Chapter 1 : Numbers and Operations
Topics covered in this snack-sized chapter:
| Natural Numbers
| The natural numbers are the counting numbers.
Examples: 1, 2, 3, 4, 5.
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| Whole Numbers
| The whole numbers are the counting numbers including 0.
Examples: 0, 1, 2, 3, 4, 5.
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| Integers
| Integers include counting numbers, zero and negative numbers.
Examples: -3, -2, -1, 0, 1, 2, 3.
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| Rational Number
| A rational number is a number that can be expressed as a fraction or ratio.
The numerator and the denominator of the fraction are both integers.
Example: 2/3.
Note that 6 can also be written as fraction.
6 = 6/1.
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| Irrational Numbers
| Numbers that cannot be written as fractions.
Example: √2 = 1.414213562…..
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| Prime Numbers
| A prime number is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the number itself.
Examples: 2, 3, 5, 7, 11, 13.
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| Complex Numbers
| Complex number is a number that can be expressed in the form of a + bi where a and b are real numbers and i is the square root of -1.
Example: We can write
= 3i
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| Composite
Number
| A composite number can be divided evenly by numbers other than 1 or itself.
Examples: 4, 6, 8, 9, 10, etc.
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| Multiples
| If a number is divisible exactly by a second number, then the first number is said to be a multiple of the second number.
Example: 15 is a multiple of 5.
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| Factors
| If one number divides a second number exactly, then the first number is said to be a factor of the second number.
Example: 5 is a factor of 15.
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| Even Numbers
| Numbers divisible by 2 are called even numbers.
Examples: 2, 4, 6, 8.
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| Odd
Numbers
| Numbers which are NOT divisible by 2 are called odd numbers.
Examples: 1, 3, 5, 7.
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| Roman
Numerals
| Roman numerals are numeral system of ancient Rome based on the letters of the alphabet, which are combined to signify the sum of their values.
| I
| 1
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| V
| 5
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| X
| 10
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| L
| 50
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| C
| 100
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| D
| 500
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| M
| 1000
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Place value determines the value of a digit in a number, based on the location of the digit.
Below image is showing the place value for 25612.5.
Addition is finding the total, or sum, by combining two or more numbers.
Example: 6 + 3 = 9
The process of taking one number or amount away from another is known as subtraction.
Example: 5 – 4 = 1
It signifies repeated addition.
Example: 3 × 4 = 12.
It can also be written as:
3 × 4 = 3 + 3 + 3 + 3 = 12
Here 3 is the multiplicand, 4 is the multiplier and 12 is the product.
Division is a way to find out how many times a number is contained in other number.
Example: Division of 19 by 4.
Here, 4 is Divisor, 19 is Dividend, 4 is quotient, and 3 is remainder.
Rules for Adding Integers with the same Signs:
Add the numbers together.
Give the answer the same sign.
Example: Add (-5) and (-10)
Both the numbers are negative. First add the numbers.
5 + 10 = 15.
As the numbers are negative so the answer would be – 15.
Rules for Adding Integers with different Signs:
Ignore the signs and find the difference.
Give the answer the sign of the larger number.
Example: Add (-5) and (10)
Ignore the signs and find the difference.
10 – 5 = 5
Here the larger number 10 is positive, so the answer is 5.
Rules for Subtracting Integers:
Change the subtraction sign to addition.
Change the sign of the second number to the opposite sign.
Follow the rules for adding integers.
Example: Subtract: -8 – (-5).
Change the subtraction sign to addition and change -5 to 5.
-8 + 5
= -3
Rules of Signs for Multiplication:
When two numbers with like signs are multiplied, the result is always positive.
When two numbers with different signs are multiplied, the result is always negative.
Examples:
Rules of Signs for Division:
When two numbers with like signs are divided, the result is always positive.
When two numbers with different signs are divided, the result is always negative.
Examples:
To ensure that anyone evaluating a mathematical expression calculates the same value, one has to follow the established order of operations:
| 1
| Find whether a bar sign is present in the expression, evaluate it first.
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| 2
| First perform any calculations inside parentheses ( ), then brackets [ ] and braces { }.
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| 3
| Next perform any exponent operations.
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| 4
| Next perform all multiplication and division, working from left to right.
If the expression contains both operations (multiplication and division), do whichever occurs first, going from left to right.
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| 5
| Lastly, perform all addition and subtraction, working from left to right.
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| Please
| First perform any calculations inside parentheses ( ), then brackets [ ] and braces { }.
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| Excuse
| Next perform any exponent operations.
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| Me
| Next perform all multiplication working from left to right.
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| Dear
| Next perform all division, working from left to right.
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| Aunt
| Next perform all addition working from left to right.
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| Susan
| Lastly, perform all subtraction, working from left to right.
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