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Would you like us to take another look at this review? No, cancel Yes, report it Thanks! A sub-set is part of a set. All the green apples form another sub-set. Now we come to the idea of a union, which is used to combine things. The symbol for union is. Here, we use it to combine two or more intervals. We use the set and interval notation and the symbols described because it is easier than having to write everything out in words.
When two numbers are added, subtracted, multiplied or divided, you are performing arithmetic 2. These four basic operations can be performed on any two real numbers. Mathematics as a language uses special notation to write things down. So instead of: mathematicians write oneplusoneisequaltotwo 1. For more advanced mathematical workings, letters are usually used to represent numbers.
A constant has a xed value. The number 1 is a constant. The speed of light in a vacuum is also a constant which has been dened to be exactly m s 1 read metres per second. The speed of light is a big number and it takes up space to always write down the entire number. Therefore, letters are also used to represent some constants. In the case of the speed of light, it is accepted that the letter c represents the speed of light.
Such constants represented by letters occur most often in physics and chemistry. In this equation, y represents the price of the item you are buying, x represents the amount of change you should get back and z is the amount of money given to the cashier. So, if the price is R10 and you gave the cashier R15, then write R15 instead of z and R10 instead of y and the change is then x.
We will learn how to solve this equation towards the end of this chapter. You can think of subtracting as being the opposite of adding since adding a number and then subtracting the same number will not change what you started with. The order in which numbers are added does not matter, but the order in which numbers are subtracted does matter.
Don't worry, its exactly the same thing. Mathematicians are ecient and like to write things in the shortest, neatest way possible. So although 4x and x4 are the same thing, it looks better to write 4x. Therefore, both addition and multiplication are described as commutative operations. This is important as you can get dierent answers depending on the order in which you do things. Likewise it doesn't matter which order you do addition and subtraction. Negative numbers can be very confusing to begin with, but there is nothing to be afraid of.
The numbers that are used most often are greater than zero. These numbers are known as positive numbers. A negative number is a number that is less than zero. So, if we were to take a positive number a and subtract it from zero, the answer would be the negative of a. Figure 1.
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Positive numbers appear to the right of zero and negative numbers appear to the left of zero Working with Negative Numbers When you are adding a negative number, it is the same as subtracting that number if it were positive. Likewise, if you subtract a negative number, it is the same as adding the number if it were positive. Numbers are either positive or negative and we call this their sign. Subtraction is actually the same as adding a negative number. Now, which do you nd easier to work out? Most people nd that the rst way is a bit more dicult to work out than the second way.
For example, most people nd 12 3 a lot easier to work out than , even though they are the same thing. So a b, which looks neater and requires less writing is the accepted way of writing subtractions. Table 1. The rst column shows the sign of the rst number, the second column gives the sign of the second number and the third column shows what sign the answer will be.
So multiplying or dividing a negative number by a positive number always gives you a negative number, whereas multiplying or dividing numbers which have the same sign always gives a positive number. Adding numbers works slightly dierently see Table 1. If you add two positive numbers you will always get a positive number, but if you add two negative numbers you will always get a negative number.
If the numbers have a dierent sign, then the sign of the answer depends on which one is bigger Living Without the Number Line The number line in Figure 1. To keep things simple, we will write down three tips that you can use to make working with negative numbers a little bit easier. These tips will let you work out what the answer is when you add or subtract numbers which may be negative, and will also help you keep your work tidy and easier to understand.
In this case, we have seen that adding a negative number to a positive number is the same as subtracting the number from the positive number. For example, a question like What is? This tip tells us that all we need to do is take the smaller number away from the larger one and remember to give the answer the sign of the larger number. In this equation, F is bigger than e. Earlier in this chapter, we wrote a general equation for calculating how much change x we can expect if we know how much an item costs y and how much we have given the cashier z.
In mathematical terms, this is known as solving an equation for an unknown x in this case. The most important thing to remember is that an equation is like a set of weighing scales. In order to keep the scales balanced, whatever is done to one side must be done to the other. In order to keep the scales balanced, you must do the same thing to both sides. So, if you add, subtract, multiply or divide the one side, you must add, subtract, multiply or divide the other side too Method: Rearranging Equations You can add, subtract, multiply or divide both sides of an equation by any number you want, as long as you always do it to both sides.
In the example, the change should be R5. In real life we can do this in our heads; the human brain is very smart and can do arithmetic without even knowing it. When you subtract a number from both sides of an equation, it looks like you just moved a positive number from one side and it became a negative on the other, which is exactly what happened. Likewise, if you move a multiplied number from one side to the other, it looks like it changed to a divide. It is wrong because we did not divide the c term by a as well.
Click here for the solution 7 2. The rst way of writing a fraction is very hard to work with, so we will use only the other two. We call the number on the top left the numerator and the number on the bottom right the denominator. The reciprocal of a fraction is the fraction turned upside down, in other words the numerator becomes the denominator and the denominator becomes the numerator.
So, the reciprocal of 2 3 is 3 2. A decimal number is a number which has an integer part and a fractional part. The integer and the fractional parts are separated by a decimal point, which is written as a comma in South African schools. For example the number can be written much more neatly as 3, All real numbers can be written as a decimal number. However, some numbers would take a huge amount of paper and ink to write out in full!
Some decimal numbers will have a number which will repeat itself, such as 0, where there are an innite number of 3's. These can be written more easily in scientic notation, which has the general form a 10 m 1. The m is an integer and if it is positive it represents how many zeros should appear to the right of a. If m is negative, then it represents how many times the decimal place in a should be moved to the left.
For example 3, represents and 3, represents 0, If a number must be converted into scientic notation, we need to work out how many times the number must be multiplied or divided by 10 to make it into a number between 1 and 10 i. We do this by counting the number of decimal places the decimal point must move. For example, write the speed of light which is ms 1 in scientic notation, to two decimal places. First, determine where the decimal point must go for two decimal places to nd a and then count how many places there are after the decimal point to determine m.
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So, the speed of light in scientic notation to two decimal places is 3, ms 1. As another example, the size of the HI virus is around 1, m. This is equal to 1, 2 0, m, which is 0, m Real Numbers Now that we have learnt about the basics of mathematics, we can look at what real numbers are in a little more detail. The following are examples of real numbers and it is seen that each number is written in a dierent way , 1, ,, 10, 2, 1, 5, 6, 35, 1.
A set diagram of the dierent number types is shown in Figure 1. The irrational numbers are the numbers not inside the set of rational numbers. All of the integers are also rational numbers, but not all rational numbers are integers Non-Real Numbers All numbers that are not real numbers have imaginary components. We will not see imaginary numbers in this book but they come from 1. Since we won't be looking at numbers which are not real, if you see a number you can be sure it is a real one Natural Numbers The rst type of numbers that are learnt about are the numbers that are used for counting.
These numbers are called natural numbers and are the simplest numbers in mathematics: 0, 1, 2, 3, 4, These are also sometimes called whole numbers. The natural numbers are a subset of the real numbers since every natural number is also a real number Integers The integers are all of the natural numbers and their negatives The integers are a subset of the real numbers, since every integer is a real number Rational Numbers The natural numbers and the integers are only able to describe quantities that are whole or complete.
For example, you can have 4 apples, but what happens when you divide one apple into 4 equal pieces and share it among your friends? Then it is not a whole apple anymore and a dierent type of number is needed to describe the apples. This type of number is known as a rational number. The following are examples of rational numbers: a b 20 9, 1 2, 20 10, 3 15 1. Mathematicians use the symbol Q to mean the set of all rational numbers. The set of rational numbers contains all numbers which can be written as terminating or repeating decimals Rational Numbers All integers are rational numbers with a denominator of 1.
You can add and multiply rational numbers and still get a rational number at the end, which is very useful. It is always best to simplify any rational number, so that the denominator is as small as possible. This can be achieved by dividing both the numerator and the denominator by the same integer. Finding a lowest common denominator means nding the lowest number that both denominators are a factor 13 of.
A factor of a number is an integer which evenly divides that number without leaving a remainder. The following numbers all have a factor of 3 and the following all have factors of 4 3, 6, 9, 12, 15, 18, 21, 24, The lowest common denominator of 3 and 4 is the smallest number that has both 3 and 4 as factors, which is For example, if we wish to add , we rst need to write both fractions so that their denominators are the same by nding the lowest common denominator, which we know is We can do this by multiplying 3 4 by 3 3 and 2 3 by and 4 4 are really just complicated ways of writing 1.
Proper fractions have a numerator that is smaller than the denominator. For example, 1 2, 3 15, 5 20 are proper fractions. Improper fractions have a numerator that is larger than the denominator. For example, 1. Improper fractions can always be written as the sum of an integer and a proper fraction Converting Rationals into Decimal Numbers Converting rationals into decimal numbers is very easy.
If you use a calculator, you can simply divide the numerator by the denominator. If you do not have a calculator, then you have to use long division. Since long division was rst taught in primary school, it will not be discussed here. If you have trouble with long division, then please ask your friends or your teacher to explain it to you Irrational Numbers An irrational number is any real number that is not a rational number. When expressed as decimals, these numbers can never be fully written out as they have an innite number of decimal places which never fall into a repeating pattern.
Identify the number type rational, irrational, real, integer of each of the following numbers: c a. Write the following in symbols: a. In mathematics, innity is often treated as if it were a number, but it is clearly a very dierent type of number" than integers or reals. When talking about recurring decimals and irrational numbers, the term innite was used to describe never-ending digits End of Chapter Exercises 1. The numbers that will be used in high school are all real numbers, but there are many dierent ways of writing any single real number.
This chapter describes rational numbers. Various authors use it in many dierent ways. We use the following denitions: natural numbers are 1, 2, 3, Denition 2. This means that all integers are rational numbers, because they can be written with a denominator of 1. A number may not be written as an integer divided by another integer, but may still be a rational number. This is because the results may be expressed as an integer divided by an integer.
The rule is, if a number can be written as a fraction of integers, it is rational even if it can also be written in another way as well.
If a is an integer, b is an integer and c is irrational, which of the following are rational numbers? If a 1 is a rational number, which of the following are valid values for a? There are two more forms of rational numbers Investigation : Decimal Numbers You can write the rational number as the decimal number 0,5. Write the following numbers as decimals: Do the numbers after the decimal comma end or do they continue?
If they continue, is there a repeating pattern to the numbers?
You can write a rational number as a decimal number. Two types of decimal numbers can be written as rational numbers: 1. The dot represents recurring 3's i. For example, the rational number 5 6 can be written in decimal notation as 0, 8 3 and similarly, the decimal number 0,25 can be written as a rational number as 1 4.
The fractional part can be written as a rational number, i. Each digit after the decimal point is a fraction with a denominator in increasing powers of ten. Write the following as fractions: a 0, 1 b 0, 12 c 0, 58 d 0, Click here for the solution 4 Table Converting Repeating Decimals into Rational Numbers When the decimal is a repeating decimal, a bit more work is needed to write the fractional part of the decimal number as a fraction.
We will explain by means of an example. In general, if you have one digit recurring, then multiply by If you have two digits recurring, then multiply by If you have three digits recurring, then multiply by Can you spot the pattern yet? The number of zeros is the same as the number of recurring digits. Not all decimal numbers can be written as rational numbers. However, when possible, you should try to use rational numbers or fractions instead of decimals. Write the following using the repeated decimal notation: a.
Write the following in decimal form, using the repeated decimal notation: a. Write the following decimals in fractional form: a. Real numbers can be either rational or irrational. A rational number is any number which can be written as a b where a and b are integers and b 0 3. The following are rational numbers: a.
Fractions with both denominator and numerator as integers. Decimal numbers that end. Decimal numbers that repeat. Write each decimal as a simple fraction: a. Show that the decimal 3, is a rational number. Click here for the solution Express 0, 7 8 as a fraction a b where a, b Z show all working. For example, instead of 5 5 5, we write 5 3 to show that the number 5 is multiplied by itself 3 times and we say 5 to the power of 3. Likewise 5 2 is 5 5 and 3 5 is We will now have a closer look at writing numbers using exponential notation.
Denition 3. The nth power of a is dened as: a n 3. We can also dene what it means if we have a negative exponent n. Some of these laws might have been seen in earlier grades, but we will list all the laws here for easy reference and explain each law in detail, so that you can understand them and not only remember them. In the days before calculators, all multiplication had to be done by hand with a pencil and a pad of paper. Multiplication takes a very long time to do and is very tedious.
Adding numbers however, is very easy and quick to do. If you look at what this law is saying you will realise that it means that adding the exponents of two exponential numbers of the same base is the same as multiplying the two numbers together. This meant that for certain numbers, there was no need to actually multiply the numbers together in order to nd out what their multiple was. For example, 3. Law 4 is just a more general way of saying the same thing. We get this law by multiplying law 3 by a m on both sides and using law 2.
An exponential of a number is just a real number. So, even though the sentence sounds complicated, it is just saying that you can nd the exponential of a number and then take the exponential of that number. You just take the exponential twice, using the answer of the rst exponential as the argument for the second one. Possible 3 answers are: 2, 1, 1, 1 3, 8. Answers may be repeated.
Khan Academy video on Exponents - 5 This media object is a Flash object. Simplify as far as possible: a b. Simplify without using a calculator. Leave your answers with positive exponents. If the nth root of a number cannot be simplied to a rational number, we call it a surd. For example, 2 and 3 6 are surds, but 4 is not a surd because it can be simplied to the rational number 2.
In this chapter we will only look at surds that look like n a, where a is any positive number, for example 7 or 3 5. It is very common for n to be 2, so we usually do not write 2 a. Instead we write the surd as just a, which is much easier to read. It is sometimes useful to know the approximate value of a surd without having to use a calculator. For example, we want to be able to estimate where a surd like 3 is on the number line. So how do we know where surds lie on the number line? From a calculator we know that 3 is equal to 1, It is easy to see that 3 is above 1 and below 2.
Challenge: Can you explain why? If you don't believe this fact, check it for a few numbers to convince yourself it is true. How do we use this fact to help us guess what 18 is? Now we have a better idea of what 18 is. Now we know that 18 is less than 5, but this is only half the story. We can use the same trick again, but this time with 18 on the right-hand side. As you can see, we have shown that 18 is between 4 and 5. You will notice that our idea used perfect squares that were close to the number Here is a quick summary of what a perfect square or cube is: note: A perfect square is the number obtained when an integer is squared.
Similarly, a perfect cube is a number which is the cube of an integer. To make it easier to use our idea, we will create a list of some of the perfect squares and perfect cubes. Exercise 4. Find the two consecutive integers such that 26 lies between them. Remember that consecutive numbers are two numbers one after the other, like 5 and 6 or 8 and 9.
The surd lies between these two numbers. Click here for the solution Find two consecutive integers such that 15 lies between them. Click here for the solution. Step 2. Step 3. Solution to Exercise 4. So 3 49 does not lie between 1 and 2.
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So 3 49 does not lie between 2 and 3. So 3 49 lies between 3 and 4. Not only is that impossible, but writing numbers out to many decimal places or a high accuracy is very inconvenient and rarely gives practical answers. For this reason we often estimate the number to a certain number of decimal places or to a given number of signicant gures, which is even better.
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This means that any number that is not a terminating decimal number or a repeating decimal number is irrational. If you are asked to identify whether a number is rational or irrational, rst write the number in decimal form. If the number is terminated then it is rational. If it goes on forever, then look for a repeated pattern of digits. If there is no repeated pattern, then the number is irrational. When you write irrational numbers in decimal form, you may if you have a lot of time and paper! However, this is not convenient and it is often necessary to round o Investigation : Irrational Numbers Which of the following cannot be written as a rational number?
Remember: A rational number is a fraction with numerator and denominator as integers. Terminating decimal numbers or repeating decimal numbers are rational. For example, if you wanted to round-o 2, to three decimal places then you would rst count three places after the decimal. You round up the nal digit if the rst digit after the was greater or equal to 5 and round down leave the digit alone otherwise. In the case that the rst digit before the is 9 and the you need to round up the 9 becomes a 0 and the second digit before the is rounded up.
So, since the rst digit after the is a 5, we must round up the digit in the third decimal place to a 3 and the nal answer of 2, rounded to three decimal places is 2, 5. For convenience irrational numbers are often rounded o to a specied number of decimal places 5. Write the following rational numbers to 2 decimal places: a. Write the following irrational numbers to 2 decimal places: a.
Use your calculator and write the following irrational numbers to 3 decimal places: a. Use your calculator where necessary and write the following irrational numbers to 5 decimal places: a. Write the following irrational numbers to 3 decimal places and then write them as a rational number to get an approximation to the irrational number. In this chapter, we learn more about the mathematics of patterns.
Patterns are recognisable as repetitive sequences and can be found in nature, shapes, events, sets of numbers and almost everywhere you care to look. For example, seeds in a sunower, snowakes, geometric designs on quilts or tiles, the number sequence 0, 4, 8, 12, 16, Investigation : Patterns Can you spot any patterns in the following lists of numbers? Here we list the most common patterns and how they are made. Examples: 1. This sequence has a dierence of 3 between each number.
The pattern is continued by adding 3 to the last number each time. This sequence has a dierence of 5 between each number. The pattern is continued by adding 5 to the last number each time. This sequence has a factor of 2 between each number. The pattern is continued by multiplying the last number by 2 each time. This sequence has a factor of 3 between each number. The pattern is continued by multiplying the last number by 3 each time Special Sequences Triangular Numbers 1, 3, 6, 10, 15, 21, 28, 36, 45, By adding another row of dots with one more dot in each row than in the previous row and counting all the dots, we can nd the next number of the sequence Square Numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, The next number is made by squaring the number of the position in the pattern.
The second number is 2 squared 2 2 or 2 2. The seventh number is 7 squared 7 2 or 7 7 etc Cube Numbers 1, 8, 27, 64, , , , , , The next number is made by cubing the number of the position in the pattern. The second number is 2 cubed 2 3 or. The seventh number is 7 cubed 7 3 or etc Fibonacci Numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, The next number is found by adding the two numbers before it together. The next number in the sequence above would be Can you gure out the next few numbers? Say you and 3 friends decide to study for Maths, and you are seated at a square table.
A few minutes later, 2 other friends join you and would like to sit at your table and help you study. Naturally, you move another table and add it to the existing one. Now 6 of you sit at the table. Another 2 of your friends join your table, and you take a third table and add it to the existing tables. Now 8 of you can sit comfortably. Figure 6. Examine how the number of people sitting is related to the number of tables. In the sequence 1; 4; 9; 16; 25; Exercise 6.
Solution on p. As before, you and 3 friends are studying for Maths, and you are seated at a square table. A few minutes later, 2 other friends join you and add another table to the existing one. Now 6 of you can sit together. A short time later 2 more of your friends join your table, and you add a third table to the existing tables. Now 8 of you can sit comfortably as shown: 6.
Then, use the general formula to determine how many people can sit around 12 tables and how many tables are needed for 20 people. It is also important to note the dierence between n and a n. Like our Study Table example above, the rst table Table 1 holds 4 people. Find the general formula for the following sequences and then nd a 10, a 50 and a : a. Work out the missing terms. A conjecture can be seen as an educated guess or an idea about a pattern.
For example: Make a conjecture about the next number based on the pattern 2; 6; 11; 17 The numbers increase by 4, 5, and 6. Conjecture: The next number will increase by 7. So, it will be or Add another two rows to the end of the pattern. Make a conjecture about this pattern. Write your conjecture in words. Generalise your conjecture for this pattern in other words, write your conjecture algebraically. Prove that your conjecture is true.
Square numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, Cube numbers 1, 8, 27, 64, , , , , , Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, Find the n th term for: 3, 7, 11, 15, Click here for the solution 2 2. Find the general term of the following sequences: a. The seating in a section of a sports stadium can be arranged so the rst row has 15 seats, the second row has 19 seats, the third row has 23 seats and so on.
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Calculate how many seats are in the row Click here for the solution 4 4. A single square is made from 4 matchsticks. Two squares in a row need 7 matchsticks and 3 squares in a row need 10 matchsticks. Determine: a. You would like to start saving some money, but because you have never tried to save money before, you have decided to start slowly. At the end of the rst week you deposit R5 into your bank account.
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Then at the end of the second week you deposit R10 into your bank account. At the end of the third week you deposit R After how many weeks do you deposit R50 into your bank account? Click here for the solution 6 6. A horizontal line intersects a piece of string at four points and divides it into ve parts, as shown below.
We can see that for 3 tables we can seat 8 people, for 4 tables we can seat 10 people and so on. We started out with 4 people and added two each time. Thus, for each table added, the number of persons increased by 2. Solution to Exercise 6. The number of people seated at n tables is: Table 6. Considering the example from the previous section, how many people can sit around say 12 tables?
Squaring a number and adding the same number gives the same result as squaring the next number and subtracting that number. We have chosen to use x here. You could choose any letter to generalise the pattern. Welcome to the Grade 10 Finance Chapter, where we apply maths skills to everyday nancial situations that you are likely to face both now and along your journey to purchasing your rst private jet.
If you master the techniques in this chapter, you will grasp the concept of compound interest, and how it can ruin your fortunes if you have credit card debt, or make you millions if you successfully invest your hard-earned money. You will also understand the eects of uctuating exchange rates, and its impact on your spending power during your overseas holidays! Before we begin discussing exchange rates it is worth noting that the vast majority of countries use a decimal currency system. This simply means that countries use a currency system that works with powers of ten, for example in South Africa we have 10 squared cents in a rand.