Welcome to Basic Computation!
Welcome to your first step in Secondary 1 Mathematics! Think of this chapter as the "toolbox" for everything else you will do in math. Just like a builder needs to know how to use a hammer and a screwdriver, a mathematician needs to be fast and accurate with numbers. We are going to learn some "shortcuts" for dividing, how to break numbers down into their smallest parts, and the correct order for solving big, messy expressions.
1. Divisibility Tests: The Mathematical VIP Pass
Have you ever looked at a large number and wondered if you could divide it evenly without using a calculator? These tests are like "VIP passes" that let you know instantly if a number is divisible by 4, 6, 8, or 9.
Divisibility by 4
Look at the last two digits of the number. If those two digits form a number that can be divided by \(4\), then the whole number is divisible by \(4\).
Example: Is \(1,524\) divisible by \(4\)? Look at the last two digits: \(24\). Since \(24 \div 4 = 6\), the answer is Yes!
Divisibility by 6
A number is divisible by \(6\) if it passes two tests: it must be even (divisible by \(2\)) AND the sum of its digits must be divisible by \(3\).
Example: Is \(162\) divisible by \(6\)? It is even (ends in \(2\)). Sum of digits: \(1 + 6 + 2 = 9\). Since \(9\) is divisible by \(3\), the answer is Yes!
Divisibility by 8
Look at the last three digits. If they are divisible by \(8\), the whole number is.
Example: For \(7,120\), check \(120\). Since \(120 \div 8 = 15\), the whole number is divisible by \(8\).
Divisibility by 9
This is like the test for \(3\). If the sum of all the digits is divisible by \(9\), then the number is divisible by \(9\).
Example: Is \(729\) divisible by \(9\)? Sum: \(7 + 2 + 9 = 18\). Since \(18 \div 9 = 2\), the answer is Yes!
Quick Tip: If a number is divisible by \(8\), it is automatically divisible by \(4\). But the opposite is not always true!
2. The Concept of Power
In math, we often multiply a number by itself many times. Instead of writing \(5 \times 5 \times 5 \times 5\), we use index notation (or powers).
In the expression \(5^4\):
- The number \(5\) is called the Base (the number being multiplied).
- The small number \(4\) is called the Index or Exponent (how many times the base appears).
Note: In this chapter, we just need to understand what the notation means. You don't need to do complex calculations with huge powers yet!
3. Prime Factorisation
Every whole number greater than \(1\) is either a Prime Number (only divisible by \(1\) and itself, like \(2, 3, 5, 7, 11\)) or a Composite Number (can be broken down further).
Prime Factorisation is the process of breaking a composite number down until it is just a string of prime numbers multiplied together.
Method A: Factor Tree
Keep splitting the number into "branches" until every branch ends in a prime number.
Example: \(12 = 2 \times 6 = 2 \times (2 \times 3)\). So, \(12 = 2 \times 2 \times 3\). Using index notation, we write \(12 = 2^2 \times 3\).
Method B: Short Division
Divide the number by the smallest prime number possible, then divide the result again, and keep going until you reach \(1\).
\(2 | \underline{60}\)
\(2 | \underline{30}\)
\(3 | \underline{15}\)
\(5 | \underline{5}\)
\(\quad 1\)
So, \(60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5\).
4. H.C.F. and L.C.M.
H.C.F. (Greatest Common Divisor / GCD): The largest number that can divide two or more numbers exactly.
L.C.M. (Least Common Multiple): The smallest number that is a multiple of two or more numbers (the first number they "meet" at when counting up).
Finding them by Prime Factorisation
If we have \(12 = 2^2 \times 3\) and \(18 = 2 \times 3^2\):
- For H.C.F.: Take the lowest power of the common prime factors. \(H.C.F. = 2^1 \times 3^1 = 6\).
- For L.C.M.: Take the highest power of all prime factors involved. \(L.C.M. = 2^2 \times 3^2 = 4 \times 9 = 36\).
Key Takeaway: H.C.F. is about "shared" factors (small), and L.C.M. is about "accumulated" multiples (large).
5. Mixed Operations: The Order of Brackets
When an expression has many parts, we must follow a specific order. You might remember BODMAS or PEMDAS. In S1, we pay special attention to different types of brackets.
The order of priority is:
- Parentheses \(( \quad )\): Solve these first.
- Square Brackets \([ \quad ]\): Solve these after the inner parentheses.
- Curly Brackets \(\{ \quad \}\): Solve these last among the brackets.
Within any bracket: Perform Multiplication and Division (from left to right) before Addition and Subtraction (from left to right).
Example: \(\{ 20 + [ 5 \times ( 6 - 2 ) ] \} \div 8\)
1. Inner \((6 - 2) = 4\)
2. Then \([5 \times 4] = 20\)
3. Then \(\{20 + 20\} = 40\)
4. Finally \(40 \div 8 = 5\)
6. Fractions and Decimals
In "Basic Computation," you will also deal with mixed operations involving fractions and decimals. Remember these golden rules:
- Fractions: To add or subtract, you must find a Common Denominator (use the L.C.M.!). To multiply, multiply tops together and bottoms together. To divide, "flip" the second fraction and multiply.
- Decimals: When adding or subtracting, line up the decimal points. For multiplication, count the total number of decimal places.
Common Mistake to Avoid: Don't try to calculate everything at once. Write your steps down line-by-line. If you skip steps, it's very easy to lose a small number!
Cross-reference: We will apply these computation skills in the next chapter, Directed Numbers, where we introduce negative values. For now, keep practicing with positive numbers!
Quick Review: Can you name a number divisible by \(9\)? Can you break \(20\) into its prime factors? If yes, you're ready for S1 Math!