

Base 12Positional NumbersNumbers 0-11 are presentT stands for ten and E stands for elevenPlace values are 1, 12, 144, 1728, etc

Subtraction is the opposite of addition. Subtraction means taking away one number from another number. Subtraction also uses algorithms to do the process.

The take-away concept uses this logic:If Nancy has 13 stickers and she gives 5 away, how many does she have left?This shows 5 things being taken away from 13 things. The answer is 8 stickers left.

The comparison concept uses this logic:If Mary has 13 stickers, and Jon has 7, how many more does Mary have than Jon?It this concept, we compare the two sets to find the difference. The answer is 6 more stickers.

The Missing Addend Concept uses this logic:If Ashley has 16 stickers, and she needs 20, how many more does she need?In this case, we start counting from 16 to 20 to find the missing number, and what we get is the difference between the two.

Division is the opposite of Multiplication. For any two whole numbers a and b, b not equal to zero, the quotient of a/b, the whole number is c, such that a=c*bWe need to know multiplicatio in order to do division.

Determine the answer by measuring off as many groups of the divisor as possible.For example:18/3Measure off as many sets of 3 as possibleThere will 6 groups of three units each.

This concept determines the answer by making as many even groups as possible, as determined by the divisor.For example:18/3Set up three groups. How many will be in each group?There will be 6 units in each group.

Addition uses algortihms to solve the problem. Algorithms involve adding the numbers, and carrying place values over to the next place valueThere are many algorithms for addition. There is one called the "Stick method", "Partial Sums method" and the standard "U.S method"

For any numbers in a given set, if their sum is also a number in the set, the set is closed. If their sum is a number not in the set, the set is open. Ex: {0,1}1+0=10+0=01+1=2this is not a closed set. 2 is not in the original set.

For any number a,a+0=a=0+a=a0 makes no change when added to a number. 0 is the only identity number for addition.

For any whole numbers a, b, and ca+(b+c)=(a+b)+cthis means that the middle number may be added to either of the other two numbers first, and the answer will not change.

When any two numbers are added, the numbers may be moved around without any change to their sum;a+b=b+a

Multiplication is often referred to as "Repeated Addition"for this reason:Any two whole numbers, a and b, the product of these two numbers is the sum of b recurring a times a*b=b+b+b+b.... a times

A factor is any number that divides another whole numberFor example:24/3---> 3 is a factor of 24, 8 is also a factor of 24. A multiple is a number that a certain number goes into. For example: Multiples of 2 are 4, 6, 8, 10, 12, 14, 16, 18 etc:

A prime number is a number that can only be divided by itself and 1ex: 5 can only divided by 1 or 5A Composite number is any number that is not prime(can be divided by more than just itself and 1)ex: 12 can be divided by 1,2,3,4,6, and 12.

Divisibility Tests are designed to make division simpler. Some divisibity tests are easy to remember, like any number that has a last digit that is even is divisible by 2. Some are harder, like the divisibility rule for 11. The website attached explains the divisibility rules in an easy to understand format.

The Greatest Common Factor (GCF) is the biggest number that is a factor for a set of two numbers.Ex: Find the GCF for 24 and 3624) 1, 2, 3, 4, 6, 8, 12, 2436) 1, 2, 3, 4, 6, 9, 12, 18, 36In this instance the GCF(24, 36) is 12. It is the biggest factor that they both share.

The Least Common Multiple (LCM) is the smallest shared multiple between a set of two numbers. Ex:What is the LCM for 6 and 9?6) 6, 12, 18, 24, 30, 36, 42, 48, 549) 9, 18, 27, 36, 45, 54, 63, 72, 81the LCM(6,9) is 18 because 18 is the smallest number that each set has.