Number Systems Natural Numbers

Where Our Numbers Came From The earliest known evidence for writing or counting are scratch marks on a bone from 150,000 years ago. About 4,000 BCE Sumerians introduced tokens to represent numbers. This gave birth to arithmetic and allowed them to calculate wealth, and collect taxes. About 500 BCE a complete number system was developed in India and our modern numbers were derived from this.

Where Our Numbers Came From

Natural Numbers Natural Numbers are an infinite (they go on forever) set of numbers We use the symbol = {1, 2, 3, 4, 5, …..} Looking at the numbers above can you tell me three rules about natural numbers? They are always positive, whole (not fractions or decimals), and greater than zero

Place Value We count in tens, probably because most people have 10 digits on their hand. We can arrange the numbers 0 to 10 in different orders of magnitude to represent small or large numbers. Each number has a different value depending where it is, as in the diagram on the right.

Number Line Another way to order our numbers is to use a number line. Each number is consecutive (in order) and goes from a low value to a high value (they ascend) You can use this to basic sums like addition, subtraction, multiplication and division.

Factors Factors (or divisors) are numbers that divide evenly into other numbers. They can be multiplied together to give larger numbers. When writing the factors of numbers it is a good idea to write them in pairs, that way you don’t forget any and you don’t repeat yourself. For example the factors of 24 are

Highest Common Factor (HCF) If you have two or more numbers then you can find their Highest Common Factor (HCF), the highest number that divides into both evenly. This is useful to know when simplifying fractions. For example if our numbers are 12 and 32 then we can find their HCF using the technique on the right.

Prime & Composite Numbers A prime number is any number that only has two factors, itself and one. Prime numbers are the building blocks of mathematics. Like the chemical elements they combine to form the universe. Riemann Hypothesis Sieve of Eratosthenes As of today the largest known prime number is over 1,000 digits long. What’s the first prime number? A composite number is a number that has multiple factors

Prime Factors Prime factors of a number are prime numbers that are also factors (they divide into it evenly). Any number can be written as a product using only its prime factors. Begin by trying to divide the number by 2 (the first prime) and continue dividing by prime numbers until you get to one.

Lowest Common Multiple (LCM) Multiples are numbers that others divide into evenly (they’re made up of factors) You would have encountered multiples before when doing your times tables. If you have two or more numbers then you can find their Lowest Common Multiple (LCM), the lowest number they both divide into evenly. This is useful to know when adding and subtracting fractions.

Lowest Common Multiple (LCM) We can use the method below to find the LCM of 3, 6 and 8. First list out the multiples in order, then see which ones are common The LCM of 3, 6 and 8 = 24

Fizz Buzz A game of multiples Count upwards from 1 and every time you get to a multiple of 3 you say fizz and a multiple of 5 you say buzz. If you say the wrong number, or don’t say fizz or buzz when appropriate then you are out.

Word Problems At the gym, Hillary swims every 6 days, runs every 4 days and cycles every 16 days. If she did all three activities today, in how many days will she do all three activities again on the same day? I want to plant 45 sunflower plants, 81 corn plants and 63 tomato plants in my garden. If I put the same number of plants in each row and each row has only one type of plant, what is the greatest number of plants I can put in one row? Look for the LCM of 6, 4 and 16 Asked to investigate when something will happen again. Looking for a multiple Answer is in 48 days Asked to investigate the largest possible amount. Looking for a factor. Look for the HCF of 45, 81 and 63. Answer is 9 plants

Powers, Exponents & Indices These are all the same operation in maths and refer to when a number is multiplied by itself. For example We say that this is 3 squared or 3 to the power of 2 This idea is linked to the area of a square, hence the name. 3 is the base number and 2 is the power number 3

Powers, Exponents & Indices Another common sum would be writing We say that this is 3 cubed or 3 to the power of 3 as it is linked to the volume of a cube. Every time that a number is multiplied by itself the power increases so

Powers, Exponents & Indices With large sums involving powers and base numbers there are a few rules we can apply. For example we can write as a single sum This shows that when you are multiplying base numbers that are the same then you can add the powers.

Powers, Exponents & Indices With large sums involving powers and base numbers there are a few rules we can apply. For example we can write as a single sum This shows that when you are dividing base numbers that are the same then you can subtract the powers.

Powers, Exponents & Indices With large sums involving powers and base numbers there are a few rules we can apply. For example we can write as a single sum

Square Roots Finding the square root of a number is the inverse (opposite) of squaring the number. For example calculate the (the square root of 9) What number do you square to get 9? The answer is 3. If you don’t immediately know the answer you can write out the factors of the number and see which one repeats.

Order of Operations This is a method we apply to a sum to make sure we arrive at the correct answer. It is known as lots of different acronyms such as; BEMDAS, BOMDAS, BIMDAS or PEMDAS All of them are correct but I will generally refer to it as BEMDAS

Order of Operations BEMDAS B rackets E xponents (powers or indices) M ultiplication D ivision A ddition S ubtraction (Note multiplication and division can be done working left to right) (Note addition & subtraction can be done working left to right)

Order of Operations Mistakes Mistake when squaring a number Mistake when multiplying a number If done correctly the answer is 100

Commutative Property This property refers to the order of a sum. When the order is relevant and when it is not. We have already seen that in sets the union or intersection of sets is commutative, This can be useful if we have a large sum and want a way to do it quickly, but it can’t always be used. Lets have a look at some examples.

Commutative Property Two of the equal signs are incorrect. Which ones are they? So we can say that addition and multiplication are commutative (the order we add or multiply doesn’t matter) but subtraction and division are not commutative.

Associative Property This property allows us to group numbers together to complete a large sum. Two of the equal signs are incorrect. Which ones are they? So we can say that addition and multiplication are associative (we can add or multiply in different groups) but subtraction and division are not associative

Distributive Property This property means that if we have a sum like the one below then the multiplication step occurs more than once (it distributes) Another way to do the sum would be This property becomes really important in algebra and in mathematical proofs

Estimating or Rounding Numbers There are two general rules Rule 1 Round 24,271 to the nearest If the first digit you remove is 4 or less, drop it and all following digits. 24,271 becomes 24,000 Rule 2 Round 24,271 to the nearest 100. If the first digit removed is 5 or greater, round up by adding 1 to the last digit kept. 24,271 becomes 24,300

Significant Figures This is another way at estimating an answer. Our number is 58,492 At the moment it has five significant figures Round 58,492 to three significant figures Count in from the left, find the third digit and apply your rounding rules 58,492 becomes 58,500 We replace the digits that were rounded with zeros so we don’t change the overall value.

All digits count Leading 0’s don’t Trailing 0’s do 0’s count in decimal form 0’s don’t count w/o decimal 0’s between digits count as well as trailing in decimal form Significant Figures