Basic Math Notes
Ernest Bywater
Basic Math Notes
Copyright © 2003 by Ernest Bywater
All rights are reserved by the author, including the right to reproduce this book or portions thereof in any form.
Cover Art
The background image is The School of Athens a fresco by Raphael and is in the public domain. The trimming, manipulation, and adding of text is by Ernest Bywater. All rights to the cover image are reserved by the copyright owners.
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22 December 2017 Edition
Published by Ernest Bywater
ISBN: 978-1-387-77199-8
The title styles in use are a chapter, a sub-chapter, and a section.
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Basic Principle and Assumptions
This document is intended as a study help guide for students of general mathematics principles and processes. It's also a set of useful reminder sheets. I started it as tutoring notes for a Reconstructing Maths course student I was tutoring. She was studying to be a Primary school teacher what they now call Years 1 to 6 or Grades 1 to 6. Soon after I started tutoring her I started tutoring her son who was in high school, because he had some issues as well. I've not covered everything that's taught in high school, just the few I tutored them on.
The following principle and assumptions apply to basic mathematics used in English.
Basic Principle
In all mathematical equations the basic principle is that the final result of all the actions and values on one side of the equals symbol, =, is exactly the same as on the other side. Thus what is on one side will always exactly balance what is on the other.
In most mathematical situations we are seeking to calculate the final result, but sometimes we know some values on both sides and this principle can be used to help find the missing values.
The concept of nothing, or no value, is called a zero and is represented with the symbol 0. Some people call the same value nought, which is another word for nothing. The zero simply means there is no value in the position it is in. So a number of 20 means two tens and no singles (refer to Positional Values below).
General Assumptions
| Numbers: | The only numbers or numerals used are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. |
| Base: | Unless specifically told otherwise (eg 58) with the base being shown beside the number, and below the line. You will generally be using Base 10, or Decimal System. |
| Extra Zeros: | A zero before the decimal dot, or first number, are assumed. A zero after the decimal dot, or last decimal place, are assumed. |
In higher levels of mathematics like algebra and calculus they will often use letters of the alphabet and other symbols to mean various things. The most common letter used is 'x' to mean an unknown value, this is often used to identify the value they want you to calculate.
Note: The mathematics in this document are in the Decimal System, unless stated otherwise.
Positional Values
The value of each number changes with its position in relation to the dot marking the difference between whole units and part units, because the face value of the number is multiplied by the value of its position. The value of the Units position is always its face value, and the other positions are calculated as a multiple of the base value.
Most mathematics with numbers you will come across work on Base 10, and this is known as the Decimal System where each number position is worth 10 times that of the one on its right. The dot between the whole units and fractions is known as the decimal dot. The system starts at zero, and the numerals used are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Thus the number ten uses the first two positions with a one and a zero to show 10. The most often used values are listed in the following order as shown in the layout form below the list, with the decimal dot in the centre of both.
million, hundred thousand, ten thousand, thousand, hundred, ten, single unit,
decimal dot,
tenth, hundredth, thousandth ten thousandth, hundred thousandth, millionth
___ ___ ___ ___ ___ ___ ___ . ___ ___ ___ ___ ___ ___
To make large whole numbers easier to read, they are shown in divisions of 3 positions that are divided by a comma (,), thus a million is shown as 1,000,000 - not as 1000000.
Another common system used is the Binary System where the base unit has a value of two units, and the only numerals used a 0 and 1. Thus a count of two is shown as 10, while three is 11, and four is 100.
Another system often seen today is the Hexadecimal System used in some computer coding. This is base sixteen, and the name is derived from the Greek for six plus ten. The count up from zero includes letters in this ascending order: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G. Where most people will see it is in computer coding for colours #000000 for white to #FFFFFF for black.
Note: The colour codes in computing are based on a cube where the six positions represent the colour based on a graph of each side shown in three pairs of two hexadecimal positions for each side as single byte of information.
Functions
The main function processes used in mathematics are:
| Addition: | Increasing the value of one number by one or more numbers. Example: 1 + 2 = 3 is one plus two equals three. |
| Subtraction: | Decreasing the value of one number by one or more numbers. Example: 3 – 2 = 1 is three minus 2 equals one. |
| Multiplication: | Increasing the value of one number by another as if you had added
the first number to itself as often as the second number is. The term times is often used to
express the multiplication function. The second number given is called the multiplier. Example: 2 x 3 = 6 is two times three equals six; equivalent to 2 + 2 + 2 = 6. |
| Division: | Finding out how often one number will go into another number. Example: 6 ÷ 3 = 2 is six divided by three equals two. |
| Square: | The square of a number is the value found when you multiply that
number by itself. This is expressed by a little 2 after and above it. Example: 32 = 9 is three squared equals nine, or 3 x 3 = 9 |
| Square Root: | The square root of a number is the value that, when multiplied by itself
will give you the number you started with. This is expressed by the root symbol √, this is usually shown with a value as
2√ (square root) or 3√ (cube root) etc; if no value it is assumed as
2√. Example: 2√9 = 3 is square root of nine equals three. |
| Cube: | The cube of a number is the value found when you multiply that number
by itself and then multiply the answer by the original number again. This is express as a little 3 after and above it. Example: 23 = 8 is two cubed equals eight, or 2 x 2 x 2 = 8 |
| Cube Root: | The cube root of a number is the value that, when multiplied by itself and
then that answer is again multiplied by the value, will give you the number you started with This is expressed by the root
symbol with the value of three, 3√. Example: 3√27 =3 is cube root of twenty-seven equals three 27 = 3 x 3 x 3 which equates to 27 = 9 x 3, and 3 x 3 x 3. |
| Powers: | The process used in squares and cubes can be extended infinitely with the small raised number beings known as the Power. Thus 22 is also known as two to the power of two. The number 26 is expressed as two to the power six, or 2 x 2 x 2 x 2 x 2 x 2 = 64. Often very large numbers are written as 1.325 x 1013, this is called Scientific Notation as it is shorter than writing 13,250,000,000,000.000. |
Note: This same power numbering process can also be applied to roots and thus you could have a number 9 √512. Luckily all you're likely to come across in daily usage are square roots and cube roots.
Later I'll mention the system known as BoDMAS, which I should've used in some of the examples above to include brackets to split parts of the equation, but I haven't used it because I've not yet explained it to you, and it doesn't change the answers above. This non-use of brackets will also appear in some of the other examples where they make no difference because I'm not mixing up the types of calculations BoDMAS clarifies. I won't use brackets until after I explain their use and why.
The final value of a calculation is after the = (equals) sign and is called the answer, total or sum. Sum was only the total of additions, but it's now used with any maths calculation. This is due to the way it was misused to identify the final answer in some early computer programs for doing calculations and spreadsheets.
Some calculations can be done on a line (2 + 2 = 4) but some are best done down the page, such as multiplication, division, and adding a list of numbers. When writing such calculations it is important to keep numbers aligned, to match their positional values to reduce errors. Notice the difference in adding readability in the three columns below (with the right-hand column having their positions aligned):
| 24 | 24 | 24 |
| 125 | 125 | 125 |
| 45 | 45 | 45 |
| 56 | 56 | 56 |
| 167 | 167 | 167 |
| 328 | 328 | 328 |
Long Multiplication
Although 2 x 6 is easy to do in the mind, 453 x 278 is not so easy, thus breaking it down into smaller steps makes it faster and easier. Long Multiplication breaks it into a series of single number times single number calculations that you can then add up, but remember to carry forward higher position values in the single digit multiplications. This process also makes the workings easier to see and check. The two main methods:
| 453 | ||
| x | 278 | |
| 3,624 | multiply each number in the top line by the unit position value of 8 | |
| 31,710 | write a zero then multiply each number in the top line by the next position value of 7 | |
| 90,600 | write 2 zeros then multiply each number in the top line by the next position value of 2 | |
| 125,934 | add up the multiplication lines |
Note: Remember to keep adding zeros for each position value you go up in the multiplier at each stage.
or calculate it as single digit entries:
| 453 | ||
| x | 278 | |
| 24 | multiply 3 x 8 | |
| 400 | write a zero then multiply 5 x 8 (equal to 50 x 8) | |
| 3,200 | write 2 zeros and multiply 4 x 8 (equal to 400 x 8) | |
| 210 | write a zero and multiply 3 x 7 (equal to 3 x 70) | |
| 3,500 | write 2 zeros and multiply 5 x 7 (equal to 50 x 70) | |
| 28,000 | write 3 zeros and multiply 4 x 7 (equal to 400 x 70) | |
| 600 | write 2 zeroes and multiply 3 x 2 (equal to 3 x 200) | |
| 10,000 | write 3 zeros and multiply 5 x 2 (equal to 50 x 200) | |
| 80,000 | write 4 zeros and multiply 4 x 2 (equal to 400 x 200) | |
| 125,934 | add up the multiplication lines |
Note: Remember to keep adding zeros for each position value you go up in the multiplier at each stage.