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Lesson Summary:
Audio GED Prep Mathematics Lesson 1
Integers and Decimal
Definition of an integer/decimal
An integer is a positive or negative number that does not have a decimal or fraction. These are what are used when counting objects. When a decimal point is present, everything to the left is an integer, and everything to the right is a decimal.
A decimal is a part of an integer. Numbers closer to the decimal point are larger, which is the opposite of how it is for integers.
Positive and negative numbers
In math, you might see what is called a number line, in which the middle point is marked 0, and numbers are listed as points on a line going out to the left and right. The number 0 is neither positive nor negative, and everything to the right of 0 is a positive number, and everything to the left of 0 is a negative number. You indicate a negative number by putting a dash/minus sign in front of it.
Negative numbers have their own set of rules. When you add a negative number, it is the same as subtracting that same number. If you subtract a negative number, it is the same as adding that same number. Just remember that these two are reversed. When you multiply two negative numbers as well, you will always get a positive number. (The minus signs cancel each other out.) If, however, you multiply a negative number and a positive number, you will always get a negative number.
Addition
Addition is when you take two numbers and combine them to create a new total. Addition is indicated by a plus sign, which looks like a small cross. When you speak an addition equation, you say “five plus six is eleven”, which is the same as 5 + 6 = 11.
Subtraction
Subtraction is when you take two numbers and take one away from another to create a new total. Subtraction is indicated by a minus sign, which looks like a small dash. When you speak an addition equation, you say “ten minus nine is one”, which is the same as 10 – 1 = 9.
Multiplication
Multiplication is when you take one number and multiply that number by another number. Multiplication is indicated by a small X or sometimes an *. When you speak a multiplication equation, you say “four times three is twelve”, which is the same as 4 x 3 = 12 or 4 * 3 = 12.
Division
Division is when you take one number and divide into the number of parts indicated by a second number. Division is indicated by a division sign (÷) or sometimes by a slash (/). When you speak a division equation, you say “ten divided by two equals five”, which is the same as 10 ÷ 2 = 5 or 10/2 = 5.
Absolute value
Absolute value refers to how many units a number is worth, no matter whether it is positive or negative. You can mark the absolute value by putting vertical lines around a number, such as |2|.
The absolute value of ten (|10|) is ten, and the absolute value of negative 10 (|-10|) is still 10.
Comparisons
Sometimes in math, you need to talk about whether something is greater than or less than, equal to, or not equal to something else.
To show that something is greater than something else, you use the sign >.
Example: 7 > 4
The sign ≥ means “greater than or equal to”.
To show that something is less than something else, you use the sign
Example: 2 < 4
The sign ≤ means “less than or equal to”.
To show that something is equal to something else, you use the sign =
Example: 5-2 = 1+2
To show that something is not equal to something else, you use the sign ≠
Example: 7-3 ≠ 9- 4
Large Addition
It is easy to do simple addition in your head, but larger problems often require either a calculator or for you to figure it out on paper. In order to do this put the larger number on top and the smaller number below it so that all the digits line up, the ones in the ones place, the tens in the tens place and so on. You then draw a line below the two numbers to indicate the equals sign. The answer will go below the line.
You then add the ones place, which is the place directly next to the decimal point to the left. If the sum of these numbers is less than ten, you simply write the total below the line directly underneath the ones place. If the result is larger than 10, you write a one above the next column over to the left and then the second digit of the total below the ones column. That one that you placed over the next column to the left needs to be figured into the total of the next column.
You repeat this process until all of the columns have been added. If one number has more places than the other, there will be some columns that only have one number. In that case, the blank space is treated like a 0. Do not forget to add any ones you had to carry over.
You can do this process with any number of numbers, even with numbers that have decimals. Just make sure that the decimal point gets put in the same place in the answer.
Large Subtraction
Large subtraction requires the same set up as addition, but be sure to put the first number on the top and second number below it. In addition, the order does not matter, but in subtraction it does.
If you subtract a column and the top number is greater than the bottom number, simply subtract one from the other and write the result below the line. If, however, the top number is smaller, you will need to “borrow” from the next column over. Subtract one from the number that is one column over to the left in the top row and then put a one in front of the top number in the column you are trying to subtract. Then write the result. You might have to do this multiple times until you have subtracted all columns.
Large Multiplication
Large multiplication requires both multiplication and addition to do. First, set up the two numbers like an addition problem. It is usually more convenient to have the smaller number on the bottom, but order does not matter.
Start with the first number to the far right on the bottom number and take turns multiplying it with every number on the top row from right to left, writing the results below the line.
One you are done with this, write a zero in the ones place directly under your first total. Then multiply everything by the second digit on the bottom and place these results next to the zero.
You must repeat this process with every digit in the bottom number, and every time, you need to add another zero before you get started. That is to compensate for each number being in a larger place, first ones, and then tens and then hundreds and so on. If you are multiplying something by 2 in the hundreds place, you are actually multiplying everything by 200, and so you need to add the two zeros at the end.
Once you have completed multiplying using all the digits in the bottom number, simply add all of the results together to a final total.
Long Division
Long division is done by placing the number to be divided inside a half box, a short vertical line to its left side and a long horizontal line above it. The number that shows how many parts the other number is to be divided into (also called the dividend) is placed on the other side of the vertical line.
You need to take the dividend and see if you can divide that number into the first digit, even if it is just one time, even if there is a remainder. You want to see how many times you can fit the dividend into that number without going over. For example, if you are dividing by three, and the first digit is 8, you can fit three in there twice (6), without going over. So, you write 2 directly over the number on the horizontal line, and then multiply the dividend by the number you just wrote, in this case 2 x 3 = 6.
You then write 6 under that first digit, which was 8 and subtract. In this case, you would get 8-6 = 2. You then look to see if there is another number next to the first number and if there is, you “bring it down” next to the result of the subtraction result you just got. If the next number next to the eight were 4, you would bring it down next to the 2 so that you now have 24. You then see how many times you can make 3 go into 24 without going over. In this case, it can go into 24 eight times, for an exact total. You multiply 3 x 8 and get 24. You then subtract 24 from 24 and get 0.
You complete this process until you have run out of digits to bring down. If you have run out of digits, you have two options. You can either write was is left over as a remainder, which is shown by a capital letter R (such as 25 R 2, or twenty-five, remainder 2), or you can divide into the decimal realm.
To continue with decimals, you can simply add a zero to whatever remainder you have and continue trying to divide until you have reached enough decimal places or you have reached 0. Be sure to put the decimal point in your answer.
©2015 Franz Amussen, All Rights Reserved
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